Operator Algebras and Their Modules: An Operator Space Approach (London Mathematical Society Monographs New Series)

L O N D O N M AT H E M AT I C A L S O C I E T Y M O N O G R A P H S Series Editors Dr Peter Neumann Professor E B Davi...

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L O N D O N M AT H E M AT I C A L S O C I E T Y M O N O G R A P H S

Series Editors Dr Peter Neumann Professor E B Davies

L O N D O N M AT H E M AT I C A L S O C I E T Y M O N O G R A P H S NEW SERIES

Previous volumes of the LMS Monographs were published by Academic Press, to whom all enquiries should be addressed. Volumes in the New Series will be published by Oxford University Press throughout the world. NEW SERIES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Diophantine inequalities R. C. Baker The Schur multiplier Gregory Karpilovsky Existentially closed groups Graham Higman and Elizabeth Scott The asymptotic solution of linear differential systems M. S. P. Eastham The restricted Burnside problem Michael Vaughan-Lee Pluripotential theory Maciej Klimek Free Lie algebras Christophe Reutenauer The restructed Burnside problem (2nd edition) Michael Vaughan-Lee The geometry of topological stability Andrew du Plessis and Terry Wall Spectral decompositions and analytic sheaves J. Eschmeier and M. Putinar An atlas of Brauer characters C. Jansen, K. Lux, R. Parker, and R. Wilson Fundamentals of semigroup theory John M. Howie Area, lattice points, and exponential sums M. N. Huxley Super-real ﬁelds H. G. Dales and W. H. Woodin Integrability, self-duality, and twistor theory L. J. Mason and N. M. J. Woodhouse Categories of symmetries and inﬁnite-dimensional groups Yu. A. Neretin Interpolation, identiﬁcation, and sampling Jonathan R. Partington Metric number theory Glyn Harman Proﬁnite groups John S. Wilson An introduction to local spectral theory K. B. Laursen and M. M. Neumann Characters of ﬁnite Coxeter groups and Iwahori-Hecke Algebras M. Geck and G. Pfeiffer Classical harmonic analysis and locally compact groups Hans Reiter and Jan D Stegeman Operator spaces E. G. Effros and Z.-J. Ruan Banach algebras and automatic continuity H. G. Dales The mysteries of the real prime M. J. Shai Haran Analytic theory of polynomials Q. I. Rahman and G. Schmeisser The structure of groups of prime power order C. R. Leedham-Green and S. McKay Maximal orders I. Reiner (reissue) Harmonic morphisms between Riemannian manifolds P. Baird and J. C. Wood Operator algebras and their modules: an operator space approach David P. Blecher and Christian Le Merdy

Operator algebras and their modules—an operator space approach David P. Blecher Department of Mathematics, University of Houston

Christian Le Merdy Laboratoire de Mathématiques, Université de Besançon

CLARENDON PRESS

•

OXFORD 2004

3

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Preface

A major trend in modern mathematics, inspired largely by physics, is toward ‘noncommutative’ or ‘quantized’ phenomena. This thrust has inﬂuenced most branches of the science. In the vast area of functional analysis, this trend has appeared notably under the name of operator spaces. This young ﬁeld lies at the border between linear analysis, operator theory, operator algebras, and quantum physics. It has useful applications in all of these directions, and in turn derives its inspiration and power from these sources. Perhaps the importance of operator space theory may be best stated as follows: it is a variant of Banach spaces, which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in ‘noncommutative mathematics’. An operator space, loosely speaking, is a linear space of bounded operators between two Hilbert spaces. Thus the category of operator spaces includes operator algebras, selfadjoint (that is, C ∗ -algebras) or otherwise. Since any normed linear space E may be regarded as a subspace of a commutative C ∗ -algebra (for example, the continuous scalar functions on the unit ball of E ∗ ), operator spaces also include all Banach spaces. In addition, most of the important modules over operator algebras are operator spaces. With this in mind, it is natural to seek to treat the subjects of C ∗ -algebras, nonselfadjoint operator algebras, and modules over such algebras (such as Hilbert C ∗ -modules), together under the umbrella of operator space theory. This is the topic of our book. In the last decade or two, it has become very apparent that it can be a useful perspective. Indeed, operator space theory, as opposed to Banach space theory, is a sensitive enough medium to reﬂect accurately important noncommutative phenomena such as the spatial tensor product. The underlying operator space structure also captures, very precisely, many of the profound relations between the algebraic and the functional analytic structures involved. Our main goal is to illustrate how a general theory of operator algebras, and their modules, naturally develops out of the operator space methodology. We emphasize both the uniform (or ‘norm’), and the dual (or ‘weak∗ ’), aspects of the theory. A second goal, or prevailing theme, is the systematic study of algebraic structure in spaces of Hilbert space operators. For example, we are interested in the structural features characterizing the objects which operator algebraists are interested in, how rigid such structures are, how they behave with respect to duality, and so on. A third goal, and this is one of the most inspiring aspects of the subject at large, is to highlight the rich interplay between spectral theory, operator theory, C ∗ -algebra and von Neumann algebra techniques, and the inﬂux

viii

Preface

of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory. Finally, our fourth goal is pedagogical: to assemble the basic concepts, theory, and methodologies, needed to equip a beginning researcher in this area. Our book falls roughly into three parts. Each chapter begins with some words of introduction, and so we will only very brieﬂy describe their contents here. Part 1—Chapters 1–3—presents the basic theory of operator spaces, operator algebras, and operator modules. We also introduce much of our notation here. Part 2—Chapters 4–7—presents more advanced topics associated with these subjects, and describes more technical results. Chapter 4 discusses, for example, the noncommutative Shilov boundary, injective envelopes, operator space multipliers, and M -ideals, and applications of these topics to ‘operator algebraic structure’. Chapter 5 is devoted to the ‘isomorphic’ (as opposed to ‘isometric’) aspects of the theory. This includes completely isomorphic characterizations of various classes (operator algebras, operator modules, Q-algebras), as well as examples of ‘operator algebra structures’. In Chapter 6, we discuss various tensor products involving operator algebras, such as the maximal tensor product, or Pisier’s δ tensor norm. We give various applications, for example to dilation theory, or to ﬁnite rank approximation (nuclearity, semidiscreteness, etc.). In Chapter 7 we collect some criteria which ensure that an operator algebra is selfadjoint. Part 3—Chapter 8—develops the theory of Hilbert C ∗ -modules and the related notion of triple systems, largely from an operator space perspective. In this chapter we also describe some of the beautiful two-way interplay between C ∗ -modules and the theory in the earlier chapters. Finally, a short appendix contains some frequently needed facts from operator theory, Banach space theory, and Banach and C ∗ -algebras. We include proofs of many of these facts. Each chapter ends with a lengthy ‘Notes and historical remarks’ section, consisting of attributions, discussion of the literature, observations, additional proofs, complementary results, and so on. We apologize for inaccuracies or omissions here, of which there are sure to be many. In all cases, the reader should consult the original papers for further details, and other perspectives. This book was begun in 1999, during a year-long visit of the second author to Houston. We wish to thank the Universities of Besan¸con and Houston, the CNRS, and the National Science Foundation, for their support. We are also indebted to several colleagues for very many kindnesses, and for teaching us much of this material, in particular William Arveson (who started it all), Edward Eﬀros, Paul Muhly, Vern Paulsen, Gilles Pisier, Zhong-Jin Ruan, Allan Sinclair, and Roger Smith. We thank Matthias Neufang, Bojan Magajna, and Damon Hay for many very helpful suggestions, and Oxford University Press and the LMS Series editors for an excellent job of processing our book. Houston Besan¸con June, 2004

D. P. B. C. L-M.

Contents

1

Operator spaces 1.1 Notation and conventions 1.2 Basic facts, constructions, and examples 1.3 Completely positive maps 1.4 Operator space duality 1.5 Operator space tensor products 1.6 Duality and tensor products 1.7 Notes and historical remarks

1 1 4 16 22 27 38 45

2

Basic theory of operator algebras 2.1 Introducing operator algebras and unitizations 2.2 A few basic constructions 2.3 The abstract characterization of operator algebras 2.4 Universal constructions of operator algebras 2.5 The second dual algebra 2.6 Multiplier algebras and corners 2.7 Dual operator algebras 2.8 Notes and historical remarks

49 49 57 62 68 78 82 88 96

3

Basic theory of operator modules 3.1 Introduction to operator modules 3.2 Hilbert modules 3.3 Operator modules over operator algebras 3.4 Two module tensor products 3.5 Module maps 3.6 Module map extension theorems 3.7 Function modules 3.8 Dual operator modules 3.9 Notes and historical remarks

102 102 109 115 119 123 128 131 136 142

4

Some ‘extremal theory’ 4.1 The Choquet boundary and boundary representations 4.2 The injective envelope 4.3 The C ∗ -envelope 4.4 The injective envelope, the triple envelope, and TROs 4.5 The multiplier algebra of an operator space 4.6 Multipliers and the ‘characterization theorems’ 4.7 Multipliers and duality

147 147 152 156 161 167 175 180

x

Contents 4.8 4.9

Noncommutative M -ideals Notes and historical remarks

183 188

5

Completely isomorphic theory of operator algebras 5.1 Homomorphisms of operator algebras 5.2 Completely bounded characterizations 5.3 Examples of operator algebra structures 5.4 Q-algebras 5.5 Applications to the isomorphic theory 5.6 Notes and historical remarks

195 195 200 209 215 224 228

6

Tensor products of operator algebras 6.1 The maximal and normal tensor products 6.2 Joint dilations and the disc algebra 6.3 Tensor products with triangular algebras 6.4 Pisier’s delta norm 6.5 Factorization through matrix spaces 6.6 Nuclearity and semidiscreteness for linear operators 6.7 Notes and historical remarks

232 232 239 241 248 254 259 265

7

Selfadjointness criteria 7.1 OS-nuclear maps and the weak expectation property 7.2 Hilbert module characterizations 7.3 Tensor product characterizations 7.4 Amenability and virtual diagonals 7.5 Notes and historical remarks

269 269 274 279 282 292

8

C ∗ -modules and operator spaces 8.1 Hilbert C ∗ -modules—the basic theory 8.2 C ∗ -modules as operator spaces. 8.3 Triples, and the noncommutative Shilov boundary 8.4 C ∗ -module maps and operator space multipliers 8.5 W ∗ -modules 8.6 A sample application to operator spaces 8.7 Notes and historical remarks

296 297 308 322 328 331 348 350

Appendix A.1 Operators on Hilbert space A.2 Duality of Banach spaces A.3 Tensor products of Banach spaces A.4 Banach algebras A.5 C ∗ -algebras A.6 Modules and Cohen’s factorization theorem

359 359 360 361 363 364 367

References

369

Index

385

1 Operator spaces

In this chapter, we present quickly the background results about operator spaces which we shall need, and also establish some notation which will be used throughout this book. The reader with a little mathematical maturity could use this chapter as a minicourse on the basics of operator space theory. Fortunately the lengthy proofs here usually belong to the very well-known results (such as Ruan’s theorem, or the extension/characterization theorems for completely positive or completely bounded maps). Thus with the exception of a few such well-known proofs (which may be found in [149, 314, 337, 385, 102]), we can be quite selfcontained. Warning: our proofs in this chapter are often only a good sketch, and some things are left as exercises. The reader should also feel free to skim through this chapter, returning later for a speciﬁc deﬁnition or fact (we try to be conscientious in later chapters about referencing these by number). Those mainly interested in the general theory of operator spaces, should consult the ﬁne aforementioned texts for a more comprehensive and leisurely development. And of course usually the original papers contain much additional material. We will take for granted facts found in any basic graduate level functional analysis text. For example, we assume that the reader is comfortable with basic spectral theory, the very basics of the theory of C ∗ -algebras and Banach algebras, and standard facts about the various topologies in Banach spaces or dual spaces. Much of this may be found in the Appendix, together with a few of the unexplained terms below. 1.1 NOTATION AND CONVENTIONS 1.1.1 Our set notation and function notation is standard. We use Ac for the complement of a set A. The term ‘scalar’ denotes a number in the complex ﬁeld C. We use n, m, i, j, k for integers, and I, J or α, β, γ for cardinal numbers. Vector spaces are almost always over the ﬁeld C unless stated to the contrary. The usual basis of Cn or 2 is written as (ei )i , and we use this notation too in the other p sequence spaces. We write IE , or sometimes I when there is no confusion, for the ‘identity map’ on a vector space E. An isomorphism, at the very least, is always assumed to be linear, one-to-one, and surjective. If T : E → F , and if W ⊂ E is a linear subspace, then we write T|W for the map from W to F obtained by restricting T to W . We often use the symbol q or qW for the canonical surjection

2

Notation and conventions

˙ , or sometimes x, from E onto E/W . We write x+W ˙ for the class of x in E/W , ˙ thus x+W = qW (x). If E is a normed space, we write Ball(E) for the set {x ∈ E : x ≤ 1}. Expressions such as ‘norm closed’, ‘norm closure’, or ‘xn → x in norm’ (or simply ‘closed’, ‘closure’, or ‘xn → x’), mean of course ‘with respect to the norm topology’. All topological spaces are assumed to be Hausdorﬀ. We use standard notation for the standard examples, for example, C(Ω) is the Banach space of scalar valued continuous functions on a compact space Ω. In the literature these are often called ‘C(K)-spaces’, and of course are exactly the commutative unital C ∗ -algebras (see A.5.4). We use the letters H, K, L for Hilbert spaces. Thus if these letters appear in the text without explanation, they will always be Hilbert spaces. We write B(E, F ) for the space of bounded linear operators from E to F , and B(E) = B(E, E). Indeed whenever C(X, Y ) is a class of operators then we use C(X) for C(X, X). We write E ∗ for the dual space of E, namely E ∗ = B(E, C), and we often write E∗ for a predual of E (if such exists). We write iE : E → E ∗∗ for the canonical embedding, but will often simply think of E as a subspace of E ∗∗ . We abbreviate ‘weak*’ to ‘w ∗ ’ usually. Thus we write w∗ w∗ -continuous, w∗ -topology, w∗ -closure, etc. Thus S denotes the w∗ -closure of a set S. We say that a net of maps Tt : E → F converges strongly (or point-norm) if Tt (x) → T (x) in the norm topology of F for all x ∈ E. If F is a dual space then Tt → T point-w∗ if Tt (x) → T (x) in the w∗ -topology of F for all x ∈ E. A multilinear map between dual spaces is said to be separately w ∗ -continuous if whenever one ﬁxes all but one of the variables, then the map is w ∗ -continuous in the remaining variable. We recommend that the reader review the facts about the w∗ -topology presented in the ﬁrst sections of the Appendix. An operator T between normed spaces, with T ≤ 1, is called a contraction. A quotient map T : E → F is a linear map which maps the ‘open unit ball of E’ onto the ‘open unit ball of F ’. A projection or idempotent on a space E is a map P : E → E satisfying P ◦ P = P . However if E is a Hilbert space then we will mean more, indeed for an operator on a Hilbert space, or more generally for an element of an operator algebra, we always use the term projection to mean an orthogonal (i.e. selfadjoint) idempotent. If K is a closed linear subspace of a Hilbert space H then PK is the canonical projection from H onto K. 1.1.2 For emphasis, we list separately here some of our major conventions. First, we usually suppose that all of our normed spaces are complete. This is not a serious restriction, since the completion of an operator space is again an operator space; and the ‘incomplete’ versions of most results ‘pass to the completion’ without diﬃculty. We make the ‘completeness’ assumption mostly to avoid having to be constantly making annoying and repetitious remarks about results ‘passing to the completion’. Another convention is our use of the notation XY for sets X, Y . Assume that we have a pairing X × Y → E where E is a Banach space. Write this pairing as the map (x, y) → xy. Then XY usually denotes the closure in the norm topology in E of the linear span of the xy, for x ∈ X and y ∈ Y . We write Span(XY ) if we are not taking the closure here.

Operator spaces

3

See also A.6.4 for some important related facts. There is an exception to this notation; if K is a subset of a Hilbert space H and if D ⊂ B(H, L) is a set of operators then we use [DK] for the norm closure in L of the span of terms xζ for x ∈ D, ζ ∈ K. If X is a subspace of B(K, H) or of a C ∗ -algebra, then we often use the symbol X (also written as X ∗ when there is no possible confusion with the dual space) for the set of ‘adjoints’ or ‘involutions’ {x∗ : x ∈ X}. 1.1.3 (Matrix notation) Fix m, n ∈ N. If X is a vector space, then so is Mm,n (X), the set of m × n matrices with entries in X. This may also be thought of as the algebraic tensor product Mm,n ⊗ X, where Mm,n = Mm,n (C). We write In for the identity matrix of Mn = Mn,n . We write Mn (X) = Mn,n (X), Cn (X) = Mn,1 (X) and Rn (X) = M1,n (X). If x is a matrix, then xij or xi,j denotes the i-j entry of x, and we write x as [xij ] or [xi,j ]i,j . We write (Eij )ij for the usual (matrix unit) basis of Mm,n (we allow m, n inﬁnite here too). We write A → At for the transpose on Mm,n , or more generally on Mm,n (X). We will frequently meet large matrices with row and column indexing that is sometimes cumbersome. For example, a matrix [a(i,k,p),(j,l,q) ] is indexed on rows by (i, k, p) and on columns by (j, l, q), and may also be written as [a(i,k,p),(j,l,q) ](i,k,p),(j,l,q) if additional clarity is needed. 1.1.4 The Hilbert space direct sum will be written as ⊕2 , or simply ⊕ (but we use the latter for some other kinds of direct sums too). We also write H (α) or 2α (H) for the Hilbert space direct sum of α copies of H. Here α is a cardinal. This is called a multiple of H. The Hilbert space tensor product is denoted H ⊗2 K. If S, T are operators on H and K respectively, then we write S ⊗ T for the usual operator on H ⊗2 K taking a rank one tensor ζ ⊗ η in H ⊗ K to S(ζ) ⊗ T (η). In particular, S ⊗ IK is often called a multiple of S. Indeed, if K is identiﬁed with 2α for some cardinal α, then we may unitarily identify H ⊗2 K with 2α (H), and S ⊗ IK with S α . Here S α ((ζi )) = (Sζi ), for (ζi ) ∈ 2α (H). The commutant of a subset S ⊂ B(H) is written as S or [S] . The C ∗ -identity is the statement T ∗T = T 2, valid for any bounded operator T between Hilbert spaces, or any element of a C ∗ -algebra. We write S p (K, H) for the Schatten p class (see also A.1.2 and A.1.3). If H = K is n-dimensional then we write this as Snp , thus Sn1 is the dual space of Mn . We use WOT for the weak operator topology (see A.1.4), although we usually prefer to use the (ﬁner) w ∗ -topology (= σ-weak topology, see A.1.2). Which of these two topologies one uses is often a matter of taste, in the situations we consider. Very frequently, we will need the polarization identity. We state one form of it: Suppose that E and F are vector spaces, and that Ψ : E × E → F is linear in the second variable and conjugate linear in the ﬁrst variable. Then 1 k i Ψ(x + ik y, x + ik y), 4 3

Ψ(y, x) =

k=0

x, y ∈ E.

(1.1)

4

Basic facts, constructions, and examples

This is frequently applied when E = F is a ∗-algebra, and Ψ(x, y) = x∗ y. 1.1.5 We also use some basic notions from algebra, such as the deﬁnitions of modules, algebras, ideals, direct sum, tensor product, etc. These may be found in any graduate algebra text. Our spaces, of course, usually have extra functional analytic structure, and in particular possess a (complete) norm. If A is an algebra, then Mn (A) is also an algebra, if one uses the usual formula for multiplying matrices. We usually refer to a closed two-sided ideal of a normed algebra simply as an ‘ideal’. One unusual usage: we use the term unital-subalgebra for a subalgebra of a unital algebra A containing the unit (identity) of A. Similarly, a unital-subspace is a subspace containing the ‘unit’ of the superspace. A unital map is one that takes the unit to the unit. We use the very basics of the language of categories, such as the notion of object, morphism, and functor. The main categories we are interested in here are those of Banach spaces and bounded linear maps, operator spaces and completely bounded linear maps, operator algebras and completely contractive homomorphisms, C ∗ -algebras and ∗-homomorphisms, and operator modules and completely bounded module maps. These notions will be introduced in detail later. However it is worth saying that each of these categories (and any others we shall meet) has its own notion of ‘isomorphism’ (i.e. when we consider two objects as being essentially the same), subobject, embedding, quotient, quotient map, direct sum, etc. When we use one of these words in later chapters, it is usually understood to be with reference to the category that we are working in. For example, in Chapter 2 we may simply write ‘A ∼ = B’, or ‘A ∼ = B as operator algebras’, and say that ‘A is isomorphic to B’, when we really mean that there is a surjective algebra homomorphism between them which is completely isometric (deﬁned below). Or we may write A → B to indicate that A is ‘embedded’ in B in the suitable sense of that chapter. For example, in Chapter 2 it means that there is a completely isometric algebra homomorphism from A to B. 1.2 BASIC FACTS, CONSTRUCTIONS, AND EXAMPLES 1.2.1 (Completely bounded maps) Suppose that X and Y are vector spaces and that u : X → Y is a linear map. For a positive integer n, we write un for the associated map [xij ] → [u(xij )] from Mn (X) to Mn (Y ). This is often called the (nth) ampliﬁcation of u, and may also be thought of as the map IMn ⊗ u on Mn ⊗ X. Similarly one may deﬁne um,n : Mm,n (X) → Mm,n (Y ). If each matrix space Mn (X) and Mn (Y ) has a given norm · n , and if un is an isometry for all n ∈ N, then we say that u is completely isometric, or is a complete isometry. Similarly, u is completely contractive (resp. is a complete quotient map) if each un is a contraction (resp. takes the open ball of Mn (X) onto the open ball of Mn (Y )). A map u is completely bounded if def ucb = sup [u(xij )]n : [xij ]n ≤ 1, all n ∈ N < ∞.

Operator spaces

5

Compositions of completely bounded maps are completely bounded, and one has the expected relation u ◦ vcb ≤ ucbvcb. If u : X → Y is a completely bounded linear bijection, and if its inverse is completely bounded too, then we say that u is a complete isomorphism. In this case, we say that X and Y are completely isomorphic and we write X ≈ Y . 1.2.2 (Operator spaces) If m, n ∈ N, and K, H are Hilbert spaces, then we always assign Mm,n (B(K, H)) the norm (written · m,n ) ensuring that Mm,n (B(K, H)) ∼ = B(K (n) , H (m) )

isometrically

(1.2)

via the natural algebraic isomorphism. Recall from 1.1.4 that H (m) = 2m (H) is the Hilbert space direct sum of m copies of H, for example. A concrete operator space is a (usually closed) linear subspace X of B(K, H), for Hilbert spaces H, K (indeed the case H = K usually suﬃces, via the canonical inclusion B(K, H) ⊂ B(H ⊕ K)). However we will want to keep track too of the norm · m,n that Mm,n (X) inherits from Mm,n (B(K, H)), for all m, n ∈ N. We write · n for · n,n ; indeed when there is no danger of confusion, we simply write [xij ] for [xij ]n . An abstract operator space is a pair (X, { · n }n≥1 ), consisting of a vector space X, and a norm on Mn (X) for all n ∈ N, such that there exists a linear complete isometry u : X → B(K, H). In this case we call the sequence { · n }n an operator space structure on the vector space X. An operator space structure on a normed space (X, ·) will usually mean a sequence of matrix norms as above, but with · = · 1 . Clearly subspaces of operator spaces are again operator spaces. We often identify two operator spaces X and Y if they are completely isometrically isomorphic. In this case we often write ‘X ∼ = Y completely isometrically’, or say ‘X ∼ = Y as operator spaces’. Sometimes we simply write X = Y . 1.2.3 (C ∗ -algebras) If A is a C ∗ -algebra then the ∗-algebra Mn (A) has a unique norm with respect to which it is a C ∗ -algebra, by A.5.8. With respect to these matrix norms, A is an operator space. This may be seen by noting that M n (A) corresponds to a closed ∗-subalgebra of B(H (n) ), when A is a closed ∗-subalgebra of B(H). We call this the canonical operator space structure on a C ∗ -algebra. If the C ∗ -algebra A is commutative, then A = C0 (Ω) for a locally compact space Ω, and then these matrix norms are determined via the canonical isomorphism Mn (C0 (Ω)) = C0 (Ω; Mn ). Explicitly, if [fij ] ∈ Mn (C0 (Ω)), then: (1.3) [fij ]n = sup [fij (t)]. t∈Ω

To see this, note that by the above one only needs to verify that (1.3) does indeed deﬁne a C ∗ -norm on Mn (C0 (Ω)). Proposition 1.2.4 For a homomorphism π : A → B between C ∗ -algebras, the following are equivalent: (i) π is contractive, (ii) π is completely contractive, and (iii) π is a ∗-homomorphism. If these hold, then π(A) is closed, and π is a

6

Basic facts, constructions, and examples

complete quotient map onto π(A); moreover π is one-to-one if and only if it is completely isometric. 2

Proof Apply A.5.8 to the ‘ampliﬁcations’ πn . ∗

1.2.5 (Norm of a row or column) Suppose that A is a C -algebra, or a space of the form B(K, H), for Hilbert spaces H, K. If X is a subspace of A, and if x1 , . . . , xn ∈ X, then we have   x1 n n 1 12 2  ..  ∗ ∗ [x1 · · · xn ] = x x and = x x   k k k k , (1.4) Rn (X) . k=1 k=1 xn C (X) n

where the product and involution mean the obvious thing (in the ambient superspace). Indeed this follows from the C ∗ -identity (see 1.1.4). 1.2.6 (Maps into a commutative C ∗ -algebra) If [aij ] ∈ Mn then

[aij ] = sup aij zj wi : z = [zj ], w = [wi ] ∈ Ball(2n ) . ij

It is easy to see that this also holds, but with ‘=’ replaced by ‘≥’, and | · | replaced by · , if aij ∈ B(H). Using these formulae, it is a simple exercise to see that any continuous functional ϕ : X → C on an operator space X is completely bounded, with ϕ = ϕcb. From this and equation (1.3), it follows that u = ucb for any bounded linear map u from an operator space into a commutative C ∗ -algebra. 1.2.7 The following trivial principle is used very often: If we are given complete contractions v : X → Y and u : Y → Z, and if uv is a complete isometry (resp. complete quotient map) then v is a complete isometry (resp. u is a complete quotient map). If, further, Z = X and uv and vu are both equal to the identity map, then both u and v are surjective complete isometries, and u = v −1 . Theorem 1.2.8 (Haagerup, Paulsen, Wittstock) Suppose that X is a subspace of a C ∗ -algebra B, that H and K are Hilbert spaces, and that u : X → B(K, H) is a completely bounded map. Then there exists a Hilbert space L, a ∗-representation π : B → B(L) (which may be taken to be unital if B is unital), and bounded operators S : L → H and T : K → L, such that u(x) = Sπ(x)T for all x ∈ X. Moreover this can be done with ST = ucb. In particular, if ϕ ∈ Ball(X ∗ ), and if B is as above, then there exist L, π as above, and unit vectors ζ, η ∈ L, with ϕ = π(·)ζ, η on X. The very last line clearly follows from the lines above it, and 1.2.6. Also note that conversely, any linear map u of the form u = Sπ(·)T as above, is completely bounded with ucb ≤ ST . This is an easy exercise using Proposition 1.2.4. We omit the well-known proof of Theorem 1.2.8 (see the cited texts above).

Operator spaces

7

1.2.9 (Injective spaces) An operator space Z is said to be injective if for any completely bounded linear map u : X → Z and for any operator space Y containing X as a closed subspace, there exists a completely bounded extension ucb = ucb. A similar deﬁnition exists for u ˆ : Y → Z such that uˆ|X = u and ˆ Banach spaces. Thus an operator space (resp. Banach space) is injective if and only if it is an ‘injective object’ in the category of operator (resp. Banach) spaces and completely contractive (resp. contractive) linear maps. The following is ‘contained’ in Theorem 1.2.8 (and the remark after it). Theorem 1.2.10 If H and K are Hilbert spaces then B(K, H) is an injective operator space. Recall that one version of the Hahn–Banach theorem may be formulated as the statement that C is injective (as a Banach space). Thus 1.2.10 is a ‘generalized Hahn–Banach theorem’. Corollary 1.2.11 An operator space is injective if and only if it is linearly completely isometric to the range of a completely contractive idempotent map on B(H), for some Hilbert space H. Proof (⇒) Supposing X ⊂ B(H), extend IX to a map from B(H) to X. (⇐) Follows from 1.2.10 and an obvious diagram chase.

2

1.2.12 (Properties of matrix norms) If K, H are Hilbert spaces, and if X is a subspace of B(K, H), then there are certain well-known properties satisﬁed by the matrix norms ·m,n described in 1.2.2. For example, adding (or dropping) a row of zeros or column of zeros does not change the norm of a matrix of operators. By this principle we really only need to specify the norms for square matrices, that is, the case m = n above. Also, switching two rows (or two columns) of a matrix of operators does not change its norm. From this we derive another useful property. Namely, the canonical algebraic isomorphisms Mn (Mm (X)) ∼ = Mm (Mn (X)) ∼ = Mmn (X)

(1.5)

are isometric, and hence, by iteration, completely isometric. Thus if X is an operator space then so is Mn (X) (or Mm,n (X)). As an exercise in operator theory, one may verify that for such X we have: (R1) αxβn ≤ αxn β, for all n ∈ N and all α, β ∈ Mn , and x ∈ Mn (X) (where multiplication of an element of Mn (X) by an element of Mn is deﬁned in the obvious way). (R2) For all x ∈ Mm (X) and y ∈ Mn (X), we have x 0 = max{xm , yn}. 0 y m+n Conditions (R1) and (R2) above are often called Ruan’s axioms. Ruan’s theorem asserts that (R1) and (R2) characterize operator space structures on a

8

Basic facts, constructions, and examples

vector space. This result is fundamental to our subject in many ways. At the most pedestrian level, it is used frequently to check that certain abstract constructions with operator spaces remain operator spaces. At a more sophisticated level, it is the foundational and unifying principle of operator space theory. Theorem 1.2.13 (Ruan) Suppose that X is a vector space, and that for each n ∈ N we are given a norm · n on Mn (X). Then X is linearly completely isometrically isomorphic to a linear subspace of B(H), for some Hilbert space H, if and only if conditions (R1) and (R2) above hold. 1.2.14 (Quotient operator spaces) If Y ⊂ X is a closed linear subspace of an operator space, then using Ruan’s theorem one can easily check that X/Y is an operator space with matrix norms coming from the identiﬁcation Mn (X/Y ) ∼ = Mn (X)/Mn (Y ). Explicitly, these matrix norms are given by the ˙ ]n = inf{[xij + yij ]n : yij ∈ Y }. Here xij ∈ X. formula [xij +Y 1.2.15 (Factor theorem) If u : X → Z is completely bounded, and if Y is a closed subspace of X contained in Ker(u), then the canonical map u ˜ : X/Y → Z induced by u is also completely bounded, with ˜ ucb = ucb. If Y = Ker(u), then u is a complete quotient map if and only if u ˜ is a completely isometric isomorphism. Indeed this follows exactly the usual Banach space case. 1.2.16 (Operator seminorms) An operator seminorm structure on a vector space X is a sequence ρ = {ρn }∞ n=1 , where ρn is a seminorm on Mn (X), satisfying axioms (R1) and (R2) discussed in 1.2.12. In this case, and if N is deﬁned to be {x ∈ X : ρ1 (x) = 0}, by (R1) we have that the kernel of ρn is Mn (N ), and ρ induces matrix norms on X/N in the obvious fashion. By Ruan’s theorem, (the completion of) X/N is then an operator space. Let X be a vector space, and let F = {Ti : i ∈ I} be a set of linear maps, where Ti maps X into an operator space Zi , for each i ∈ I. We suppose that supi Ti (x) < ∞, for all x ∈ X. Let N = ∩i Ker(Ti ). For each n ∈ N, we deﬁne a seminorm on Mn (X) by [xpq ] −→ sup [Ti (xpq )]. i

This is fairly clearly an operator seminorm structure for X. Using the facts in the last paragraph, these seminorms become matrix norms on X/N , and with these norms (the completion of) X/N is an operator space. Similarly, if I is a directed set then the expressions lim supi [Ti (xpq )] deﬁne an operator seminorm structure on X. This yields an operator space as before. 1.2.17 (The ∞-direct sum) This is the simplest direct sum of a family of operator spaces {Xλ : λ ∈ I}, and we will write this operator space as ⊕λ Xλ (or ⊕∞ λ Xλ if more clarity is needed). If I = {1, . . . , n} then we usually write this sum as X1 ⊕∞ · · · ⊕∞ Xn . If Xλ ⊂ B(Hλ ) then ⊕λ Xλ may be regarded as the obvious subspace of B(⊕2λ Hλ ). A tuple (xλ ) is in ⊕∞ λ Xλ if and only if xλ ∈ Xλ

Operator spaces

9

for all λ, and supλ xλ < ∞. We may identify Mn (⊕λ Xλ ) with ⊕λ Mn (Xλ ) isometrically (and by iteration, completely isometrically). Thus if x ∈ Mn (⊕λ Xλ ), then we have xn = supλ xλ Mn (Xλ ) . Clearly the canonical inclusion and projection maps between ⊕λ Xλ and its ‘λth summand’ are complete isometries and complete quotient maps respectively. If Xλ are C ∗ -algebras then this direct sum is the usual C ∗ -algebra direct sum. If the Xλ are W ∗ -algebras then this direct sum is a W ∗ -algebra too, and is easy to work with in terms of the canonical central projections corresponding to the summands. The ∞-direct sum has the following universal property. If Z is an operator space and uλ : Z → Xλ are completely contractive linear maps, then there is a canonical complete contraction Z → ⊕λ Xλ taking z ∈ Z to the tuple (uλ (z)). If Xλ = X for all λ ∈ I, then we usually write ∞ I (X) for ⊕λ Xλ . If I = N we may simply write ∞ (X). Note that we have ∼ ∞ Mn (∞ I (X)) = I (Mn (X)). If I = N then one may deﬁne a c0 -direct sum operator space of operator spaces X1 , X2 , . . . . This is simply the subspace of ⊕∞ n Xn consisting of tuples (xn ) with limn xn = 0. We write c0 (X) for this space if Xn = X for all n. 1.2.18 (Operator valued continuous functions) Let Ω be a compact space, let X be an operator space, and consider the space C(Ω; X) of continuous X-valued functions on Ω (see A.3.2). This is an operator space with matrix norms coming from the identiﬁcation Mn (C(Ω; X)) = C(Ω; Mn (X)). Clearly this ‘canonical’ operator space structure is given by the same formula as (1.3), and the natural ∗ embedding C(Ω; X) ⊂ ∞ Ω (X) is a complete isometry. Note that if A is a C ∗ algebra, then C(Ω; A) is a C -algebra, with product as pointwise multiplication and with f ∗ (t) = (f (t))∗ for any f ∈ C(Ω; A) and t ∈ Ω. Similarly if Ω is merely a locally compact space, then C0 (Ω; X) is an operator space as well, with Mn (C0 (Ω; X)) = C0 (Ω; Mn (X)) for all n. 1.2.19 (Mapping spaces) If X, Y are operator spaces, then the space CB(X, Y ) of completely bounded linear maps from X to Y , is also an operator space, with matrix norms determined via the canonical isomorphism between Mn (CB(X, Y )) and CB(X, Mn (Y )). Equivalently, if [uij ] ∈ Mn (CB(X, Y )), then (1.6) [uij ]n = sup [uij (xkl )]nm : [xkl ] ∈ Ball(Mm (X)), m ∈ N . Here the matrix [uij (xkl )] is indexed on rows by i and k and on columns by j and l. Applying the above with n replaced by nN , to the space of matrices MN (Mn (CB(X, Y ))) = MnN (CB(X, Y )), yields Mn (CB(X, Y )) ∼ = CB(X, Mn (Y ))

completely isometrically.

(1.7)

One may see that (1.6) deﬁnes an operator space structure on CB(X, Y ) by appealing to Ruan’s theorem 1.2.13 (directly or in the form of 1.2.16). Alternatively, one may see it as follows. Consider the set I = ∪n Ball(Mn (X)), and for

10

Basic facts, constructions, and examples

x ∈ Ball(Mm (X)) ⊂ I set nx = m. Consider the operator space direct sum (see 1.2.17) ⊕∞ x∈I Mnx (Y ). Then the map from CB(X, Y ) to ⊕x∈I Mnx (Y ) taking u to the tuple ((Inx ⊗ u)(x))x ∈ ⊕x Mnx (Y ) is (almost tautologically) a complete isometry. Thus CB(X, Y ) is an operator space. 1.2.20 (The dual of an operator space) The special case when Y = C in 1.2.19 is particularly important. In this case, for any operator space X, we obtain by 1.2.19 an operator space structure on X ∗ = CB(X, C). The latter space equals B(X, C) isometrically by 1.2.6. We call X ∗ , viewed as an operator space in this way, the operator space dual of X. This duality will be studied further in Sections 1.4–1.6. By (1.7) we have Mn (X ∗ ) ∼ = CB(X, Mn )

completely isometrically.

(1.8)

Note that the map implementing this isomorphism is exactly the canonical map (described in A.3.1) from Mn ⊗ X ∗ to B(X, Mn ). 1.2.21 (Minimal operator spaces) Let E be a Banach space, and consider the canonical isometric inclusion of E in the commutative C ∗ -algebra C(Ball(E ∗ )). Here E ∗ is equipped with the w∗ -topology. This inclusion induces, via 1.2.3, an operator space structure on E, which is denoted by Min(E). By (1.3), the resulting matrix norms on E are given by (1.9) [xij ]n = sup [ϕ(xij )] : ϕ ∈ Ball(E ∗ ) for [xij ] ∈ Mn (E). Thus every Banach space may be canonically considered to be an operator space. Since Min(E) ⊂ C(Ball(E ∗ )), we see from 1.2.6 that for any bounded linear u from an operator space Y into E, we have u : Y −→ Min(E)cb = u : Y −→ E.

(1.10)

From this last fact one easily sees that Min(E) is the smallest operator space structure on E. Also, if Ω is any compact space and if i : E → C(Ω) is an isometry, then the matrix norms inherited by E from the operator space structure of C(Ω), coincide again with those in (1.9). This may be seen by applying 1.2.6 to i and i−1 . Summarizing: ‘minimal operator spaces’ are exactly the operator spaces completely isometrically isomorphic to a subspace of a C(K)-space. According to A.3.1, another way of stating (1.9) is to say that ˇ Mn Min(E) = Mn ⊗E (1.11) isometrically via the canonical isomorphism. 1.2.22 (Maximal operator spaces) If E is a Banach space then Max(E) is the largest operator space structure we can put on E. We deﬁne the matrix norms on Max(E) by the following formula [xij ]n = sup [u(xij )] : u ∈ Ball(B(E, Y )), all operator spaces Y . This may be seen to be an operator space structure on X by using 1.2.16 say; and from this formula it is also clear that it is the largest such. Since every Banach

Operator spaces

11

space is isometric to an operator space (see 1.2.21), · 1 is evidently the usual norm on E. It is clear from this formula that Max(E) has the property that for any operator space Y , and for any bounded linear u : E → Y , we have u : Max(E) −→ Y cb = u : E −→ Y .

(1.12)

1.2.23 (Hilbert column and row spaces) If H is a Hilbert space then there are two canonical operator space structures on H most commonly considered. The ﬁrst is the Hilbert column space H c . Informally one should think of H c as a ‘column in B(H)’. Thus if H = 2n then H c = Mn,1 , thought of as the matrices in Mn which are ‘zero except on the ﬁrst column’. We write this operator space also as Cn , and the ‘row’ version as Rn . For a general Hilbert space H there are several simple ways of describing H c more precisely. For example, one may identify H c with the concrete operator space B(C, H). Another equivalent description is as follows (we leave the equivalence as an exercise). If η is a ﬁxed unit vector in H, then the set H ⊗η of rank one operators ζ ⊗η is a closed subspace of B(H) which is isometric to H via the map ζ → ζ ⊗ η. (By convention, ζ ⊗ η maps ξ ∈ H to ξ, ηζ.) Thus we may transfer the operator space structure on H ⊗ η inherited from B(H) over to H. The resulting operator space structure is independent of η and coincides with H c . Indeed from the C ∗ -identity in Mn (B(H)) applied to [ζij ⊗ η], one immediately obtains n 21 [ζij ]Mn (H c ) = ζkj , ζki ,

[ζij ] ∈ Mn (H).

(1.13)

k=1

If T ∈ B(H, K) then T = T cb, where the latter is the norm taken in CB(H c , K c ). Indeed let [ζij ] ∈ Mn (H c ), and let α ∈ B(2n , 2n (H)) correspond to this matrix via the identity Mn (H c ) = Mn (B(C, H)) = B(2n , 2n (H)). Likewise let β ∈ B(2n , 2n (K)) corresponding to [T ζij ]. Then β = (I2n ⊗ T ) ◦ α, and hence β ≤ T α. This shows that T cb ≤ T . More generally, we have B(H, K) = CB(H c , K c )

completely isometrically

(1.14)

We give a quick proof of this identity in the Notes for this section. A subspace K of a Hilbert column space H c is again a Hilbert column space, as may be seen by considering (1.13). Similarly the quotient H c /K c is a Hilbert column space completely isometric to (H K)c , as may be seen by applying 1.2.15 to the canonical (completely contractive by (1.14)) projection P from H c onto (H K)c . ¯ is a Hilbert We deﬁne Hilbert row space similarly. Recalling that H ∗ ∼ = H r ¯ space too, we identify H with the concrete operator space B(H, C). Analogues of the above results for H c hold, except that there is a slight twist in the corresponding version of (1.14). Namely, although B(H, K) = CB(H r , K r ) isometrically, this is not true completely isometrically. Instead there is a canonical ¯ r ). We have ¯ r, H completely isometric isomorphism B(H, K) ∼ = CB(K

12

Basic facts, constructions, and examples ¯r (H c )∗ ∼ = H

and

¯c (H r )∗ ∼ = H

(1.15)

completely isometrically using the operator space dual structure in 1.2.20. The ﬁrst relation is obtained by setting K = C in (1.14). Similarly, the second relation follows e.g. from the line above (1.15). We write C and R for 2 with its column and row operator space structures respectively. 1.2.24 (The operator space R ∩ C) We let R ∩ C be 2 with the operator space structure deﬁned by the embedding 2 → R ⊕∞ C which takes any x ∈ 2 to the pair (x, x). Let (ek )k≥1 denote the canonical basis of 2 . Then it follows from (1.4) that for any N ≥ 1 and any x1 , . . . , xn in MN , we have 1 1 (1.16) xk ⊗ ek MN (R∩C) = max x∗k xk 2 , xk x∗k 2 . k

k

k

We also note that if X is an operator space and u : X → 2 is a bounded linear map, then u is completely bounded from X into R ∩ C if and only if it is both completely bounded from X into R and from X into C. Moreover we have (1.17) u : X → R ∩ Ccb = max u : X → Rcb , u : X → Ccb . 1.2.25 (Opposite and adjoint) If X is an operator space, in B(K, H) say, then we deﬁne the adjoint operator space to be the space X = {x∗ : x ∈ X} (see 1.1.2). As an abstract operator space X is independent of the particular representation of X on H and K. Indeed we can alternatively deﬁne X as the ¯ ∗ , and with set of formal symbols x∗ for x ∈ X, with scalar product λx∗ = (λx) ∗ matrix norms [xij ]n = [xji ]n , where the latter norm is taken in Mn (X). The adjoint operator space is sometimes denoted by X by some authors. However we warn the reader that X is not the same as the conjugate operator space considered in [337]. If X is an operator space then we deﬁne the opposite operator space X op to be the Banach space X with the ‘transposed matrix norms’ [xij ]op n = [xji ]n . Note that if A is a C ∗ -algebra, then these matrix norms on Aop coincide with the canonical matrix norms on the C ∗ -algebra which is A with its reversed multiplication. If X is a subspace of a C ∗ -algebra A, then X op may be identiﬁed completely isometrically with the associated subspace of the C ∗ -algebra Aop . If u : X → Y , then we write uop for u considered as a map from X op to Y op , and u for the map from X to Y deﬁned by u (x∗ ) = u(x)∗ . These maps are completely bounded, completely contractive, completely isometric, etc., if u has these properties. There is a ‘conjugate linear complete isometry’ from X op to X , namely the map x → x∗ . 1.2.26 (Matrix spaces) If X is an operator space, and I, J are cardinal numbers or sets, then we write MI,J (X) for the set of I × J matrices whose ﬁnite submatrices have uniformly bounded norm. Such a matrix is normed by the supremum

Operator spaces

13

of the norms of its ﬁnite submatrices. Similarly there is an obvious way to deﬁne a norm on Mn (MI,J (X)) by equating this space with MI,J (Mn (X)), and one has Mn (MI (X)) ∼ = Mn.I (X), for n ∈ N. We are being deliberately careless here, and indeed in the rest of the book we often abusively blur the distinction between cardinals and sets. Technically if I, J are cardinals, we should ﬁx sets I0 and J0 of cardinality I and J respectively, consider matrices [xij ] indexed by i ∈ I0 and j ∈ J0 , and write MI0 ,J0 (X) instead of MI,J (X). However if one chooses diﬀerent sets I1 and J1 of these cardinalities, then there is an obvious completely isometric isomorphism MI0 ,J0 (X) ∼ = MI1 ,J1 (X), so that with a little care our convention should not lead us into trouble. Or we may protect ourselves by ﬁxing one well-ordered set associated with each cardinal. We write MI (X) = MI,I (X), CIw (X) = MI,1 (X), and RIw (X) = M1,I (X). If I = ℵ0 we simply denote these spaces by M(X), C w (X) and Rw (X) respectively. Also, Mﬁn I,J (X) will denote the vector subspace of MI,J (X) consisting of ‘ﬁnitely supported matrices’, that is, those matrices with only a ﬁnite number of nonzero entries. We write KI,J (X) for the norm closure in MI,J (X) of Mﬁn I,J (X). We set KI (X) = KI,I (X), CI (X) = KI,1 (X), and RI (X) = K1,I (X). Again we merely write K(X), R(X) and C(X) for these spaces if I = ℵ0 . If X = C then CIw (C) = CI (C) = (2I )c (see 1.2.23 for this notation), and we usually write this column Hilbert space as CI . Similarly, RI = RI (C) = (2I )r . We write KI,J for KI,J (C), and MI,J for MI,J (C). It is fairly obvious that if u : X → Y is completely bounded, then so is the obvious ampliﬁcation uI,J : MI,J (X) → MI,J (Y ), and uI,J cb = ucb. Clearly uI,J also restricts to a completely bounded map from KI,J (X) to KI,J (Y ). If u is a complete isometry, then so is uI,J . Thus the MI,J (·) and KI,J (·) constructions are ‘injective’ in some sense. For cardinals I, J, we leave it as an exercise that MI,J ∼ = B(2J , 2I ). Via this ∞ 2 2 identiﬁcation, KI,J = S (J , I ). Thus for any Hilbert spaces K, H we have that B(K, H) ∼ = MI0 ,J0 for some cardinals I0 , J0 . We leave it as another exercise that MI,J (MI0 ,J0 ) ∼ = MI×I0 ,J×J0

(1.18)

completely isometrically. Putting these two exercises together, we have established that for any cardinals I, J, we have MI,J (B(K, H)) ∼ = B(K (J) , H (I) )

completely isometrically.

(1.19)

If X is an operator space then so is MI,J (X). This may be seen by choosing a completely isometric embedding X ⊂ B(H), and noting that by the ‘injectivity’ mentioned a few paragraphs back, and formula (1.19), we have MI,J (X) ⊂ MI,J (B(H)) ∼ = B(H (J) , H (I) ) completely isometrically. If X is complete then so is MI,J (X), since it is clearly norm closed in MI,J (B(K, H)). For any operator space X, we have MI,J (X) = CIw (RJw (X)) = RJw (CIw (X)).

(1.20)

14

Basic facts, constructions, and examples

One way to see this is to ﬁrst check (1.20) in the case X = B(H) using (1.19), and then use this fact to do the general case. By a similar argument, MI,J (MI0 ,J0 (X)) ∼ = MI×I0 ,J×J0 (X) for any operator space X, generalizing (1.18). 1.2.27 (Inﬁnite sums) Suppose that X, Y are subspaces of a (complete) operw ator algebra or C ∗ -algebra A ⊂ B(H). Let I be an inﬁnite set. If x ∈ RI (X) and y ∈ CI (Y ), then the ‘product’ xy (deﬁned to be i xi yi if x and y have ith entries xi and yi respectively) actually converges in norm to an element of A. To see this, we use the following notation. If z is an element of RIw (X) or CI (Y ), and if ∆ ⊂ I, write z∆ for z but with all entries outside ∆ ‘switched to zero’. Since y ∈ CI (Y ), given > 0 there is a ﬁnite set ∆ ⊂ I, such that y − y∆ = y∆c < . If ∆ is a ﬁnite subset of I not intersecting ∆ then xi yi = x∆ y∆ ≤ x∆ y∆ ≤ xy∆ < x. i∈∆

Hence the sum converges in norm as claimed. Thus we have RIw (X) CI (Y ) ⊂ XY

and

RI (X) CIw (Y ) ⊂ XY,

(1.21)

where XY is as deﬁned in 1.1.2, a closed subset of A. Also xy ≤ xy for x, y as above, as may be seen from a computation identical to the ﬁrst part of the second last centered equation. Proposition 1.2.28 For any operator space X and cardinal I, we have that CB(CI , X) ∼ = RIw (X) and CB(RI , X) ∼ = CIw (X) completely isometrically. Proof We prove just the ﬁrst relation. Deﬁne L : RIw (X) → CB(CI , X) by w L(x)(z) = i xi zi , for x ∈ RI (X), z ∈ CI . This map is well deﬁned, by the argument for (1.21) for example. It is also easy to check, by looking at the partial sums of this series as in (1.21), that L is contractive. Conversely, for u in CB(CI , X), let x ∈ RIw (X) have ith entry u(ei ), where (ei ) is the canonical basis. It is not hard to see that xRw ≤ ucb, and L(x) = u. Thus L is a I (X) surjective isometry. This together with (1.7) yields Mm (CB(CI , X)) ∼ = CB(CI , Mm (X)) ∼ = RIw (Mm (X)) ∼ = Mm (RIw (X)) isometrically. From this one sees that L is a complete isometry.

2

Proposition 1.2.29 If X and Y are operator spaces then there are canonical complete isometries KI,J (CB(X, Y )) → CB(X, KI,J (Y )) → CB(X, MI,J (Y )) ∼ = MI,J (CB(X, Y )). In particular, if Y = C, we have CB(X, MI,J ) ∼ = MI,J (X ∗ ). Proof Since KI,J (Y ) ⊂ MI,J (Y ), the middle inclusion is evident. There is a canonical map Θ : MI,J (CB(X, Y )) → CB(X, MI,J (Y )), which takes an element

Operator spaces

15

[uij ] from MI,J (CB(X, Y )), to the map x → [uij (x)]. Also there is a canonical map CB(X, MI,J (Y )) → MI,J (CB(X, Y )), which takes u to the matrix [πij ◦ u], where πij is the projection of MI,J (Y ) onto its i-j entry. It is rather easy to check that these maps are mutual inverses, and are both completely contractive. Hence they are complete isometries. Thus MI,J (CB(X, Y )) ∼ = CB(X, MI,J (Y )). Finally, the isometry above taking MI,J (CB(X, Y )) into CB(X, MI,J (Y )), clearly takes Mﬁn I,J (CB(X, Y )) into CB(X, KI,J (Y )). By density, KI,J (CB(X, Y )) em2 beds completely isometrically in CB(X, KI,J (Y )). 1.2.30 (Interpolation) We recall the complex interpolation method for Banach spaces (e.g. see [33, Chapter 4]). Suppose that (X0 , X1 ) is a compatible couple of Banach spaces. This means that we are given a topological vector space Z, and one-to-one continuous linear mappings from X0 to Z and X1 to Z. Regard X0 and X1 as subspaces of Z. Their ‘sum’ X0 + X1 ⊂ Z is, by deﬁnition, the space of all x0 + x1 , with x0 ∈ X0 and x1 ∈ X1 . This is a Banach space with norm x = inf{x0 X0 + x1 X1 : x0 ∈ X0 , x1 ∈ X1 , x = x0 + x1 }. We let S denote the strip of all complex numbers z with 0 ≤ Re(z) ≤ 1 and we let F = F(X0 , X1 ) be the space of all bounded and continuous functions f : S → X0 + X1 such that the restriction of f to the interior of S is analytic, and such that the maps t → f (it) and t → f (1 + it) belong to C0 (R; X0 ) and C0 (R; X1 ) respectively. Then F is a Banach space for the norm f F = max sup f (it)X0 , sup f (1 + it)X1 . t

(1.22)

t

For any 0 ≤ θ ≤ 1, the interpolation space Xθ = [X0 , X1 ]θ is the subspace of X0 + X1 formed by all x such that x = f (θ) for some f ∈ F. This turns out to be a Banach space for the norm xXθ = inf f F : f ∈ F, f (θ) = x . If we let F θ = F θ (X0 , X1 ) be the subspace of all f ∈ F for which f (θ) = 0, we see that the mapping f → f (θ) induces an isometric isomorphism Xθ = F(X0 , X1 )/F θ (X0 , X1 ).

(1.23)

Assume now that X0 and X1 are operator spaces. Then each interpolation space Xθ has a ‘natural’ operator space structure. Indeed note from (1.22) that the mapping which takes any f ∈ F(X0 , X1 ) to the pair of its restrictions to the lines {Re(z) = 0} and {Re(z) = 1}, induces an isometric embedding F(X0 , X1 ) ⊂ C0 (R; X0 ) ⊕∞ C0 (R; X1 ). By 1.2.17 and 1.2.18, we may consider F(X0 , X1 ) as an operator space, the norm on Mn (F(X0 , X1 )) being inherited from C0 (R; Mn (X0 )) ⊕∞ C0 (R; Mn (X1 )).

16

Completely positive maps

Then taking the resulting quotient operator space structure on F/F θ and applying (1.23), makes Xθ an operator space. More explicitly, the matrix norms on the operator space Xθ are given for any [xjk ] ∈ Mn (Xθ ) by [xjk ]n = inf max sup [fjk (it)]Mn (X0 ) , sup [fjk (1 + it)]Mn (X1 ) , t

t

the inﬁmum taken over all fjk ∈ F(X0 , X1 ) such that fjk (θ) = xjk , for all j, k. Observe that for each n ≥ 1, we have natural one-to-one continuous linear maps Mn (X0 ) → Mn (Z) and Mn (X1 ) → Mn (Z) . Hence (Mn (X0 ), Mn (X1 )) is a compatible couple of Banach spaces. It follows easily from the above discussion that Mn (F (X0 , X1 )) = F(Mn (X0 ), Mn (X1 )) isometrically, and hence (1.24) Mn (Xθ ) = Mn (X0 ), Mn (X1 ) θ . This formula readily implies that a key interpolation theorem (see [33, Theorem 4.1.2]) extends to completely bounded maps. Namely, let (Y0 , Y1 ) be another compatible couple of operator spaces, and let u : X0 + X1 → Y0 + Y1 be a linear map. If u is completely bounded as a map from X0 into Y0 , and from X1 into Y1 , then u is completely bounded from Xθ into Yθ , for any θ ∈ (0, 1), with θ u : Xθ −→ Yθ cb ≤ u : X0 −→ Y0 1−θ cb u : X1 −→ Y1 cb .

1.2.31 (Ultraproducts) Let U be an ultraﬁlter on a set I and let (Xi )i∈I be a family of operator spaces. We let N U ⊂ ⊕∞ i Xi be the space of all (xi )i such that limU xi Xi = 0. By deﬁnition, the ultraproduct of the family (Xi )i∈I along U is the quotient operator space Xi /U = ⊕∞ NU i Xi i∈I

from 1.2.17 and 1.2.14. It is easy to check that for any x = (xi )i in ⊕i Mn (Xi ), the norm of its class x˙ modulo Mn (N U ) is equal to x ˙ = lim xi Mn (Xi ) . U This implies the ‘injectivity’ of ultraproducts. Namely if (Yi )i∈I is another family of operator spaces such that Xi ⊂ Yi completely isometrically for each i ∈ I, then i∈I Xi /U ⊂ i∈I Yi /U completely isometrically. Finally, we observe that if (Ai )i∈I is a family of C ∗ -algebras, then N U is an ∗ ideal of ⊕∞ i Xi and hence their ultraproduct is fairly clearly a C -algebra. Note ∞ ˙ that if (ai )i and (bi )i belong to ⊕i Ai , the product a˙ b of their classes modulo N U is the class of (ai bi )i . 1.3 COMPLETELY POSITIVE MAPS 1.3.1 (Unital operator spaces) Recall that a C ∗ -algebra A is called unital if it contains an identity element 1. We say that an operator space X is unital if it

Operator spaces

17

has a distinguished element usually written as e or 1, called the identity of X, such that there exists a complete isometry u : X → A into a unital C ∗ -algebra with u(e) = 1. A unital-subspace of such X is a subspace containing e. 1.3.2 (Operator systems) An operator system is a unital-subspace S of a unital C ∗ -algebra A which is selfadjoint, that is, x∗ ∈ S if and only if x ∈ S. A subsystem of an operator system S is a selfadjoint linear subspace of S containing the ‘identity’ 1 of S. If S is an operator system, a subsystem of a C ∗ -algebra A, then S has a distinguished ‘positive cone’ S+ = {x ∈ S : x ≥ 0 in A}. We also write Ssa for the real vector space of selfadjoint elements x (i.e. those satisfying x = x∗ ) in S. Then S has an associated ordering ≤, namely we say that x ≤ y if x, y are selfadjoint and y −x ∈ S+ . By the usual trick, any element of an operator system S is of the form h + ik for h, k ∈ Ssa . Also, if h ∈ Ssa then h1 + h and h1 − h are positive. Thus Ssa = S+ − S+ . A linear map u : S → S between operator systems is called ∗-linear if u(x∗ ) = u(x)∗ for all x ∈ S. Some authors say that such a map is selfadjoint. We say that u is positive if u(S+ ) ⊂ S+ . By facts at the end of the last paragraph it is easy to see that a positive map is ∗-linear. The operator system M n (S), which is a subsystem of Mn (A), has a ‘positive cone’ too, and thus it makes sense to talk about completely positive maps between operator systems. These are the maps u such that un = IMn ⊗ u : Mn (S) → Mn (S ) is positive for all n ∈ N. Indeed the morphisms in the category of operator systems are often taken to be the unital completely positive maps. Suppose that S is a subsystem of a unital C ∗ -algebra. By the Hahn–Banach theorem and A.4.2 (resp. (A.11)), it follows that Ssa (resp. S+ ) is exactly the set of elements x ∈ S such that ϕ(x) ∈ R (resp. ϕ(x) ≥ 0) for all ϕ ∈ (Span{1, x})∗ with ϕ(1) = ϕ = 1. From this it is clear that if u : S1 → S2 is a contractive unital linear map between operator systems, then u is a positive and ∗-linear map. Applying this principle to un , we see that a completely contractive unital linear map between operator systems is completely positive. Clearly an isomorphism between operator systems which is unital and completely positive, and has completely positive inverse, preserves all the ‘order’. Such a map is called a complete order isomorphism. The range of a completely positive unital map between operator systems is clearly also an operator system; we say that such a map is a complete order injection if it is a complete order isomorphism onto its range. The following simple fact relates the norm to the matrix order, and is an elementary exercise using the deﬁnition of a positive operator. Namely, if x is an element of a unital C ∗ -algebra or operator system A, or if x ∈ B(K, H), then

1 x x∗ 1

≥ 0

⇐⇒

x ≤ 1.

Here ‘≥ 0’ means ‘positive in M2 (A)’ (or ‘positive in B(H ⊕ K)’).

(1.25)

18

Completely positive maps

1.3.3 It is easy to see from (1.25) that a completely positive unital map u between operator systems is completely contractive. (For example, to see that u is contractive, take x ≤ 1, and apply u2 to the associated positive matrix in (1.25). This is positive, so that using (1.25) again we see that u(x) ≤ 1.) Putting this together with some facts from 1.3.2 we see that a unital map between operator systems is completely positive if and only if it is completely contractive; and in this case the map is ∗-linear. If, further, u is one-to-one, then by applying the above to u and u−1 one sees immediately that a unital map between operator systems is a complete order injection if and only if it is a complete isometry. We omit the well-known proofs of the following two results. Theorem 1.3.4 (Stinespring) Let A be a unital C ∗ -algebra. A linear map u : A → B(H) is completely positive if and only if there is a Hilbert space K, a unital ∗-homomorphism π : A → B(K), and a bounded linear V : H → K such that u(a) = V ∗ π(a)V for all a ∈ A. This can be accomplished with u cb = V 2 . Also, this equals u. If u is unital then we may take V to be an isometry; in this case we may view H ⊂ K, and we have u(·) = PH π(·)|H . Theorem 1.3.5 (Arveson’s extension theorem) If S is a subsystem of a unital C ∗ -algebra A, and if u : S → B(H) is completely positive, then there exists a completely positive map u ˆ : A → B(H) extending u. Indeed, if u is unital, then 1.3.5 may be easily seen from 1.2.10 and 1.3.3 (although usually one proves 1.2.10 using the completely positive variant). Lemma 1.3.6 (Arveson) Suppose that X is a unital-subspace of an operator system, and suppose that u : X → B(K) is a unital contraction (resp. complete contraction, complete isometry) with range Y . Then there exists a positive map (resp. completely positive map, complete order isomorphism) u ˜ between the operator systems X + X and Y + Y , which extends u, namely the map x1 + x∗2 → u(x1 ) + u(x2 )∗ , for x1 , x2 ∈ X. Proof Note that the ‘contraction’ result here applied to the ampliﬁcations u n will imply the ‘complete contraction’ result; and the complete isometry case will then follow by considering u−1 . Suppose that u is a contraction, and consider u restricted to the operator subsystem X ∩ X . This is a unital contraction and therefore is positive and ∗-linear by a fact in 1.3.2. From this it is easy to check directly that the formula above for u ˜ is well deﬁned. Suppose that x1 + x∗2 ∈ X + X is positive, and that ζ is a unit vector in K, so that ϕ = ·ζ, ζ is a state on B(K). Then ϕ ◦ u extends by the Hahn–Banach theorem to a contractive unital functional ψ on X + X . By the aforementioned fact from 1.3.2, ψ is therefore positive and ∗-linear. Thus (u(x1 ) + u(x2 )∗ )ζ, ζ = ψ(x1 ) + ψ(x2 ) = ψ(x1 + x∗2 ) ≥ 0. Hence u(x1 ) + u(x2 )∗ ≥ 0.

2

Operator spaces

19

1.3.7 (The diagonal) Because of this last result, if X is a unital operator space (see 1.3.1) then there is an essentially unique operator system, written as X +X , which is spanned by X and its adjoint space X . Indeed, if u : X → B(H) is any unital complete isometry into B(H) (or into an operator system), then by 1.3.6 the operator system u(X) + u(X) is (up to unital complete order isomorphism) independent of the particular u. We usually identify two unital operator spaces up to unital completely isometric isomorphism. By the same principle, any such X contains a canonical operator system, namely ∆(X) = {x ∈ X : u(x)∗ ∈ u(X)}, where u is any unital complete isometry as in the last paragraph. This is well deﬁned independently of u. We call ∆(X) the diagonal of X. The following follows immediately from a fact in 1.3.2 and the last deﬁnition: Corollary 1.3.8 If S is an operator system, if Y is a unital operator space, and if u : S → Y is a unital contraction, then Ran(u) ⊂ ∆(Y ), and u is positive. Proposition 1.3.9 (A Kadison–Schwarz inequality) If u : A → B is a unital completely positive (or equivalently unital completely contractive) linear map between unital C ∗ -algebras, then u(a)∗ u(a) ≤ u(a∗ a), for all a ∈ A. Proof By 1.3.4 we have u = V ∗ π(·)V , with V ≤ 1 and π a ∗-homomorphism. 2 Thus u(a)∗ u(a) = V ∗ π(a)∗ V V ∗ π(a)V ≤ V ∗ π(a)∗ π(a)V = u(a∗ a). Corollary 1.3.10 Let u : A → B be a completely isometric unital surjection between unital C ∗ -algebras. Then u is a ∗-isomorphism. Proof By 1.3.9 applied to both u and u−1 we have u(x)∗ u(x) = x∗ x for all x ∈ A. Now use the polarization identity (see (1.1)). 2 Proposition 1.3.11 Let u : A → B be as in 1.3.9. Suppose that c ∈ A, and that c satisﬁes u(c)∗ u(c) = u(c∗ c). Then u(ac) = u(a)u(c) for all a ∈ A. Proof Suppose that B ⊂ B(H). We write u = V ∗ π(·)V as in Stinespring’s theorem, with V ∗ V = IH . Let P = V V ∗ be the projection onto V (H). By hypothesis V ∗ π(c)∗ P π(c)V = V ∗ π(c)∗ π(c)V . For ζ ∈ H, set η = π(c)V ζ. Then P η2 = V ∗ π(c)∗ P π(c)V ζ, ζ = η2 . Thus P η = η, and V V ∗ π(c)V = π(c)V . 2 Therefore u(a)u(c) = V ∗ π(a)V V ∗ π(c)V = V ∗ π(a)π(c)V = u(ac). 1.3.12 (Completely positive bimodule maps) An immediate consequence of 1.3.11: Suppose that u : A → B is as in 1.3.9, and that there is a C ∗ -subalgebra C of A with 1A ∈ C, such that π = u|C is a ∗-homomorphism. Then u(ac) = u(a)π(c)

and

u(ca) = π(c)u(a)

(a ∈ A, c ∈ C).

Theorem 1.3.13 (Choi and Eﬀros) Suppose that A is a unital C ∗ -algebra, and that Φ : A → A is a unital, completely positive (or equivalently by 1.3.3, completely contractive), idempotent map. Then we may conclude: (1) R = Ran(Φ) is a C ∗ -algebra with respect to the original norm, involution, and vector space structure, but new product r1 ◦Φ r2 = Φ(r1 r2 ).

20

Completely positive maps

(2) Φ(ar) = Φ(Φ(a)r) and Φ(ra) = Φ(rΦ(a)), for r ∈ R and a ∈ A. (3) If B is the C ∗ -subalgebra of A generated by the set R, and if R is given the product ◦Φ , then Φ|B is a ∗-homomorphism from B onto R. Proof (2) By linearity and the fact that a positive map is ∗-linear (see 1.3.2), we may assume that a, r are selfadjoint. Set 0 r ∗ d = d = . r∗ a Then Φ2 (d2 ) ≥ (Φ2 (d))2 by the Kadison–Schwarz inequality 1.3.9, so that 2 r rΦ(a) Φ(r2 ) Φ(ra) ≥ . Φ(ar) ∗ Φ(a)r ∗ Here ∗ is used for a term we do not care about. Applying Φ2 gives Φ(r2 ) Φ(ra) Φ(r2 ) Φ(rΦ(a)) ≥ . Φ(ar) ∗ Φ(Φ(a)r) ∗

Thus

0 Φ(ra) − Φ(rΦ(a)) Φ(ar) − Φ(Φ(a)r) ∗

≥ 0,

which implies that Φ(ra) − Φ(rΦ(a)) = 0 and Φ(ar) − Φ(Φ(a)r) = 0. (1) By (2) we have for r1 , r2 , r3 ∈ R that (r1 ◦Φ r2 ) ◦Φ r3 = Φ(Φ(r1 r2 )r3 ) = Φ(r1 r2 r3 ). Similarly, r1 ◦Φ (r2 ◦Φ r3 ) = Φ(r1 r2 r3 ), which shows that the multiplication is associative. It is easy to check that R (with original norm, involution, and vector space structure, but new multiplication) satisﬁes the conditions necessary to be a C ∗ -algebra. For example: (r1 ◦Φ r2 )∗ = Φ(r1 r2 )∗ = Φ(r2∗ r1∗ ) = r2∗ ◦Φ r1∗ . We check the C ∗ -identity using the Kadison–Schwarz inequality 1.3.9: r∗ ◦Φ r = Φ(r∗ r) ≥ Φ(r)∗ Φ(r) = r∗ r = r2 , and conversely, r2 = r∗ r ≥ Φ(r∗ r) = r∗ ◦Φ r. (3) This will follow if we can prove that Φ(r1 r2 · · · rn ) = r1 ◦Φ r2 · · · ◦Φ rn , for ri ∈ R. This follows in turn by induction on n. Supposing that it is true for n = k, we see that r1 ◦Φ r2 · · · ◦Φ rk+1 equals Φ((r1 ◦Φ r2 · · · ◦Φ rk )rk+1 ) = Φ(Φ(r1 r2 · · · rk )rk+1 ) = Φ(r1 r2 · · · rk rk+1 ), using (2) in the last equality.

2

Operator spaces

21

It is important to note, and easy to check, that the canonical matrix norms from 1.2.3 for the C ∗ -algebra Φ(A) in the result above, coincide with its canonical matrix norms as a subspace of A. This may be seen by the uniqueness of a complete C ∗ -norm on a ∗-algebra (which in turn is immediate from A.5.8), and an application of Theorem 1.3.13 to the canonically associated projection Φn on Mn (A), for each n ∈ N. 1.3.14 (The Paulsen system) If X is a subspace of B(H), we deﬁne the Paulsen system to be the operator system CIH X λ x S(X) = : x, y ∈ X, λ, µ ∈ C = X CIH y∗ µ in M2 (B(H)), where the entries λ and µ in the last matrix stand for λIH and µIH respectively. The following important lemma shows that as an operator system (i.e. up to complete order isomorphism) S(X) only depends on the operator space structure of X, and not on its representation on H. Lemma 1.3.15 (Paulsen) Suppose that for i = 1, 2, we are given Hilbert spaces Hi , Ki , and linear subspaces Xi ⊂ B(Ki , Hi ). Suppose that u : X1 → X2 is a linear map. Let S i be the following operator system inside B(Hi ⊕ Ki ): CIHi Xi . Si = Xi CIKi If u is contractive (resp. completely contractive, completely isometric), then λ u(x) λ x → Θ: ∗ u(y)∗ µ y µ is positive (resp. completely positive and completely contractive, a complete order injection) as a map from S 1 to S 2 . Proof Suppose that z is a positive element of S1 . Thus a x z = x∗ b where a and b are positive. Since z ≥ 0 if and only if z + 1 ≥ 0 for all > 0, we may assume that a and b are invertible. Then −1 −1 1 1 a 2 0 a 2 0 a x 1 a− 2 xb− 2 = ≥ 0. 1 1 1 1 x∗ b 0 b− 2 0 b− 2 b− 2 x∗ a− 2 1 1

1

Hence by (1.25), we have that a− 2 xb− 2 ≤ 1. Applying u we obtain that 1 1 a− 2 u(x)b− 2 ≤ 1. Reversing the argument above now shows that Θ(z) ≥ 0. So Θ is positive, and a similar argument shows that it is completely positive if u is completely contractive. By 1.3.3 we have that Θ is completely contractive in that case. If in addition u is a complete isometry, then applying the above to u 2 and u−1 we obtain the ﬁnal assertion.

22

Operator space duality

1.4 OPERATOR SPACE DUALITY An operator space Y is said to be a dual operator space if Y is completely isometrically isomorphic to the operator space dual (see 1.2.20) X ∗ of an operator space X. We also say that X is an operator space predual of Y , and sometimes we write X as Y∗ . If X, Y are dual operator spaces then we write w ∗ CB(X, Y ) for the space of w∗ -continuous completely bounded maps from X to Y . Unless otherwise indicated, in what follows the symbol X ∗ denotes the dual space together with its dual operator space structure as deﬁned in 1.2.20. Of course X ∗∗ is considered as the dual operator space of X ∗ . Proposition 1.4.1 If X is an operator space then X ⊂ X ∗∗ completely isometrically via the canonical map iX . Proof Let X ⊂ B(H). By the deﬁnitions, iX is completely contractive. To see that iX is completely isometric, it suﬃces to ﬁnd for a given n ∈ N, > 0, and [xkl ] ∈ Mn (X), an integer m and a completely contractive u : B(H) → Mm such that [u(xkl )] ≥ [xkl ] −. For such[xkl ] and , by (1.2) we may choose ζ1 , . . . , ζn , η1 , . . . , ηn ∈ H with l ζl 2 = k ηk 2 = 1 and xkl ζl , ηk ≥ [xkl ] − . k,l

Let K = Span {ζ1 , . . . , ζn , η1 , . . . , ηn }, and let u : B(H) → B(K) be the map u(x) = PK x|K . If m = dim(K), then we may view u as mapping into Mm . It is completely contractive, by the remark after 1.2.8 for example. Finally, [u(xkl )] ≥ u(xkl )ζl , ηk = xkl ζl , ηk ≥ [xkl ] − , k,l

k,l

2

which is the desired inequality. 1.4.2 (Remarks) From 1.4.1 we have for any [xij ] ∈ Mn (X) that [xij ]n = sup{[ϕkl (xij )] : m ∈ N, [ϕkl ] ∈ Ball(Mm (X ∗ ))}

(1.26)

In other words, the following canonical isomorphism is a complete isometry: Mn (X) ∼ = w∗ CB(X ∗ , Mn ) ⊂ CB(X ∗ , Mn ).

(1.27)

Another consequence of 1.4.1, is that if X is an operator space which as a Banach space is reﬂexive, then X ∼ = X ∗∗ completely isometrically. 1.4.3 (The adjoint map) The ‘adjoint’ or ‘dual’ u∗ of a completely bounded map u : X → Y between operator spaces is completely bounded from Y ∗ to X ∗ , with u∗ cb ≤ ucb, as may be seen from the obvious computation. Indeed using (1.26) during this computation or applying 1.4.1, one sees that u cb = u∗ cb . Direct computations from the deﬁnitions also show that if u is a complete quotient map then u∗ is a complete isometry. It is slightly harder to see that

Operator spaces

23

u is completely isometric if and only if u∗ is a complete quotient map. Indeed, if u is a complete isometry, then we may regard X ⊂ Y , and then an element in the open ball of Mn (X ∗ ) ∼ = CB(X, Mn ) may be ‘extended’ by 1.2.10 to an element in the open ball of CB(Y, Mn ) = Mn (Y ∗ ). This shows that u∗ is a complete quotient map. Conversely, if u∗ is a complete quotient map then u∗∗ is a complete isometry, so that u is a complete isometry (using 1.4.1). Finally, one may see as in the Banach space case, that for complete operator spaces, if u ∗ is a complete isometry then u is a complete quotient map (e.g. see A.2.3 in [149]). Thus u is a complete isometry if and only if u∗∗ is a complete isometry. 1.4.4 (Duality of subspaces and quotients) The operator space versions of the usual Banach duality of subspaces and quotients follow easily from 1.4.3. If X is a subspace of Y , then applying 1.4.3 to the inclusion map X → Y yields the fact that X ∗ ∼ = Y ∗ /X ⊥ completely isometrically via the canonical map. Similarly, the dual of the canonical quotient map Y → Y /X is the canonical complete isometry (Y /X)∗ ∼ = X ⊥ . The predual versions go through too with the same proofs as in the Banach space case: if X is a w ∗ -closed subspace of a dual operator space Y , then (Y∗ /X⊥ )∗ ∼ = (X⊥ )⊥ = X as dual operator spaces. Also, ∗ ∼ ⊥ (X⊥ ) = Y /(X⊥ ) = Y /X completely isometrically. 1.4.5 (The trace class operator space) If H is a Hilbert space then B(H) is a dual operator space. More precisely, let us equip its predual Banach space S 1 (H) (e.g. see A.1.2) with the operator space structure it inherits from B(H)∗ via the canonical isometric inclusion S 1 (H) → B(H)∗ . Then B(H) = S 1 (H)∗ completely isometrically. Indeed the canonical map from B(H) to S 1 (H)∗ is completely contractive by deﬁnition. That this map is completely isometric follows from the fact, included in the proof of 1.4.1, that for any n ∈ N, > 0, and [xkl ] ∈ Mn (B(H)), we can ﬁnd an integer m and a w ∗ -continuous completely contractive u : B(H) → Mm such that [u(xkl )] ≥ [xkl ] − . Similarly, B(K, H) is the dual operator space of the space S 1 (H, K) of trace class operators, the latter regarded as a subspace of B(K, H)∗ . Henceforth, when we write S 1 (H, K) we will mean the operator space predual of B(K, H) described above. Similarly, we will henceforth also view Sn1 = Mn∗ as an operator space. Lemma 1.4.6 Any w∗ -closed subspace X of B(H) is a dual operator space. Indeed, if Y = S 1 (H)/X⊥ is equipped with its quotient operator space structure inherited from S 1 (H), then X ∼ = Y ∗ completely isometrically. Proof This follows from 1.4.4 and 1.4.5.

2

In particular this shows that any W ∗ -algebra equipped with its ‘natural’ operator space structure (see 1.2.3) is a dual operator space. The converse of 1.4.6 is true too, as we see next, so that ‘dual operator spaces’, and the w∗ -closed subspaces of some B(H), are essentially the same thing. Lemma 1.4.7 Any dual operator space is completely isometrically isomorphic, via a homeomorphism for the w ∗ -topologies, to a w∗ -closed subspace of B(H), for some Hilbert space H.

24

Operator space duality

Proof Suppose that W is a dual operator space, with predual X. Let Y = C, and recall from 1.2.19 the construction of a complete isometry W = CB(X, Y ) −→ ⊕∞ x∈I Mnx (Y ) = ⊕x∈I Mnx , namely the map J taking w ∈ W to the tuple ([w, xij ])x in ⊕x Mnx . Since ∗ the maps w → w, xij are w∗ -continuous, and since ⊕ﬁn x Mnx is dense in the 1 ∗ Banach space predual ⊕x Mnx of ⊕x Mnx , it is easy to see that J is w∗ -continuous too. Thus by A.2.5, W is completely isometrically and w ∗ -homeomorphically isomorphic to a w∗ -closed subspace of ⊕x Mnx . If the latter is regarded as a von Neumann subalgebra of B(H) say, then W is completely isometrically and 2 w∗ -homeomorphically isomorphic to a w ∗ -closed subspace of B(H). 1.4.8 (W ∗ -continuous extensions) If X and Y are two operator spaces and if u : X → Y ∗ is completely bounded, then its (unique) w ∗ -continuous extension ucb = ucb. u ˜ : X ∗∗ → Y ∗ provided by A.2.2 is completely bounded, with ˜ Indeed recall from A.2.2 that u ˜ = i∗Y ◦ u∗∗ ; and this extension clearly satisﬁes the asserted norm equality (using the ﬁrst paragraph in 1.4.3). Note that since w∗ u ˜ is w∗ -continuous, we have u˜(X ∗∗ ) ⊂ u(X) . The above also shows that CB(X, Y ∗ ) = w∗ CB(X ∗∗ , Y ∗ )

(1.28)

isometrically via the mapping u → u ˜. In fact (1.28) can easily be made a complete isometry with the help of the later item 1.6.2. By 1.4.5, the last paragraph applies in particular to B(K, H) valued maps. 1.4.9 (The second dual) Let X be an operator space, and ﬁx n ∈ N. We wish to compare the spaces Mn (X ∗∗ ) (equipped with its ‘operator space dual’ matrix norms as in 1.2.20), and Mn (X)∗∗ . First note that they can be canonically identiﬁed as topological vector spaces, as may Mn (X ∗ ) and Mn (X)∗ (this is just the simple fact that if E = F ⊕ · · ·⊕ F is a ﬁnite direct sum of copies of a Banach space F , which has been assigned a norm compatible with the norm on F , then E ∗ is canonically algebraically and topologically isomorphic to F ∗ ⊕ · · · ⊕ F ∗ , the latter with any norm compatible with the norm on F ∗ ). We will prove in 1.4.11 below that this identiﬁcation is an isometry. As a ﬁrst easy step, let us check that the identity mapping from Mn (X)∗∗ to Mn (X ∗∗ ) implementing this identiﬁcation, is a contraction. For this purpose, let η = [ηij ] ∈ Mn ⊗ X ∗∗ , and assume that its norm in Mn (X)∗∗ is less than or equal to 1. By Goldstine’s lemma A.2.1, there is a net (xs )s in Ball(Mn (X)) such that xs → η in the w∗ -topology of Mn (X)∗∗ . Let ϕ = [ϕpq ] be an element of Ball(Mm (X ∗ )) for some m ≥ 1. The fact that xs → η in the w∗ -topology of Mn (X)∗∗ , is equivalent to the fact that xsij → ηij in the w∗ -topology of X ∗∗ for all 1 ≤ i, j ≤ n. Hence we deduce that [ηij , ϕpq ] = lim [ϕpq , xsij ]. s

By (1.6) or (1.26), the norm of the latter matrix is dominated by 1. Thus [ηij , ϕpq ] ≤ 1. By (1.6) again, we deduce that [ηij ]Mn (X ∗∗ ) ≤ 1, which proves the result.

Operator spaces

25

1.4.10 (The second dual of a C ∗ -algebra) If A is a C ∗ -algebra, then the second dual A∗∗ has two canonical operator space structures. The ﬁrst is its ‘operator space dual’ matrix norms (see 1.2.20); when we write Mn (A∗∗ ) in the lines below, we will be using these norms. The second are those from 1.2.3, arising from the fact that A∗∗ is a C ∗ -algebra (see A.5.6). We claim that these two operator space structures are the same. To see this we will need to use notation and facts from A.5.6. In particular we let πu : A → B(Hu ) denote the universal representation u (A∗∗ ) (see the proof of A.5.6). The of A, and we write A†† for the W ∗ -algebra π claim will follow if we can prove for any ﬁxed n ≥ 1 that Mn (A)∗∗ ∼ = Mn (A†† ) = Mn (A∗∗ ) ∼

isometrically

(1.29)

via the canonical maps. The ﬁrst of these maps is the contraction from M n (A)∗∗ to Mn (A∗∗ ) discussed in 1.4.9. The second map in (1.29) is IMn ⊗ π u , which is a contraction since according to 1.4.8, the mapping π u is a complete contraction. To establish (1.29), we need only prove that the resulting contraction ρ : Mn (A)∗∗ −→ Mn (A†† ) is isometric. It is clearly one-to-one. We regard Mn (A)∗∗ as a C ∗ -algebra by applying A.5.6 to Mn (A). It therefore suﬃces to (n) check that ρ is a ∗-homomorphism. Regarding ρ as valued in B(Hu ), we have (n) ρ(η) ζ, ξ = i,j πu (ηij )ζj , ξi , for ζ = [ζi ], ξ = [ξi ] ∈ Hu , and η = [ηij ] as in ∗ 1.4.9. Since π u , and the maps η → ηij , are w -continuous, it follows that ρ is w∗ -continuous too. By the w∗ -continuity properties of the involution and product in a W ∗ -algebra (see A.5.1), it suﬃces to prove that the restriction of ρ to Mn (A) is a ∗-homomorphism. Since the latter equals IMn ⊗ πu , we are done. The last result has many consequences. For example, we can use it to see that S ∞ (H)∗ = S 1 (H) completely isometrically, complementing the observation in 1.4.5. Also we obtain: Theorem 1.4.11 If X is an operator space then Mm,n (X)∗∗ ∼ = Mm,n (X ∗∗ ) completely isometrically for all m, n ∈ N (via an isomorphism extending the identity map on Mm,n (X)). Proof First suppose that m = n, and choose a C ∗ -algebra A with X ⊂ A completely isometrically. Then X ∗∗ ⊂ A∗∗ completely isometrically by 1.4.3, hence we have both Mn (X)∗∗ ⊂ Mn (A)∗∗ , and Mn (X ∗∗ ) ⊂ Mn (A∗∗ ), isometrically. Under the identiﬁcations between Mn (A)∗∗ and Mn (A∗∗ ) and between Mn (X)∗∗ and Mn (X ∗∗ ) discussed above, these two embeddings are easily seen to be the same. Hence the isometry Mn (A∗∗ ) = Mn (A)∗∗ provided by 1.4.10, implies that we also have Mn (X ∗∗ ) = Mn (X)∗∗ isometrically. The complete isometry follows by iterating the isometric case. The case n = m may be derived from the above, viewing Mm,n (X) as a 2 subspace of Mk (X) where k = max{m, n}. We leave this as an exercise. 1.4.12 (Duality of Min and Max) For any Banach space E, we have Min(E)∗ = Max(E ∗ )

and

Max(E)∗ = Min(E ∗ ).

(1.30)

26

Operator space duality

To see this, note that by using (1.8), (1.11) and (1.12), and the basic properties ˇ seen in A.3.1, we have of ⊗ ˇ ∗∼ Mn (Max(E)∗ ) ∼ = CB(Max(E), Mn ) = B(E, Mn ) ∼ = Mn ⊗E = Mn (Min(E ∗ )). That is, Max(E)∗ = Min(E ∗ ). Therefore Max(E ∗ )∗ = Min(E ∗∗ ). However we claim that Min(E ∗∗ ) = Min(E)∗∗ . This claim may be seen using the fact that ‘minimal operator spaces’ are completely isometric to subspaces of unital commutative C ∗ -algebras (i.e. of C(K)-spaces), the fact that the second dual of a complete isometry is a complete isometry (see 1.4.4), and 1.4.10. Hence Max(E ∗ ) and Min(E)∗ are two operator space structures on E ∗ with the same operator space dual, and therefore they are completely isometric, by 1.4.1. 1.4.13 (The 1-direct sum) For a family {Xλ : λ ∈ I} of operator spaces, let ⊕ﬁn λ Xλ be the set of ‘ﬁnitely supported’ elements of the algebraic direct sum ∞ ∗ ∗ of the Xλ . There is a canonical one-to-one map µ : ⊕ﬁn λ Xλ → (⊕λ Xλ ) , 1 and we may deﬁne the 1-direct sum ⊕λ Xλ by identifying it with the closure of ∞ ∗ ∗ 1 µ(⊕ﬁn λ Xλ ) in the dual operator space (⊕λ Xλ ) . Evidently, the 1norm on ⊕λ Xλ 1 is exactly the usual ‘ -norm’ λ xλ ; and a tuple (xλ ) is in ⊕λ Xλ if and only if xλ ∈ X for all λ, and λ xλ < ∞. If Xλ = X for all λ ∈ I then we write 1I (X) for ⊕1λ X. It is easy to argue that the canonical inclusion and projection maps λ and πλ between ⊕1λ Xλ and its ‘λth summand’ are complete isometries and complete quotient maps respectively. Also, ∗ (⊕1λ Xλ )∗ ∼ = ⊕∞ λ Xλ

as dual operator spaces.

(1.31)

Similarly, the dual of an ∞-direct sum of ﬁnitely many operator spaces (or the dual of a c0 -direct sum of inﬁnitely many) is completely isometric to the 1-direct sum of the duals. We leave these assertions to the reader to check. In particular, ∗ 1I (X)∗ = ∞ I (X )

and

c0 (X)∗ = 1 (X ∗ ).

The 1-direct sum has the following useful universal property. Namely, if Z is another operator space and uλ : Xλ → Z are completely contractive linear maps, then there is a canonical complete contraction u : ⊕1λ Xλ → Z such that u ◦ λ = uλ . One may see this with a diagram chase: u∗λ maps Z ∗ to Xλ∗ , so that by the universal property of the ∞-direct sum we obtain a map v : Z ∗ → ⊕λ Xλ∗ . Then v ∗ : (⊕λ Xλ∗ )∗ → Z ∗∗ . Composing v with the canonical map ⊕1λ Xλ → (⊕λ Xλ∗ )∗ gives a map u : ⊕1λ Xλ → Z ∗∗ . One easily checks that u ◦ λ = uλ , whence u actually maps into Z, and we obtain the desired result. This universal property may be rephrased as an isometric isomorphism CB(⊕1i Xi , Z) ∼ = ⊕∞ i CB(Xi , Z). Replacing Z by Mn (Z) and using (1.7) shows that this relation also holds completely isometrically.

Operator spaces

27

∗ The canonical inclusion and projection maps λ and πλ between ⊕∞ λ Xλ and ∗ its ‘λth summand’ are w -continuous. This may be seen by dualizing the inclusion and projection maps between ⊕1λ Xλ and its summands. Any operator space is a complete quotient of a 1-sum of spaces of the form Sn1 = Mn∗ . This may be seen, for example, by applying the second last fact in 1.4.3 to the map J in the proof of 1.4.7. Another frequently used property of these direct sums is as follows. Suppose that uλ : Xλ → Yλ is a complete contraction for each λ ∈ I. Then this family of maps clearly induces a single contraction u = (uλ ) : ⊕1λ Xλ → ⊕1λ Yλ . It ∞ also clearly induces a complete contraction (uλ ) : ⊕∞ λ Xλ → ⊕λ Yλ ; which is completely isometric if every uλ is. Applying the ﬁrst fact in the last sentence to ∗ ∞ ∗ the family of maps u∗λ , gives a complete contraction from ⊕∞ λ Yλ to ⊕λ Xλ , and it is easy to see from the norm density of the ‘ﬁnitely supported’ tuples in a 1-direct sum, that this map is the dual of the map u above, and hence is w ∗ -continuous. From this and 1.4.3 we deduce that u is a complete contraction. We have also shown that families (vλ ) of w∗ -continuous complete contractions between dual spaces canonically induce a single w ∗ -continuous complete contraction between the ∞-direct sums of the spaces.

1.5 OPERATOR SPACE TENSOR PRODUCTS As we said in the introduction, the reader should feel free to skim through these results, returning later when necessary for a deﬁnition or fact. In this section and the next we tend to leave more details than in previous sections as exercises for the interested reader. 1.5.1 (Minimal tensor product) Let X and Y be operator spaces, and let X ⊗Y denote their algebraic tensor product. Any ﬁnite rank bounded map between operator spaces is ‘composed’ of scalar functionals, and hence is automatically completely bounded by 1.2.6. Thus the correspondences between tensor products and ﬁnite rank mappings discussed in A.3.1 yield embeddings X ⊗ Y → CB(Y ∗ , X) and X ⊗ Y ∗ → CB(Y, X). The minimal tensor product X ⊗min Y may then be deﬁned to be (the completion of) X ⊗ Y in the matrix norms inherited from the operator space structure on CB(Y ∗ , X) described in 1.2.19. That is, X ⊗min Y → CB(Y ∗ , X) completely isometrically. (1.32) n That is, if u = k=1 xk ⊗ yk ∈ X ⊗ Y , then the norm of u in X ⊗min Y equals sup xk ψij (yk ) , (1.33) k

Mn (X)

the supremum taken over all ﬁnite matrices [ψij ] of norm 1, where ψij ∈ Y ∗ . A similar formula holds for u ∈ Mn (X ⊗ Y ). From this similar form of (1.33) and (1.26), we see that the matrix norms on X ⊗min Y are also given by the formula [wrs ]n = sup [(ϕkl ⊗ ψij )(wrs )] (1.34)

28

Operator space tensor products

for [wrs ] ∈ Mn (X ⊗ Y ), where the supremum is taken over all ﬁnite matrices [ϕkl ] and [ψij ] of norm 1, where ϕkl ∈ X ∗ and ψij ∈ Y ∗ , and where ϕkl ⊗ ψij denotes the obvious functional on X ⊗ Y formed from ϕkl and ψij . We see from (1.34) that ⊗min is commutative, that is X ⊗min Y = Y ⊗min X as operator spaces. It is also easy to see from (1.34) that ⊗ min is functorial. That is, if Xi and Yi are operator spaces for i = 1, 2, and if ui : Xi → Yi are completely bounded, then the map x ⊗ y → u1 (x) ⊗ u2 (y) on X1 ⊗ X2 has a unique continuous extension to a map u1 ⊗ u2 : X1 ⊗min X2 → Y1 ⊗min Y2 , with u1 ⊗ u2 cb ≤ u1 cb u2 cb . As an exercise, the reader could check that this is actually an equality, but we shall not need this. If, further, the ui are completely isometric, then so is u1 ⊗ u2 . This latter fact is called the injectivity of the tensor product. To prove it, since u1 ⊗ u2 = (u1 ⊗ I) ◦ (I ⊗ u2 ), we may by symmetry reduce the argument to the case that Y2 = X2 , u2 = IX2 , X1 ⊂ Y1 and that u1 is this inclusion map. Then the result we want follows from (1.32) and the obvious fact that CB(X2∗ , X1 ) ⊂ CB(X2∗ , Y1 ) completely isometrically. For any operator spaces X, Y , we have X ⊗min Y ∗ → CB(Y, X)

completely isometrically.

(1.35)

To prove this, note by the injectivity discussed above that we may assume that X = B(H). However, CB(Y, B(H)) = w ∗ CB(Y ∗∗ , B(H)) ⊂ CB(Y ∗∗ , B(H)) isometrically by (1.28). Since the norm on B(H) ⊗min Y ∗ is induced by the embedding of B(H) ⊗ Y ∗ in the latter space, B(H) ⊗min Y ∗ ⊂ CB(Y, B(H)) isometrically. The complete isometry follows by changing H into H (n) , and using (1.2), (1.7), and the fact that Mn (X ⊗min Y ) = Mn (X)⊗min Y for operator spaces X, Y (which in turn follows from (1.32) and (1.7)). 1.5.2 (Further properties of ⊗min ) Suppose that H1 , H2 are Hilbert spaces, and consider the canonical map π : B(H1 ) ⊗ B(H2 ) → B(H1 ⊗2 H2 ). This is the map taking a rank one tensor S⊗T in B(H1 )⊗B(H2 ) to the map on H1 ⊗2 H2 denoted also by S ⊗ T in 1.1.4. We claim that π actually is a complete isometry when B(H1 )⊗B(H2 ) is given its norm as a subspace of B(H1 )⊗min B(H2 ). To see this, we choose a cardinal I such that H2 = 2I , so that we both have MI ∼ = B(H2 ) and H1 ⊗ 2 H 2 ∼ = 2I (H1 ). By (1.35) and 1.4.5, B(H1 )⊗min B(H2 ) → CB(S 1 (H1 ), MI ). However, by 1.2.29, and (1.19), we have CB(S 1 (H1 ), MI ) ∼ = MI (B(H1 )) ∼ = B(H1 ⊗2 H2 ). This proves the claim. Thus if X and Y are subspaces of B(H1 ) and B(H2 ) respectively, then by the injectivity of this tensor product, we have that X ⊗min Y is completely isometrically isomorphic to the closure in B(H1 ⊗2 H2 ) of the span of the operators x ⊗ y on H1 ⊗2 H2 , for x ∈ X, y ∈ Y . We remark, in passing, that the above says that

Operator spaces

29

the minimal tensor product coincides with the tensor product of the same name used in C ∗ -algebra theory, or with the so-called spatial tensor product. Indeed note that if A and B are C ∗ -subalgebras of B(H1 ) and B(H2 ) respectively, then A ⊗min B may be identiﬁed with the closure of a ∗-subalgebra of B(H1 ⊗2 H2 ). Thus A ⊗min B is a C ∗ -algebra. If A and B are also commutative, then so is A ⊗min B, since it is the closure of a commutative ∗-subalgebra. From the last paragraph and 1.2.2 it is clear that for any operator space X, ∼ Mn (X) (1.36) Mn ⊗min X = completely isometrically. By similar reasoning, using also (1.19), we have KI,J ⊗min X ∼ (1.37) = KI,J (X). Indeed, in the case I = J, both sides of (1.37) correspond to the closure of (I) ), if X ⊂ B(H). Mﬁn I (X) in B(H Similarly, it follows from the second last paragraph, and from the fact that B((H1 ⊗2 H2 ) ⊗2 H3 ) ∼ = B(H1 ⊗2 (H2 ⊗2 H3 )), that ⊗min is associative. That is, (X1 ⊗min X2 ) ⊗min X3 = X1 ⊗min (X2 ⊗min X3 ).

(1.38)

Accordingly, this space will be merely denoted by X1 ⊗min X2 ⊗min X3 . Similarly, one may consider the N -fold minimal tensor product X1 ⊗min · · · ⊗min XN of any N -tuple of operator spaces. Proposition 1.5.3 Let E, F be Banach spaces and let X be an operator space. ˇ as Banach spaces. (1) Min(E) ⊗min X = E ⊗X ˇ ) as operator spaces. (2) Min(E) ⊗min Min(F ) = Min(E ⊗F (3) For any compact space Ω we have (with notation as in 1.2.18), C(Ω) ⊗min X = C(Ω; X)

completely isometrically.

(1.39)

∗

Proof We have isometric embeddings Min(E) ⊗min X ⊂ CB(X , Min(E)) and ˇ ⊂ B(X ∗ , E) by (1.32) and (A.1). However CB(X ∗ , Min(E)) = B(X ∗ , E) E ⊗X by (1.10), which proves (1). The isometry in (2) follows from (1). Thus the complete isometry in (2) will follow if Min(E)⊗min Min(F ) is a minimal operator space. But this is clear if we use the ‘injectivity’ of ⊗min , the fact that ‘minimal operator spaces’ are the subspaces of C(K)-spaces, and the observation in 1.5.2 that the minimal tensor product of commutative C ∗ -algebras is a commutative C ∗ -algebra, and hence is a C(K)-space. We now prove (3). Since C(Ω) is a minimal operator space, the relation C(Ω) ⊗min X = C(Ω; X) holds isometrically by (1) and (A.2). By deﬁnition (see 1.2.18), Mn (C(Ω; X)) = C(Ω; Mn (X)). Replacing X by Mn (X) in the previous relation, we deduce that C(Ω) ⊗min Mn (X) = Mn (C(Ω; X)) isometrically for all n ∈ N. However by (1.36), and also by the associativity and commutativity of ⊗min , we have ∼ C(Ω) ⊗min Mn ⊗min X = ∼ Mn C(Ω) ⊗min X . C(Ω) ⊗min Mn (X) = This shows that Mn (C(Ω) ⊗min X) = Mn (C(Ω; X)) isometrically.

2

30

Operator space tensor products

1.5.4 (Haagerup tensor product) Before we deﬁne this tensor product, we introduce an intimately related class of bilinear maps. Suppose that X, Y , and W are operator spaces, and that u : X × Y → W is a bilinear map. For n, p ∈ N, deﬁne a bilinear map Mn,p (X) × Mp,n (Y ) → Mn (W ) by (x, y) −→

p

u(xik , ykj ) ,

k=1

i,j

where x = [xij ] ∈ Mn,p (X) and y = [yij ] ∈ Mp,n (Y ). If p = n we write this map as un . If the norms of these bilinear maps (as deﬁned in A.3.3) are uniformly bounded over p, n ∈ N, then we say that u is completely bounded, and write the supremum of these norms as ucb. It is easy to see (by adding rows and columns of zeroes to make p = n) that ucb = supn un . We say that u is completely maps of three varicontractive if ucb ≤ 1. Completely bounded multilinear ables have a similar deﬁnition (involving the expression [ k,l u(xik , ykl , zlj )]), and similarly for four or more variables. We remark that if v : X → B(H) and w : Y → B(H) are completely bounded linear maps, then it is easy to see that the bilinear map (x, y) → v(x)w(y) is completely bounded in the sense above, and has completely bounded norm dominated by vcbwcb . Indeed this boils down to the fact that B(H (n) ) is a Banach algebra. Let X, Y be operator spaces. For n ∈ N and z ∈ Mn (X ⊗ Y ) we deﬁne zh = inf{xy},

(1.40)

where the inﬁmum is taken over all p ∈ N, and all ways to write z = x y, where x ∈ Mn,p (X), y ∈ Mp,n (Y ). Here x y denotes the formal matrix p product of x and y using the ⊗ sign as multiplication: namely x y = [ k=1 xik ⊗ ykj ]. We leave it as an exercise (see 9.2.1 in [149] for more details if needed) that the expressions in (1.40) for all n ∈ N, deﬁne an operator seminorm structure on X ⊗ Y in the sense of 1.2.16. Now let u : X × Y → W be a bilinear map which is completely contractive in the sense above. Let u˜ : X ⊗ Y → W be the canonically associated linear map. For z ∈ Mn (X ⊗ Y ), we have by the deﬁnitions above that ˜ un (z) ≤ zh ,

(1.41)

where the latter quantity is as deﬁned in (1.40). If ϕ and ψ are contractive functionals on X and Y respectively, then using 1.2.6 and the fact at the end of the second last paragraph, we see that the bilinear map (x, y) → ϕ(x)ψ(y) is completely contractive. Thus from (1.41) and the deﬁnition of the Banach space injective tensor norm of z (see A.3.1), we deduce that the latter norm of an element z ∈ X ⊗ Y is dominated by zh. Hence indeed · h is a norm. By the fact at the end of the last paragraph, together with Ruan’s theorem, we see that the completion X ⊗h Y of X ⊗ Y with respect to · h is an operator space. This operator space is called the Haagerup tensor product. Note that the canonical bilinear map ⊗ : X × Y → X ⊗h Y is completely contractive in the sense above.

Operator spaces

31

Using (1.41) we see that if u : X × Y → W is a bilinear map with associated linear map u ˜ : X ⊗Y → W , then u is completely bounded if and only if u ˜ extends to a completely bounded linear map on X ⊗h Y . Moreover we have ˜ : X ⊗h Y −→ W cb ucb = u in that case. The above property means that the Haagerup tensor product linearizes completely bounded bilinear maps. A moments thought shows that this is a universal property. That is, suppose that (W, µ) is a pair consisting of an operator space W , and a completely contractive bilinear map µ : X × Y → W , such that the span of the range of µ is dense in W , and which possesses the following property: Given any operator space Z and given any completely bounded bilinear map u : X × Y → Z, then there exists a linear completely bounded u˜ : W → Z such that u ˜(µ(x, y)) = u(x, y) for all x ∈ X, y ∈ Y , and such that ˜ ucb ≤ ucb. W via a complete isometry v satisfying v ◦ ⊗ = µ. Then X ⊗h Y ∼ = 1.5.5 (Further properties of the Haagerup tensor product) By the deﬁnition of the Haagerup tensor product, or by its universal property, it is easy to see that this tensor product is functorial. That is, if ui : Xi → Yi are completely bounded maps between operator spaces, then u1 ⊗ u2 : X1 ⊗h X2 → Y1 ⊗h Y2 is completely bounded, and u1 ⊗ u2 cb ≤ u1 cb u2 cb . It is also easy to show that the Haagerup tensor product is associative (e.g. see [318]). That is, (X1 ⊗h X2 )⊗h X3 ∼ = X1 ⊗h (X2 ⊗h X3 ) completely isometrically. Accordingly, this space will be merely denoted by X1 ⊗h X2 ⊗h X3 . The induced norms on Mn (X1 ⊗ X2 ⊗ X3 ) may be described by the ‘3-variable’ version of (1.40). From this one may see that X1 ⊗h X2 ⊗h X3 has the universal property of ‘linearizing’ completely bounded trilinear maps (see discussion at the end of 1.5.4). Similar assertions clearly hold for the N -fold Haagerup tensor product X1 ⊗h · · · ⊗h XN of any N -tuple of operator spaces. Also, the Haagerup tensor product is injective. That is, if u1 and u2 in the last paragraph are completely isometric, then so is u1 ⊗u2 . See the Notes for directions to a selection of proofs of this fact. It is much easier to see that the Haagerup tensor product is projective, that is, if u1 and u2 in the last paragraph are complete quotient maps, then so is u1 ⊗ u2 . To see this, note that by the functoriality, the map u1 ⊗ u2 is a complete contraction. Let z ∈ Mn (Y1 ⊗ Y2 ), with zh < 1. By deﬁnition, we may write z = y1 y2 , where y1 ∈ Mn,p (Y1 ), y2 ∈ Mp,n (Y2 ) both have norm < 1. Then y1 = (u1 )n,p (x1 ) and y2 = (u2 )p,n (x2 ) for x1 ∈ Mn,p (X1 ), x2 ∈ Mp,n (X2 ), both of norm < 1. Let w = x1 x2 ∈ Mn (X1 ⊗h X2 ), this matrix has norm < 1, and (u1 ⊗ u2 )n (w) = z. By an obvious density argument, this shows that u1 ⊗ u2 above is a complete quotient map. The Haagerup tensor product is not commutative. That is, in general X ⊗ h Y and Y ⊗h X are not isometric. We shall see some examples of this later. One ﬁnal property of the Haagerup tensor product: there are convenient norm expressions for ·h . Suppose that A and B are C ∗ -algebras, or spaces of the form

32

Operator space tensor products

B(K, H). If X and Y are subspaces of A and B respectively, and if z ∈ X ⊗ Y , then by the deﬁnition in 1.5.4 and (1.4) we may write p p 12 12 zh = inf ak a∗k b∗k bk k=1

(1.42)

k=1

p where the inﬁmum is taken over all ways to write z = k=1 ak ⊗ bk in X ⊗ Y (or equivalently, by the injectivity of this tensor product, in A ⊗ B). Proposition 1.5.6 may write z as a norm sum (1) If we ∞ convergent z∞∈ X ⊗h Y with zh < 1 then ∞ ∗ ∗ a ⊗ b in X ⊗ Y , with a a < 1 and b b < 1, and k k h k k k k=1 k=1 k=1 k where the last two sums converge in norm. That is, [a1 a2 · · · ] ∈ R(X) and [b1 b2 · · · ]t ∈ C(Y ). (2) If x = [a1 a2 · · · ] ∈ R(X) and y = [b1 b2 · · · ]t ∈ C w (Y ), that is if ∞ ∞ ∗ ∗ a a converges in norm and if the partial sums of k=1 k k k=1 bk bk are uni∞ formly bounded in norm, then k=1 ak ⊗ bk converges in norm in X ⊗h Y . Similarly if x ∈ Rw (X) and y ∈ C(Y ). X ⊗ Y with z − w Proof (1) If z is as stated, choose w1 ∈ 1 h < 2 and n1 ∗ w1 h < 1. By (1.42) we may write w1 = k=1 xk ⊗ yk with k xk xk ≤ 1 and k yk∗ yk ≤ 1. Repeating this argument, we may choose w ∈ X ⊗ Y with 2 n2 − w < , and w < . By (1.42) we write w = z − w 1 2 h 2 k=n1 +1 xk ⊗ yk 22 2∗ h 2 with k xk x∗k ≤ 2 and k yk yk ≤ 2 . Continuing so, we obtain for every m ∈ N nm a ﬁnite rank tensor wm = k=n xk ⊗ yk with z − w1 − . . . − wm < 2m , m−1 +1 nm n m ∗ yk∗ yk ≤ 2m−1 . Now it is clear that k=nm−1 +1 xk xk ≤ k=n 2m−1 , and m−1 +1 ∞ ∞ ∗ ∗ the partial sums of k=1 xk xk and k=1 yk yk converge in norm to elements ∞ with norm close to 1, and that z = k=1 xk ⊗ yk as desired. a ⊗ b are Cauchy, note that from (2) To see that the partial sums of ∞ mk=1 k ∗ 1 k m ∗ m 1 2 2 (1.42) we have k=n ak ⊗ b k h ≤ k=n ak ak k=n bk bk . Now use the ∞ 2 fact that the partial sums of k=1 ak a∗k are Cauchy.

The following important result is due to Christensen and Sinclair, and Paulsen and Smith. We will refer to it as the ‘CSPS theorem’. See the Notes section for references, and directions to a sample of the available proofs. Note that the second part of this result follows from the ﬁrst part and 1.2.8. Theorem 1.5.7 (CSPS theorem) Suppose that X and Y are operator spaces, and that u : X × Y → B(K, H) is a bilinear map. (1) u is completely contractive (as a bilinear map) if and only if there is a Hilbert space L, and there are completely contractive linear maps v : X → B(L, H) and w : Y → B(K, L), with u(x, y) = v(x)w(y) for all x ∈ X and y ∈ Y . (2) If X and Y are subspaces of unital C ∗ -algebras A and B respectively, and if the conditions in (1) hold, then there exist Hilbert spaces K 1 and K2 , unital

Operator spaces

33

∗-representations π : A → B(K1 ) and ρ : B → B(K2 ), and contractions T ∈ B(K, K2 ), S ∈ B(K2 , K1 ) and R ∈ B(K1 , H), such that u(x, y) = Rπ(x)Sρ(y)T,

x ∈ X, y ∈ Y.

1.5.8 (Remarks on the CSPS theorem) An analoguous result to (1) holds for multilinear completely bounded maps of three or more variables, and can be proved using 1.5.7 and the discussion on associativity in 1.5.5. Thus if X1 , . . . , XN are operator spaces and if vj : Xj → B(Hj , Hj−1 ) are completely contractive linear maps then the N -linear mapping taking (x1 , . . . , xN ) ∈ X1 × · · · × XN to v1 (x1 ) · · · vN (xN ) ∈ B(HN , H0 ) is completely contractive. Conversely, any completely contractive N -linear map has this form. Likewise, (2) has analoguous formulations for multilinear maps. For example, let u : X × Y × Z → B(K, H) be any completely contractive trilinear map, and assume that X and Z are subspaces of unital C ∗ -algebras A and B respectively. Then u has a factorization of the form u(x, y, z) = Rπ(x)Ψ(y)ρ(z)T , where π : A → B(K1 ) and ρ : B → B(K2 ) are unital ∗-representations, Ψ is a complete contraction from Y to B(K2 , K1 ), and R, T are contractions. If u : X × Y → B(K, H) is a completely contractive bilinear map, then the Hilbert space L in (1) may be replaced by the possibly smaller space which is densely spanned by v(X)∗ (H) and w(Y )(K). In particular L may be chosen to be ﬁnite-dimensional if X, Y, K, H are all ﬁnite-dimensional. As a ﬁnal remark, we stress the case of completely bounded bilinear forms. If we have a completely contractive u : X × Y → C, the CSPS theorem ensures that u may be written as u(x, y) = w(y), v(x)H , where H is a Hilbert space and v : X → H r and w : Y → H c are completely contractive maps. 1.5.9 (Self-duality of ⊗h ) Let X and Y be operator spaces. Then X ∗ ⊗h Y ∗ → (X ⊗h Y )∗

completely isometrically

(1.43)

via the canonical map. Indeed, ﬁrst assume that X and Y are ﬁnite-dimensional. We let J be the canonical map from X ∗ ⊗h Y ∗ into (X ⊗h Y )∗ , which is a surjection in our ﬁnite-dimensional case. By (1.8), it suﬃces to show that Jn is an isometric isomorphism from Mn (X ∗ ⊗h Y ∗ ) to CB(X ⊗h Y, Mn ), for any n ≥ 1. We consider z ∈ Mn (X ∗ ⊗h Y ∗ ) and we let u = Jn (z). By 1.5.7 (1) and the third paragraph in 1.5.8, there exist an integer p ≥ 1 and two linear maps v : X → Mn,p and w : Y → Mp,n such that vcbwcb ≤ ucb and u(x ⊗ y) = v(x)w(y) for any x ∈ X and y ∈ Y . We let ϕ = [ϕij ] ∈ Mn,p (X ∗ ) and ψ = [ψij ] ∈ Mp,n (Y ∗ ) be the two matrices corresponding to v and w respectively. Then ϕik (x)ψkj (y) , x ∈ X, y ∈ Y. u(x ⊗ y) = v(x)w(y) = [ϕij (x)][ψij (y)] = k

Thus z = ϕ ψ, and hence zh ≤ ϕψ = vcb wcb ≤ ucb. The converse inequality ucb ≤ zh may be obtained by reversing the arguments.

34

Operator space tensor products

In the general case, ﬁx [uij ] ∈ Mn (X ∗ ⊗ Y ∗ ). Write each uij ∈ X ∗ ⊗ Y ∗ in m the form k=1 ϕk ⊗ ψk , for functionals ϕk ∈ X ∗ , ψk ∈ Y ∗ . Let W (resp. Z) be the span of all these (ﬁnite number of) functionals in X ∗ (resp. Y ∗ ), over all i and j too. We may view [uij ] in Mn (W ⊗h Z), and by the injectivity of ⊗h (see 1.5.5), its norm in the latter space equals its norm in Mn (X ∗ ⊗h Y ∗ ). By the last paragraph, and by 1.4.4, its norm also equals the norm of the associated element in Mn (((X/W⊥ ) ⊗h (Y /Z⊥ ))∗ ). However, by the projectivity of ⊗h (see 1.5.5), and 1.4.3, we may view ((X/W⊥ ) ⊗h (Y /Z⊥ ))∗ ⊂ (X ⊗h Y )∗ completely isometrically. The composition of these identiﬁcations is easily seen to take [u ij ] to the desired matrix in Mn ((X ⊗h Y )∗ ), and so we are done. 1.5.10 (Some relations to Banach space tensor products) The Haagerup tensor product of minimal and maximal operator spaces corresponds to well-known Banach space tensor products (see A.3.5 for a review). Let E, F be Banach ˇ spaces. Then for any m ≥ 1, we have isometric identities Rm (Min(E)) = 2m ⊗E 2 ˇ and Cm (Min(F )) = m ⊗F , as may be seen from (1.11). Then it is immediate from (1.40) and (A.5) that Min(E) ⊗h Min(F ) = E ⊗γ2 F

isometrically.

(1.44)

If G is an arbitrary Banach space and H is a Hilbert space then CB(Min(G), H r ) = Π2 (G, H)

isometrically,

(1.45)

where the latter denotes the Banach space of 2-summing maps from G into H (see A.3.4). To prove this consider u : G → H and assume that u is 2-summing. ˇ → 2n (H) has norm less than or According to A.3.4, the mapping I2n ⊗ u : 2n ⊗G equal to π2 (u) for any n ≥ 1. Hence I2 ⊗ I2 ⊗ u : 2n ⊗ ˇ 2n ⊗G ˇ −→ 2n ⊗ ˇ 2n (H) ≤ π2 (u). n

n

Using (1.2) and (A.1), we have isometric identities ¯ C)) ∼ ¯ 2n ) ∼ ˇ 2n (H). Mn (H r ) ∼ = Mn (B(H, = B(2n (H), = 2n ⊗ ∼ Mn by (A.1) again, so 2 ⊗ ˇ 2n = ˇ 2n ⊗G ˇ = Mn (Min(G)) On the other hand, 2n ⊗ n by (1.11). Thus the second last centered equation exactly means that the map un from Mn (Min(G)) to Mn (H r ) has norm ≤ π2 (u). Thus ucb ≤ π2 (u). The converse inequality is easily obtained by reversing the arguments. Suppose that x in Rm (Max(E)), viewed as an element of Rm (Max(E)∗∗ ) ∼ = CB(Max(E)∗ , Rm ) = CB(Min(E ∗ ), Rm ), the last isomorphisms using 1.2.29 and (1.30). Applying (1.45) with G = E ∗ we see that if w : E ∗ → 2m is the linear mapping corresponding to x, then x is equal to π2 (w). Therefore the argument before (1.44), and (A.6), show that Max(E) ⊗h Min(F ) = E ⊗g2 F

isometrically.

(1.46)

isometrically.

(1.47)

Likewise by (A.7) we have Max(E) ⊗h Max(F ) = E ⊗γ2∗ F

Operator spaces

35

1.5.11 (Operator space projective tensor product) As with the Haagerup tensor product, it is convenient to ﬁrst deﬁne an intimately related class of bilinear maps. Suppose that X, Y , and W are operator spaces and that u : X × Y → W is a bilinear map. We say that u is jointly completely bounded if there exists a constant K ≥ 0 such that [u(xij , ykl )](i,k),(j,l) ≤ K[xij ][ykl ] for all m, n and [xij ] ∈ Mn (X), and [ykl ] ∈ Mm (Y ). The least such K is written as ujcb. We say that u is jointly completely contractive if ujcb ≤ 1. Jointly completely bounded multilinear maps of three or more variables are deﬁned similarly. We also observe that any completely contractive (in the sense of 1.5.4) bilinear map u is jointly completely contractive. This is immediate from the simple relation [u(xij , ykl )] = unm ([xij ] ⊗ Im , In ⊗ [ykl ]). Let X, Y be operator spaces, and let n ∈ N. For z ∈ Mn (X ⊗ Y ) deﬁne z = inf{αxyβ},

(1.48)

the inﬁmum taken over p, q ∈ N, and all ways to write z = α(x ⊗ y)β, where α ∈ Mn,pq , x ∈ Mp (X), y ∈ Mq (Y ), and β ∈ Mpq,n . Here we wrote x ⊗ y for the ‘tensor product of matrices’, namely x ⊗ y = [xij ⊗ ykl ](i,k),(j,l) . We omit the easy proof (see [149] if necessary) that this deﬁnes an operator seminorm structure on X ⊗ Y . For z ∈ Mn (X ⊗ Y ), and for any jointly completely contractive bilinear map u : X × Y → W , it is easy to see from the deﬁnitions that ˜ un (z) ≤ z ,

(1.49)

where u ˜ : X ⊗ Y → W is the associated linear map. From the observation at the end of the last paragraph, it follows that this is also true if u is completely contractive. Taking u = ⊗ : X × Y → X ⊗h Y , we deduce that · h ≤ · . Thus the quantities in (1.48) are norms. By Ruan’s theorem the completion of X ⊗ Y with respect to these matrix norms is an operator space, which we call the operator space projective tensor product, and write as X ⊗ Y . By (1.48) and (1.49) we see that if u : X × Y → W is a bilinear map with associated linear map u˜ : X ⊗ Y → W , then u is jointly completely bounded if and only if u˜ extends to a completely bounded linear map on X ⊗ Y . Moreover, ˜ : X ⊗ Y −→ W cb ujcb = u in that case. The above property means that the operator space projective tensor product linearizes jointly completely bounded bilinear maps. As for the Haagerup tensor product this is a universal property, and characterizes X ⊗ Y uniquely up to complete isometry. From this it is easy to argue that

CB(X ⊗ Y, Z) ∼ = CB(X, CB(Y, Z)) ∼ = CB(Y, CB(X, Z))

(1.50)

36

Operator space tensor products

isometrically, and indeed (by replacing Z by Mn (Z)) completely isometrically. In particular,

(X ⊗ Y )∗ ∼ = CB(X, Y ∗ ) ∼ = CB(Y, X ∗ )

completely isometrically.

(1.51)

We now list a sequence of properties of the operator space projective tensor product. Again it is very easy to see, copying the idea of the analoguous proofs in 1.5.5, that ⊗ is functorial, and projective. It is also an easy exercise to see that ⊗ is associative, and (1.51) shows that ⊗ is commutative. We use these words in the sense that we have used them for the other two tensor products. From the universal properties of ⊗ and ⊕1i (see above and 1.4.13), it is easy to check that

Y ⊗ (⊕1i Xi ) ∼ = ⊕1i (Y ⊗ Xi ),

(1.52)

for operator spaces Y and {Xi }. Proposition 1.5.12 Let E, F be Banach spaces and let Y be an operator space.

ˆ as Banach spaces. (1) Max(E) ⊗ Y = E ⊗Y ˆ ) = Max(E) ⊗ Max(F ) as operator spaces. (2) Max(E ⊗F Proof The ﬁrst item follows by computing the duals of these two tensor products, using (1.51), (1.12), and (A.4). Then (2) follows from (1) if we can show that Max(E) ⊗ Max(F ) is a maximal operator space. To do this, observe that

ˆ W ) = B(E, B(F, W )), B(Max(E) ⊗ Max(F ), W ) = B(E ⊗F, for any operator space W , by (1) and (A.3). The latter space equals

CB(Max(E), CB(Max(F ), W )) = CB(Max(E) ⊗ Max(F ), W )

by (1.12) and (1.50). Thus Max(E) ⊗ Max(F ) is ‘maximal’.

2

Proposition 1.5.13 (Comparison of tensor norms) If X and Y are operator spaces then the various tensor norms on X ⊗ Y are ordered as follows: · ∨ ≤ · min ≤ · h ≤ · ≤ · ∧ .

Indeed the ‘identity’ is a complete contraction X ⊗ Y → X ⊗h Y → X ⊗min Y . ˇ (see Proof The ﬁrst inequality follows from (1.34) and the deﬁnition of ⊗ ˆ A.3.1). The fact that · ≤ · ∧ follows from the universal property of ⊗ (see A.3.3), since the bilinear map ⊗ : X × Y → X ⊗ Y is jointly completely contractive, and hence contractive. We saw in 1.5.11 the complete contraction X ⊗ Y → X ⊗h Y . For the remaining relation we may assume by the injectivity of these tensor norms that X and Y are unital C ∗ -algebras, and then X ⊗min Y

Operator spaces

37

is a C ∗ -algebra as observed in 1.5.2. We may write x ⊗ y = (x ⊗ 1)(1 ⊗ y), which may be viewed as the product π(x)ρ(y) of two (completely contractive) ∗-homomorphisms. Thus the bilinear map ⊗ : X × Y → X ⊗min Y is completely contractive, which by (1.41) proves the desired relation. 2 Proposition 1.5.14 If X, Y are operator spaces, if H, K are Hilbert spaces, and if m, n ∈ N, then we have the following complete isometries:

(1) H r ⊗h X = H r ⊗ X, and X ⊗h H c = X ⊗ H c . (2) H c ⊗h X = H c ⊗min X, and X ⊗h H r = X ⊗min H r . (3) Cn (X) ∼ = Cn ⊗h X = Cn ⊗min X, and Rn (X) ∼ = X ⊗h Rn = X ⊗min Rn . ¯ r ⊗h X ⊗h K c )∗ ∼ ¯ r ⊗ X ⊗ K c )∗ = (H (4) (H = CB(X, B(K, H)). ∞ c ∼ ¯ r and S ∞ (K, H) ⊗min X ∼ ¯ r. (5) S (K, H) = H ⊗min K = H c ⊗h X ⊗h K ∼ Cm ⊗h X ⊗h Rn . (6) Mm,n (X) = (7) Mm,n (X ⊗h Y ) ∼ = Cm (X) ⊗h Rn (Y ).

(8) H c ⊗ K c = H c ⊗h K c = H c ⊗min K c = (H ⊗2 K)c , and similarly for row Hilbert spaces. ¯ r ⊗ H c. (9) S 1 (K, H) ∼ =K (10) CB(S 1 (2I , 2J ), X) ∼ = MI,J (X), if I, J are cardinals.

Proof To prove that X ⊗h H c = X ⊗ H c completely isometrically, it suﬃces by Proposition 1.5.13 to show that I : X ⊗h H c → X ⊗ H c is completely contractive. This will follow if we can show that any jointly completely contractive map u : X × H c → B(L) is completely contractive (in the sense of 1.5.4). Assume that H = 2J for some cardinal J. Let v : X → RJw (B(L)) = B(L(J) , L) be the linear map deﬁned by v(x) = (u(x, ei ))i∈J for any x ∈ X. Then v corresponds to u via the identiﬁcations

CB(X ⊗ CJ , B(L)) = CB(X, CB(CJ , B(L))) = CB(X, RJw (B(L))) provided by (1.50) and Proposition 1.2.28. Thus vcb = ujcb. Then we deﬁne a map w : CJ → CJ (B(L)) ⊂ B(L, L(J) ) by w(ζ) = (ζj IL ) for ζ = (ζj ) ∈ 2J . It is clear that wcb = 1 and moreover we have a factorization u(x, ζ) = v(x)w(ζ) for any x ∈ X, ζ ∈ H = 2J . By the easy part of the CSPS theorem 1.5.7, we deduce the desired inequality ucb ≤ ujcb. The ﬁrst part of (1) is similar. To prove (2), we may assume by injectivity of ⊗min and ⊗h that H and X are ﬁnite-dimensional. In that case the result follows from (1) by duality, using (1.15), 1.5.9, (1.51), and (1.32). Item (3) follows from (2) and (1.37). The ﬁrst equality in (4) is clear from (1), and the rest is clear from the string

¯ r ⊗ X ⊗ K c )∗ ∼ (H = CB(X, CB(K c , H c )), = CB(X ⊗ K c , H c ) ∼ which equals CB(X, B(K, H)). Here we have used (1.51), (1.50), (1.15), and (1.14). The ﬁrst equality in (5) is clear from (1.32), (1.15), and (1.14). Then

38

Duality and tensor products

¯ r by commutativity of ⊗min , and so S ∞ (K, H) ⊗min X = H c ⊗min X ⊗min K the second part of (5) follows from (2). Item (6) is a special case of (5), and (7) follows from (6) by (3) and the associativity of the Haagerup tensor product. The middle equality in (8) follows from (2), and the ﬁrst equality from (1). Writing ¯ r = RI for cardinals I, J, the last equality in (8) may be seen H c = CJ and K from (1.37) and (1.18). Clearly (9) follows from (4) with X = C. Lastly, for (10), CB(S 1 (2I , 2J ), X) is completely isometric to

CB(RI ⊗ CJ , X) ∼ = CB(CJ , CB(RI , X)) ∼ = RJw (CIw (X)) ∼ = MI,J (X), 2

using (9), (1.50), 1.2.28, and (1.20). 1.6 DUALITY AND TENSOR PRODUCTS This material is usually only used in the ﬁnal section of later chapters.

1.6.1 (Mapping spaces as duals) If Y is a dual operator space then so is CB(X, Y ), for any operator space X. Indeed by (1.51) an explicit predual for CB(X, Y ) is X ⊗ Y∗ . From this, together with the density of the ﬁnite rank tensors in X ⊗ Y∗ , it follows that a bounded net (ut )t in CB(X, Y ) converges in the w∗ -topology to a u ∈ CB(X, Y ) if and only if ut (x) converges in the w∗ -topology to u(x) in Y for all x ∈ X. In particular, setting Y = B(K, H) for Hilbert spaces H, K, and using the same principle again, we see that the above equivalent conditions are also equivalent to ut (x)ζ, η → u(x)ζ, η

for all x ∈ X, ζ ∈ K, η ∈ H.

(1.53)

1.6.2 (Dual matrix spaces) If X is a dual operator space then so is Mn (X). Indeed by (1.51) and (1.8) we have

(Sn1 ⊗ X∗ )∗ ∼ = Mn (X). = CB(X∗ , Mn ) ∼ More generally the same proof, but substituting 1.2.29 for (1.8), shows that for cardinals I, J, MI,J (X) is a dual operator space with operator space predual S 1 (2I , 2J ) ⊗ X∗ , and also MI,J (X) ∼ = CB(X∗ , MI,J ). If I, J are sets, and if I0 and J0 are ﬁnite subsets of I and J respectively, write ∆ = I0 × J0 . The set Λ of such ∆ is a directed set under the usual ordering. For such ∆, and for x ∈ MI,J (X), we write x∆ for the matrix x, but with entries xij switched to zero if (i, j) ∈ / ∆. Then (x∆ )∆ is a net indexed by ∆ ∈ Λ, which we call the net of ﬁnite submatrices of x. Corollary 1.6.3 (Eﬀros and Ruan) Let X be a dual operator space, and let I, J be cardinals. (1) If (xt )t is a bounded net in MI,J (X), then xt → x ∈ MI,J (X) in the w∗ topology in MI,J (X), if and only if each entry in xt converges in the w∗ topology in X to the corresponding entry in x.

Operator spaces

39

(2) If Y is a dual operator space, and if u : X → Y is a w ∗ -continuous completely bounded map, then the ampliﬁcation uI,J : MI,J (X) → MI,J (Y ) is w∗ -continuous. If, further, u is completely isometric, then uI,J is a w∗ homeomorphic complete isometry onto a w ∗ -closed subspace. ∗ (3) Mﬁn I,J (X) is w -dense in MI,J (X). Indeed if I, J are sets, and x ∈ MI,J (X), then the net of ﬁnite submatrices of x converges to x in the w ∗ -topology. Proof Using the fact that MI,J (X) = CB(X∗ , MI,J ) = CB(X∗ , B(2J , 2I )) (see 1.6.2), (1) follows from (1.53), and the density of Span{ei } in 2I and 2J . Items (2) and (3) follow immediately from (1) and A.2.5. 2 Note from 1.6.3 that for a w ∗ -closed subspace X ⊂ B(K, H), one may deﬁne (J) MI,J (X) to be the w∗ -closure of Mﬁn , H (I) ). I,J (X) in MI,J (B(K, H)) = B(K 1.6.4 (A characterization of dual operator spaces) An operator space X which is a dual Banach space, is a dual operator space if and only if Mn (X) is a dual Banach space, and the n2 canonical inclusion maps from X into Mn (X) are w∗ continuous, for all n ≥ 2. This follows immediately from the following criterion (which is the one which will be used in the sequel). If X is the dual Banach space of W , and if W is equipped with its natural matrix norms as a subspace of X ∗ via the natural inclusion, then X is the dual operator space of W if and only if the unit ball of Mn (X) is σ(X, W )-closed for every positive integer n. That is, if and only if whenever (xt )t is a net in Ball(Mn (X)), x ∈ Mn (X), and the i-j entry of xt converges in the σ(X, W ) topology to the i-j entry of x for all i, j = 1, . . . , n, then x ∈ Ball(Mn (X)). The ‘only if’ in the last criterion follows from 1.6.2 and 1.6.3 (1). For the other direction, assume that the unit ball of Mn (X) is σ(X, W )-closed for every n ≥ 1. By 1.5.14 (6) and the selfduality of the Haagerup tensor product (see 1.5.9), we have Mn (X)∗ = Rn ⊗h X ∗ ⊗h Cn . Since ⊗h is injective we deduce that Rn ⊗h W ⊗h Cn ⊂ Mn (X)∗ isometrically. Hence it is easy to argue that the image of Ball(Mn (X)) inside (Rn ⊗h W ⊗h Cn )∗ has polar set in Rn ⊗h W ⊗h Cn equal to the unit ball of Rn ⊗h W ⊗h Cn . Also, this image is w∗ -closed by hypothesis. Therefore by the bipolar theorem, Ball(Mn (X)) is equal to the unit ball of (Rn ⊗h W ⊗h Cn )∗ . By 1.5.14 (4) and (1.8), the latter space is isometrically equal to CB(W, Mn ) = Mn (W ∗ ). Hence Ball(Mn (X)) = Ball(Mn (W ∗ )). Since this holds for any n ≥ 1, this shows that X ∗ = W completely isometrically. 1.6.5 (Normal spatial tensor product) If X and Y are dual operator spaces, with operator space preduals X∗ and Y∗ , then CB(Y∗ , X) is the dual operator space of X∗ ⊗ Y∗ by 1.6.1. As in (1.35), we regard X ⊗min Y → CB(Y∗ , X), and we deﬁne the normal minimal tensor product X ⊗ Y to be the w ∗ -closure of X ⊗Y (or of X ⊗min Y ) in CB(Y∗ , X). Equivalently, if X and Y are w∗ -closed subspaces of B(H) and B(K) respectively, then we may deﬁne X ⊗ Y to be the w ∗ -closure in B(H ⊗2 K) of the copy of X ⊗ Y . This is sometimes referred to as the normal spatial tensor product. If M and N are W ∗ -algebras, then M ⊗ N as described above is the usual von Neumann algebra tensor product (e.g. see [407, IV; 5])

40

Duality and tensor products

and in particular, B(H) ⊗ B(K) = B(H ⊗2 K). To see that these two deﬁnitions of X ⊗ Y are the same (up to w∗ -homeomorphic complete isometry), we use the following argument. Since X and Y are w ∗ -closed subspaces of B(H) and B(K) respectively, we know by 1.4.6 that X∗ and Y∗ are quotients of S 1 (H) and S 1 (K) respectively. By the ‘projectivity’ property of ⊗, we obtain a complete quotient map Q : S 1 (H) ⊗ S 1 (K) → X∗ ⊗ Y∗ . Using the identiﬁcation (1.51) we see that Q∗ may be viewed as a w∗ -continuous completely isometric embedding CB(Y∗ , X) → CB(S 1 (K), B(H)) ∼ = B(H ⊗2 K), the last relation from the ﬁrst paragraph of 1.5.2. Via the canonical identiﬁcation of X ⊗ Y with a subset of CB(Y∗ , X), we see that the w∗ -closure of X ⊗ Y in B(H ⊗2 K) may be identiﬁed with the w∗ -closure of X ⊗ Y in CB(Y∗ , X). In general, CB(Y∗ , X) (or equivalently, (X∗ ⊗ Y∗ )∗ ) is not equal to X ⊗ Y . However they are equal when X and Y are W ∗ -algebras, as was shown by Eﬀros and Ruan (see the Notes section for a few more details). This is also the case when X = B(H). Indeed, for any dual operator space Y and cardinals I, J, MI,J ⊗ Y ∼ = MI,J (Y )

(1.54)

as dual operator spaces. This follows by the remark after 1.6.3, and an argument similar to the one used for (1.37). Also, MI,J (Y ) ∼ = CB(Y∗ , MI,J ) by 1.2.29. Setting H = 2I gives B(H) ⊗ Y ∼ = CB(Y∗ , B(H)). We leave it as an exercise that the normal spatial tensor product is ‘associative’, and ‘functorial’ for w∗ -continuous completely bounded maps. 1.6.6 (Operator valued measurable functions) Let (Ω, µ) be a measure space (σ-ﬁnite for speciﬁcity), and let Y be a dual operator space. Since L∞ (Ω) is a commutative W ∗ -algebra, ucb = u for any L∞ (Ω)-valued bounded map (see 1.2.6). Using natural conditional expectations, there is a net vλ of ﬁnite rank contractive maps on L∞ (Ω) such that vλ converges to IL∞ point-w∗ . If u ∈ B(Y∗ , L∞ (Ω)) then (vλ u)λ is a net of ﬁnite rank operators Y∗ → L∞ (Ω) converging point-w∗ to u. Since this net is bounded, this implies that vλ u → u in the w∗ -topology of CB(Y∗ , L∞ (Ω)) by 1.6.1. Each vλ u is ﬁnite rank and so ‘belongs’ to Y ⊗ L∞ (Ω). This together with the ﬁrst few lines of 1.6.5 shows that B(Y∗ , L∞ (Ω)) = CB(Y∗ , L∞ (Ω)) ∼ = L∞ (Ω) ⊗ Y.

(1.55)

Assume that Y∗ is separable. Then the latter operator space has an interesting description in terms of Y -valued functions deﬁned on Ω. We say that a function f : Ω → Y is w∗ -measurable if f (· ), ϕ : Ω → C is measurable for any ϕ ∈ Y∗ . Using the separability assumption it is easy to see that for such an f , the scalarvalued function f (· )Y is measurable. We set f ∞ = f (· )Y L∞ (Ω) and we say that f is essentially bounded if f ∞ < ∞. Then f ∞ is a seminorm on the space of such functions and we let L∞ (Ω; Y ) be the normed space obtained after

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41

taking the quotient by N0 = {f : f ∞ = 0}. As in the scalar-valued case, we identify any w∗ -measurable essentially bounded function f with its class modulo N0 . Using again the separability of Y∗ , we see that for any f ∈ L∞ (Ω; Y ), f ∞ = sup f (· ), ϕL∞ : ϕ ∈ Ball(Y∗ ) . Thus we may deﬁne a bounded map Jf : Y∗ → L∞ (Ω), by Jf (ϕ) = f (· ), ϕ, and we have Jf = f ∞ . It turns out that the resulting isometric embedding ˆ ∗ )∗ is onto. This is proved e.g. in [380, L∞ (Ω; Y ) → B(Y∗ , L∞ (Ω)) ∼ = (L1 (Ω)⊗Y Theorem 1.22.13] when Y is a von Neumann algebra and the proof given there extends to the general case. (See also Proposition IV.7.16 and its proof in [407].) The resulting identity B(Y∗ , L∞ (Ω)) ∼ = L∞ (Ω; Y ) shows that the latter space is complete and by (1.55), we obtain an isometric identiﬁcation L∞ (Ω) ⊗ Y = L∞ (Ω; Y ). By the reasoning in 1.5.3 (3), this identity is a complete isometry if L ∞ (Ω; Y ) has matrix norms obtained by equating Mn (L∞ (Ω; Y )) and L∞ (Ω; Mn (Y )). Thus L∞ (Ω; Y ) is a dual operator space. If jY : L∞ (Ω) ⊗ Y → L∞ (Ω; Y ) is the w∗ continuous mapping providing this identiﬁcation, and if we consider ﬁnite families (fk )k in L∞ (Ω) and (yk )k in Y , then jY maps k fk ⊗ yk to k fk (·)yk . 1.6.7 (W ∗ -continuous extensions: the bilinear case) Let X, Y be operator spaces, let W be a dual operator space, and let u : X × Y → W be a completely bounded bilinear map. Then u admits a (necessarily unique) separately w ∗ continuous extension u˜ : X ∗∗ × Y ∗∗ → W , this extension is completely bounded, and ˜ ucb = ucb. To prove this, we may assume by Lemma 1.4.7 that W is a w∗ -closed subspace of some B(H). By the CSPS theorem (1.5.7), we ﬁnd a Hilbert space L and two completely bounded maps v : X → B(L, H) and w : Y → B(H, L) such that u(x, y) = v(x)w(y) for all x ∈ X, y ∈ Y . According to 1.4.8, v and w admit w ∗ -continuous extensions v˜ : X ∗∗ → B(L, H) and w ˜ : Y ∗∗ → B(H, L) with ˜ v cb = vcb and w ˜ cb = wcb . Deﬁne a ˜(η, ν) = v˜(η)w(ν) ˜ for any bilinear map u ˜ : X ∗∗ × Y ∗∗ → B(H) by letting u η ∈ X ∗∗ , ν ∈ Y ∗∗ . Then the easy part of the CSPS theorem ensures that u˜ is completely bounded with ˜ ucb ≤ ˜ v cb w ˜ cb = vcb wcb ≤ ucb. Since u is ˜ is valued in W as well. valued in W , and W is w∗ -closed, its extension u 1.6.8 (Normal Haagerup tensor product) If X and Y are two dual operator spaces, we let (X ⊗h Y )∗σ denote the subspace of (X ⊗h Y )∗ corresponding to the completely bounded bilinear forms X × Y → C which are separately w ∗ continuous. Then we deﬁne the normal Haagerup tensor product X ⊗ σh Y to be the operator space dual of (X ⊗h Y )∗σ . From 1.6.7 one may deduce the following two consequences. First, if X, Y are two (not necessarily dual) operator spaces then (X ⊗h Y )∗ ∼ = (X ∗∗ ⊗h Y ∗∗ )∗σ . To see this, consider the map u → u ˜ in 1.6.7, in the case that W = Mn . Note that

42

Duality and tensor products

a map u into Mn is separately w∗ -continuous if and only if each ‘entry’ uij is separately w∗ -continuous. It follows that (X ⊗h Y )∗∗ = X ∗∗ ⊗σh Y ∗∗

as dual operator spaces.

(1.56)

Second, if X, Y are two dual operator spaces, then the natural action of X ⊗ Y on (X ⊗h Y )∗σ yields a completely isometric embedding X ⊗h Y ⊂ X ⊗σh Y.

(1.57)

To see (1.57), let j : X ⊗ Y → X ⊗σh Y be the canonical mapping, and ﬁx z in X ⊗ Y . It is clear from the deﬁnitions that j(z) ≤ zh. The converse inequality relies on the selfduality of the Haagerup tensor product. We let > 0. By (1.43), we have X ⊗h Y ⊂ (X∗ ⊗h Y∗ )∗ isometrically. Hence there exists a map Φ ∈ X∗ ⊗ Y∗ such that Φh = 1 and zh ≤ |z, Φ| + . Writing Φ as ∈ X∗ and ψk ∈ Y∗ , then the bilinear form u : X ×Y → C k ϕk ⊗ψk for some ϕk ∗ deﬁned by u(x, y) = k ϕk (x)ψk (y) is clearly separately w -continuous and we have z, Φ = j(z), u. Hence |z, Φ| ≤ j(z)ucb. By (1.43) again, and the injectivity of ⊗h , we have X∗ ⊗h Y∗ ⊂ X ∗ ⊗h Y ∗ ⊂ (X ⊗h Y )∗ , so that ucb = Φh = 1. Since was arbitrary, this shows that zh ≤ j(z). Hence j is an isometry, and a similar argument shows that it is a complete isometry. Indeed, X ⊗ Y is w∗ -dense in X ⊗σh Y . Let q : (X ⊗h Y )∗∗ → X ⊗σh Y be the adjoint mapping of the embedding (X ⊗h Y )∗σ ⊂ (X ⊗h Y )∗ , and let i : X ⊗h Y → (X ⊗h Y )∗∗ be the canonical embedding into the second dual. Then the embedding in (1.57) is qi. That its range is w ∗ -dense follows from Goldstine’s lemma A.2.1, and the fact that q is a quotient map, and is w ∗ -continuous. 1.6.9 (Weak* Haagerup tensor product) This is another ‘dual version’ of the Haagerup tensor product. Since will not use this tensor product very much, we content ourselves with an abridged development of it. See [70, 150] for much more information, for example, there are several other ways to describe this tensor product. If X and Y are operator spaces then we deﬁne X ∗ ⊗w∗ h Y ∗ = (X ⊗h Y )∗ .

(1.58)

We now discuss why this is a ‘tensor product’. By the last paragraph of 1.5.8, and writing the space H there as 2I , we have that w ∈ (X ⊗h Y )∗ if and only if w may be written in the form w(x ⊗ y) = ϕ(x), ψ(y), for a cardinal I, ϕ ∈ CB(X, RI ) and ψ ∈ CB(Y, CI ). Now CB(Y, CI ) ∼ = CIw (Y ∗ ) and CB(X, RI ) = RIw (X ∗ ), using 1.2.29. Writing ϕ = [ϕi ] and ψ = [ψi ], we have shown the following asw ∗ sertion. Namely, w ∈ (X ⊗h Y )∗ if and only if there exist [ϕ i ] ∈ RI (X ) and w ∗ [ψi ] ∈ CI (Y ) such that w may be written as w(x ⊗ y) = i ϕi (x)ψi (y) for x ∈ X, y ∈ Y . The last sum converges absolutely in C, as may be seen by the Cauchy–Schwarz inequality. We may therefore think of elements w ∈ X ∗ ⊗w∗ h Y ∗

Operator spaces

43

as tensors, and write w = i ϕi ⊗ ψi , or w = ϕ ψ. We call this a weak representation for w, and equate two weak representations if they agree ‘as functionals on X ⊗h Y ’. By the above, ∗ [ψi ]C w (Y ∗ ) }, w = min{[ϕi ]Rw I (X ) I

where the minimum is over all weak representations for w. Note that X ⊗h Y ⊂ X ⊗w∗ h Y completely isometrically. This is just a restatement of (1.43). Thus if X is ﬁnite-dimensional then X ⊗w∗ h Y = X ⊗h Y . It follows readily from the ‘associativity’ of the Haagerup tensor product (see 1.5.4), and (1.58), that the weak* Haagerup tensor product is associative too. It is ‘functorial’ too in an appropriate sense. That is, w ∗ -continuous completely bounded maps will tensor. In fact more is true: Given dual operator spaces Xk and Yk , for k = 1, 2, and completely bounded maps uk : Xk → Yk , there ∗ ∗ is a canonical completely bounded map u1 ⊗ u2 : X1 ⊗w h X2 → Y1 ⊗w h Y2 with (u ⊗ u )( x ⊗ w ) = u (x ) ⊗ u (w ), for any weak representation 1 2 i i 1 i 2 i i i i xi ⊗ wi . Moreover, u1 ⊗ u2 cb ≤ u1 cb u2 cb . We omit the proof of this perhaps surprising assertion, which may be found in the original paper [70] (or see [150] for a diﬀerent proof). Many of the identities in 1.5.14 have versions appropriate to ⊗w∗ h . For example, for a dual operator space X and a cardinal I we have ¯ ∼ CI ⊗w∗ h X = CI ⊗X = CIw (X)

(1.59)

and a similar relation holds for RIw (X). Indeed, the ‘∼ =’ here follows from (1.54).

The predual of CIw (X) is RI ⊗ X∗ , by 1.2.28 and (1.51). However by 1.5.14 (1), the latter space equals RI ⊗h X∗ , which is the predual of CI ⊗w∗h X. Another useful relation is

RI ⊗w∗ h X = RI ⊗h X = RI ⊗ X ∼ = CI (X∗ )∗

(1.60)

(and there is a matching formula for RI (X∗ )∗ ). To see (1.60), note that if w is in RI ⊗w∗h X, then w has a weak representation i ri ⊗ xi , with [ri ] ∈ RJw (RI ) and [xi ] ∈ CJw (X), for a cardinal J. However RJw (RI ) = RJ (RI ), so that by the proof of 1.5.6 (2) we have that w ∈ RI ⊗h X. This gives the ﬁrst equality in (1.60). The second equality is 1.5.14 (1), whereas the third follows from (1.58), since CI (X∗ ) = CI ⊗h X∗ by 1.5.14 (2). By analogy with 1.5.14 (7), for cardinals I, J we have MI,J (X ⊗w∗ h Y ) ∼ = CIw (X) ⊗w∗ h RJw (Y ).

(1.61)

To see this, notice that the ﬁrst space here equals CI ⊗w∗ h (X ⊗w∗ h Y ) ⊗w∗h RJ ∼ = (CI ⊗w∗ h X) ⊗w∗ h (Y ⊗w∗ h RJ ), using (1.20), (1.59) and its matching ‘row version’, and the associativity of ⊗ w∗ h . By (1.59) again, this equals the right-hand side of (1.61).

44

Duality and tensor products As an application, we show that for any operator space X we have KI,J (X)∗∗ ∼ = MI,J (X ∗∗ )

as dual operator spaces.

(1.62)

By 1.5.14 (5), (1.37), (1.58), and (1.60) and its matching ‘column’ formulation, KI (X)∗ ∼ = (CI ⊗h X ⊗h RI )∗ ∼ = (CI ⊗h X)∗ ⊗h CI ∼ = RI ⊗h X ∗ ⊗h CI . By 1.5.14 (4) and 1.2.29 we have (RI ⊗h X ∗ ⊗h CI )∗ ∼ = MI (X ∗∗ ). = CB(X ∗ , MI ) ∼ ∗ This proves (1.62). Another proof may be given using C -algebraic principles. 1.6.10 (A principle for separately w ∗ -continuous maps) Suppose, for example, that we have a completely bounded trilinear map Φ : X × Y × Z → B(K, H), where X, Y, Z are dual operator spaces. Suppose that Φ is separately w ∗ -continuous. From the CSPS theorem 1.5.7 we have Φ(x, y, z) = u(x)v(y)w(z)

(1.63)

for completely contractive maps u : X → B(K1 , H), v : Y → B(K2 , K1 ), and w : Z → B(K, K2 ). Here of course H, K, K1 , K2 are Hilbert spaces. Then we claim that K1 , K2 , and the maps u, v, w may be slightly adjusted, to each be w∗ -continuous and still satisfy (1.63). To see this, we ﬁrst state an obvious principle, which we leave as an exercise. Namely, if u : X → B(K, H) is a completely contractive w ∗ -continuous map, if L is a subspace of H with associated projection PL , and if G is a subspace of K, then the map PL u(·)|G is w∗ -continuous and completely contractive too. Because of this principle, the following procedure applied to the maps one at a time, from right to left, will make them w ∗ -continuous, without destroying w∗ -continuity of any of the previously adjusted maps. We begin on the right, with w. We note that for ζ ∈ K, η ∈ H, Φ(x, y, z)ζ, η = w(z)ζ, v(y)∗ u(x)∗ η = PL w(z)ζ, v(y)∗ u(x)∗ η,

(1.64)

where L = [v(Y )∗ u(X)∗ H], and PL is the projection onto L. Replacing w by PL w, and v by its restriction to L, we may assume that K2 = L. If (zt )t is a bounded net in Z converging in the w∗ -topology to z, and if x ∈ X and y ∈ Y are ﬁxed, then Φ(x, y, zt ) → Φ(x, y, z) in the w∗ -topology. Thus from (1.64), w(zt )ζ, v(y)∗ u(x)∗ η → w(z)ζ, v(y)∗ u(x)∗ η. From this, and norm density considerations in L, it is clear that w(zt ) → w(z) in the WOT. Since this is a bounded net, it converges in the w ∗ -topology (see A.1.4). Thus by A.2.5, w is w∗ -continuous. Finally, we replace K2 by its subspace [w(Z)H], and replace v by its restriction to this subspace. This ends the procedure. We now begin the same procedure again, to make the next map v w∗ -continuous. Thus we replace K1 by its subspace [u(X)∗ H], and use an argument similar to the above. Once v is w ∗ -continuous, we replace K1 again by [v(Y )w(Z)H], and then continue, to make u w ∗ -continuous.

Operator spaces

45

1.7 NOTES AND HISTORICAL REMARKS One might say that operator space theory grew out of Stinespring’s theorem [398], via Arveson’s landmark papers [21, 22] and the fundamental work in the seventies (continuing into the early 1980s) of Choi and Eﬀros, Haagerup, Kirchberg, Wittstock, and others. The eighties saw much activity in the area of completely bounded maps; the vision of Eﬀros (e.g. see [133]) and Paulsen’s text [307] were particularly inﬂuential in this development. See also [94], and the much overlooked work of Hamana (referenced in our bibliography). However operator spaces did not become a ﬁeld in its own right until Ruan’s theorem 1.2.13. This result was proved by Ruan in his thesis [369] (see also [370]) under the direction of Eﬀros, and was inspired by [90,318]. Ruan used this theorem to deﬁne operator space quotients, operator seminorm structures, and so on. See also the inspiring papers [142, 143], for example. A couple of years after that, operator space duality, and the basic operator space tensor theory, were developed independently by Blecher and Paulsen, and Eﬀros and Ruan (e.g. see [42, 66, 145, 146]). (The Haagerup tensor product of operator spaces had been developed earlier (e.g. in the important paper [318]).) In the early 1990s Pisier lent his considerable strength and expertise to the subject, as is fabulously testiﬁed to in [332, 337]. After this the ﬁeld grew quite considerably, with many brilliant mathematicians making remarkable contributions. For a more complete account of basic operator space theory, and for more references to the literature to complement the list in these Notes and historical remarks sections, the reader should also consult the texts [149, 314, 337, 385]. See also [433], which is currently being rewritten in an expanded printed form. There are other good surveys of part of the subject, such as [338, 366]. 1.2: Parts of 1.2.4 are folklore, no doubt known from the beginning. A generalization is proved in the later result 8.3.2. Observation 1.2.6 is due to Loebl. A related useful fact is that for a bounded linear map into Mn , we have T cb = Tn (see [389]). The canonical example of a map which is not completely bounded is the ‘transpose map’ on K. Injectivity of operator systems was studied deeply by Choi and Eﬀros (e.g. see [90]), partly inspired by Arveson’s extension theorem (see 1.3.5 and [21]). Later, Theorems 1.2.8 and 1.2.10 were proved by Wittstock [432, 431], Haagerup [177], and Paulsen [305]. See those papers, and [149, 314, 337, 385], for additional historical details. Paulsen’s proofs of these two results is now the standard route. See also [325, 335] for a diﬀerent approach extending Theorem 1.2.8 to a Banach space setting. The ﬁrst isomorphism in (1.5) has been dubbed the ‘canonical shuﬄe’ by Paulsen. It is not true that a one-to-one and surjective completely bounded linear map T : X → Y between complete operator spaces has a completely bounded inverse. Indeed, there is no ‘open mapping theorem’ in this sense. Of course if the countably inﬁnite ampliﬁcation of T maps K(X) onto K(Y ), then there is no obstacle. That CB(X, Y ) is an operator space was observed ﬁrst in [143]. The operator space dual was called the standard dual by Blecher and Paulsen, and the asso-

46

Notes and historical remarks

ciated duality theory was developed independently by those authors and Eﬀros and Ruan. Minimal and maximal operator spaces were ﬁrst considered in [142] and [66] respectively. Formally, Hilbert row and column spaces may ﬁrst appear in [432]. They played a role in [135,66] and were further developed in [147]. Some of the results in the latter paper had also been noticed by Blecher, and [43] gives alternative routes to these and to other results from [147]. There are other interesting operator space structures on a Hilbert space, most notably the (selfdual) operator Hilbert space OH of Pisier [328, 337] which will be mentioned again in Chapter 5 (see 5.3.4). Nearly all of the deﬁnitions/results on inﬁnite matrix spaces here may be found in a series of papers by Hamana, and Eﬀros and Ruan (e.g. see [186,143,146]). Interpolation and ultraproducts of operator spaces were ﬁrst considered by Pisier [328, 331, 327]. See also [436] for related work. We sketch a simple proof of (1.14). We may write H c = CJ , K c = CI , for cardinals I, J. Setting X = CI in 1.2.28 yields CB(CJ , CI ) ∼ = RJw (CI ) ∼ = MI,J completely isometrically, where the last equality follows from any of the centered equations in 1.2.26. 1.3: Many of the results in this section are from Arveson’s original papers [21, 22]. These papers established, for example, the role of completely positive maps in the unitary equivalence theory of irreducible sets of compact operators, a generalization of the Shilov boundary to arbitrary operator algebras, and generalizations of the Sz. Nagy–Foias dilation theorem to representations of arbitrary function algebras. Arveson’s result 1.3.6 is the decisive relation between unital operator spaces and operator systems. Paulsen’s lemma 1.3.15 (originally from [305]) yields the decisive relation between general operator spaces and operator systems. Note that we did not use the fact that the map was unital in 1.3.9, indeed many of the results in this section are true without this assumption (e.g. see [36]). There are also direct proofs of 1.3.9 not using Stinespring’s theorem. The result 1.3.10 is an easy variant of the Banach–Stone–Kadison theorem [212]. Item 1.3.11 seems to appear ﬁrst in [301], but is related to Choi’s multiplicative domain [87]. Other sources for results in this section are [88,90,211,213,307,399]. 1.4: The facts in this section are due to Blecher (see [42], which was written close to the date of [66,145], although it appeared much later), with the following main exceptions. The fact that X ⊂ X ∗∗ completely isometrically was independently noticed in [66, 145]. Eﬀros and Ruan had noticed 1.4.7 via a diﬀerent route [146]. 1.5: The minimal tensor product may have been ﬁrst considered for operator spaces in [307]. An account of this tensor product from the perspective of C ∗ -algebra theory can be found in [407], for example. Relations such as the complete isometry X ⊗min Y → CB(Y ∗ , X) were ﬁrst noticed in [66]. The approach presented here to the minimal tensor product is taken from the latter paper, with some simpliﬁcations inﬂuenced by the exposition in Chapter 8 of [149]. The functoriality property of the minimal (or spatial) tensor product is one of the most important properties of completely bounded maps: they may be tensored. This may have been ﬁrst observed in [117], which dates to 1982 or earlier. This

Operator spaces

47

property is not shared by general bounded maps with respect to the natural tensor products found in C ∗ -algebra theory. Haagerup deﬁned the tensor product that bears his name in [176], at least for C ∗ -algebras. It was studied in the 1980s in papers of Eﬀros, some with coauthors (see [149] for detailed references). In [318] Paulsen and Smith showed that the Haagerup tensor product in the generality presented here, is an operator space (this is usually seen now using Ruan’s theorem, as was ﬁrst noted in [369]), and proved many of the other facts presented here. The class of bilinear or multilinear maps described here as ‘completely bounded’ are given a diﬀerent name in [149]; instead Eﬀros and Ruan use this term for the class which we call ‘jointly completely bounded’. The injectivity of the Haagerup tensor product is also from [318]. Other proofs were found later by Eﬀros, Paulsen, and others (e.g. see [385, 149, 314, 337] or 2.10 in [43], for a selection). The projectivity of ⊗h and ⊗ was observed in [147]. Part (2) of 1.5.6 is also true for uncountable sums with the same proof. The original references for Theorem 1.5.7 are [93] (for C ∗ -algebras) and [318] (for general operator spaces). Other proofs may be found in [149, 307, 337]. The selfduality relation (1.43) was conjectured by Blecher and proved in full in [147]. Diﬀerent proofs appear in [43,70]. The latter paper sketches a route based on Smith’s important notion of strong independence [391], through many of the basic results in this section. Observation (1.44) was perhaps ﬁrst in [40]. The relation (1.45) is from [147], see also [325] and [310] for related results and developments. For further information on g2 , we refer the reader to [118] (e.g. see 12.7 there). Relations (1.44), (1.46), and (1.47) are just three examples of relationships between the Haagerup tensor product and Pisier’s ‘gamma-norms’ introduced in [326]. See [328] and [243] for related developments. The operator space projective tensor product and its basic properties were developed in [66,145]. Formula (1.48) is from the latter paper. There is a similar formula valid for all z ∈ Mn (X ⊗ Y ), writing z = α(x ⊗ y)β, where the latter matrices are now countably inﬁnite (see [149]). In [66] it is shown without using formula (1.48), that there is an operator space X ⊗ Y having all the other properties mentioned in 1.5.11 and the paragraphs after it. Various forms of the relations in 1.5.14 may be found in [135,66,147,139,43], but [147] is the primary reference. Note that 1.2.10 follows from 1.5.14 (4), and the injectivity of ⊗ h . 1.6: Nearly all the facts about inﬁnite matrices in 1.6.2 and 1.6.3 are explicitly in [186, 143, 146]. Neufang has generalized 1.6.3 in [292]. The result 1.6.4 is due to Le Merdy [244], who also showed in that paper that if X is an operator space which is a dual Banach space with predual Y , then there may not exist an operator space structure on Y such that Y ∗ ≈ X completely isomorphically (see also 2.7.15). The equivalence between the two descriptions of ⊗ in 1.6.5 is related to the fact that (X∗ ⊗ Y∗ )∗ = CB(Y∗ , X) equals the normal Fubini product of X and Y , which was proved independently by Blecher and Ruan (see [43, Theorem 2.5], [372]), inspired by the von Neumann algebra case [146]. The Fubini product for operator spaces was ﬁrst studied in [186] though. Since the normal Fubini

48

Notes and historical remarks

product is deﬁned in terms of slice maps, such maps are key to understanding when X ⊗ Y = CB(Y∗ , X) holds for dual operator spaces X and Y . For example, the fact that this holds for two von Neumann algebras, which was proved in [146], follows from the slice map theorem of Tomiyama [410]. This fact is of great importance in many recent papers on Fourier analysis (e.g. see [378]), since, for example, one may deduce from it that A(G1 × G2 ) ∼ = A(G1 ) ⊗ A(G2 ) for locally compact groups G1 , G2 (see [146]). A dual operator space X is said to have the dual slice mapping property if X ⊗ Y = CB(Y∗ , X) holds for all dual operator spaces Y . This is equivalent to X∗ possessing the ‘operator space approximation property’ (see [146,139,233]). The argument at the beginning of 1.6.6 is only part of this result in the case when X is a commutative W ∗ -algebra. It extends to all semidiscrete (=injective) von Neumann algebras. The description of L ∞ (Ω; Y ) as a space of Y -valued w∗ -measurable functions is essentially taken from [380, Section 1.22]. The deﬁnitions and results in 1.6.8 are due to Eﬀros and Ruan (see [138, 150], and references therein). The weak* Haagerup tensor product was developed by Blecher and Smith in [70] for dual operator spaces. Later this tensor product, and its basic properties, were generalized to all operator spaces (e.g. see [150], and to a lesser extent [5]), under the name extended Haagerup tensor product. Spronk has shown in [397] that the original arguments of [70] immediately yield many of these generalizations from [150]. Result 1.6.10 is from [150].

2 Basic theory of operator algebras

2.1 INTRODUCING OPERATOR ALGEBRAS AND UNITIZATIONS By deﬁnition, a concrete operator algebra is a closed subalgebra of B(H), for some Hilbert space H. When the context is clear, we often simply write A ⊂ B(H) or A → B(H) to denote this. As for operator spaces, we will have to consider operator algebras from an abstract point of view. We will start from the obvious observation that any operator algebra A is both an operator space and a Banach algebra. Conversely, if A is both an operator space and a Banach algebra, then we call A an (abstract) operator algebra if there exist a Hilbert space H and a completely isometric homomorphism π : A → B(H). In that situation, we often identify A and the concrete operator algebra π(A) and say that A is represented as a subalgebra of B(H), or that A is an operator algebra on H. We often identify any two operator algebras A and B which are completely isometrically isomorphic, that is, there exists a complete isometric algebra homomorphism from A onto B. In this case, we write ‘A ∼ = B completely isometrically isomorphically’, or ‘as operator algebras’; or simply ‘A = B’. Some more notation that will be used throughout: We say that an operator algebra A is unital if it has an identity (i.e. a unit) of norm 1. In this case, the unit is often written as e or 1 (or sometimes eA or 1A if necessary). However in this book we will usually focus on the larger class of operator algebras which are approximately unital; that is, which possess a contractive approximate identity (cai). The main reason for this choice of focus is because this class includes all C ∗ -algebras. We repeat from 1.1.5 the convention that a unital-subalgebra is a subalgebra A of a unital algebra B with 1B ∈ A. Of course operator algebras may be thought of as the (closed) subalgebras of C ∗ -algebras. The closure of a subalgebra of an operator algebra is obviously an operator algebra. By a representation of an operator algebra A on a Hilbert space H, we mean a homomorphism π : A → B(H). Of course we are almost always interested in the completely contractive representations. It should be noticed that unlike ∗-representations between C ∗ -algebras, bounded and completely bounded homomorphisms between operator algebras have no automatic rigidity property in general. For instance, a completely contractive homomorphism need not have a closed range. If A and B are operator algebras, then we will often write A → B when there exists a completely isometric homomorphism j : A → B. Again, a unital map is one that takes the identity to the identity.

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∗ 2.1.1 (C ∗ -covers) If S is a subset of a C ∗ -algebra B, then CB (S) denotes the ∗ ∗ C -subalgebra of B generated by S (that is, the smallest C -subalgebra of B containing S). A C ∗ -cover of an operator algebra A is a pair (B, j) consisting of a C ∗ -algebra B, and a completely isometric homomorphism j : A → B, such ∗ (j(A)) = B. that j(A) generates B as a C ∗ -algebra. That is, such that CB

2.1.2 (The diagonal) We deﬁned this for unital operator spaces in 1.3.7. If A is an operator algebra (possibly nonunital), represented as a subalgebra of B(H) say, then we deﬁne the diagonal of A to be the C ∗ -algebra ∆(A) = {a ∈ A : a∗ ∈ A}. Note that if A is a w∗ -closed subalgebra of B(H) then so is ∆(A) (by the w ∗ continuity properties mentioned in A.1.2), and hence ∆(A) is a W ∗ -algebra. To see that the deﬁnition of ∆(A) is (up to ∗-isomorphism) independent of the particular representation of A, consider any isometric homomorphism π : A → B(K). Then it follows from A.5.8 that π(a)∗ = π(a∗ ) for any a ∈ ∆(A). Thus π maps ∆(A) ∗-isomorphically (and therefore also completely isometrically, by 1.2.4) onto the corresponding space {b ∈ π(A) : b∗ ∈ π(A)}. From this, one deduces that a contractive homomorphism from a C ∗ -algebra into an operator algebra B, actually maps into ∆(B), and is a ∗-homomorphism. Also, from this we see that a closed subalgebra of a C ∗ -algebra B may have at most one involution with respect to which it is a C ∗ -algebra. If there exists such an involution, then the subalgebra is a ∗-subalgebra of B. 2.1.3 (Recognizing selfadjoint elements, etc.) Certain important elements in an operator algebra may be characterized intrinsically, and such results are very often useful. For the discussion that follows, we ﬁx a closed subalgebra A of B(H). Then the projections in B(H) which happen to be also in A, are exactly the idempotent elements in A of norm 1 (see A.1.1). The selfadjoint operators in B(H) which happen to be also in A, are exactly the selfadjoint elements in the diagonal ∆(A). If A is a unital operator algebra, then these selfadjoint elements are exactly the Hermitian elements of A in the sense of A.4.2. The diagonal ∆(A) is the span of these selfadjoint elements of course. If a, b are elements in the unit ball of A, and if ab = 1A , then a = b∗ by A.1.1, and a, b ∈ ∆(A). If in fact ab = IH , then b is an isometry on H, and a is a coisometry. If in addition ba = 1A , then b is a unitary in ∆(A), and is a unitary operator on H if 1A = IH . 2.1.4 Let A ⊂ B(H) be a unital operator algebra, with unit denoted by e. Then e2 = e and e = 1, consequently e : H → H is a projection, as noted in 2.1.3. Let K ⊂ H denote the range of e. If a ∈ A, then since eae = a, we have a(K) ⊂ K and a(K ⊥ ) = {0}. We may therefore regard A ⊂ B(K) as an operator algebra on K, and in that representation, the unit e coincides with the identity operator IK . Also [aij ] = [eaij e] = [(aij )|K ]Mn (B(K)) ,

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51

for aij ∈ A. This shows that any unital operator algebra may be represented (completely isometrically) as a unital-subalgebra of some B(H). In 2.1.10 below, we will prove a similar result for approximately unital operator algebras. In fact the next several results establish various useful properties of contractive approximate identities in operator algebras. 2.1.5 (Nondegeneracy) Let A be a Banach algebra, let H be a Hilbert space, and let π : A → B(H) be a contractive homomorphism. We say that π is nondegenerate if [π(A)H] = H. This terminology is consistent with that in A.6.1. Indeed if we consider H as a left Banach A-module by letting aζ = π(a)ζ, for any a ∈ A and any ζ ∈ H, then π is obviously nondegenerate if and only if H is a nondegenerate A-module. This is also equivalent, as we mentioned in A.6.1, to saying that π(et ) → IH strongly, if (et )t is a cai for A. If A ⊂ B(H) is a concrete operator algebra, then we say that A is a nondegenerate subalgebra of B(H), or that ‘A ⊂ B(H) nondegenerately’, if the embedding from A into B(H) is nondegenerate. Lemma 2.1.6 Suppose that a is an element of a subspace of B(K, H), and that (et )t is a net of contractions in B(K) such that aet → a. Then aet e∗t → a, ae∗t et → a, and ae∗t → a. Proof If aet → a then aet e∗t a∗ → aa∗ , so that 0 ≤ a(I − et e∗t )a∗ →0, where ∗ I = IK . Thus by the C -identity, a I − et e∗t → 0. Multiplying by I − et e∗t we see that a(I − et e∗t ) → 0 as required for the ﬁrst assertion. Also, ae∗t − a ≤ ae∗t − aet e∗t + aet e∗t − a → 0 since ae∗t − aet e∗t ≤ a − aet → 0. Finally, ae∗t et − a ≤ ae∗t et − aet + aet − a ≤ ae∗t − a + aet − a → 0 by what we just proved.

2

Lemma 2.1.7 Let A be an operator algebra and suppose that B is a C ∗ -cover of A. If A is approximately unital, then B is a nondegenerate A-bimodule in the sense of A.6.1. In this case (viewing A ⊂ B): (1) If b ∈ B then there exists an element b0 ∈ B and elements a1 , a2 ∈ A with b = a1 b0 a2 . Moreover if b < 1 then b0 , a1 , a2 may be chosen of norm < 1. (2) Every cai for A is a cai for B. Proof Notice that once (2) is established, then B is a nondegenerate Banach A-bimodule, and then (1) follows from Cohen’s factorization theorem (see A.6.2). Let (et )t be a cai for A, and let a ∈ A. Then et a → a, and using 2.1.6 we have a∗ et → a∗ . Thus if b is in the dense ∗-subalgebra of B generated by A, then 2 bet → b. By density, bet → b for every b ∈ B. Similarly, et b → b. 2.1.8 It follows from the last result that any C ∗ -cover (B, j) of an approximately unital operator algebra A is a unital C ∗ -algebra if and only if A is unital. Indeed

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if A is unital with unit 1A , then j(1A ) is a unit for B by 2.1.7 (2). Conversely, if B is unital with unit 1B , then by 2.1.7 (2) we see that any cai (et ) for A satisﬁes j(et ) = j(et ) 1B −→ 1B . Since j(A) is closed, we have 1B ∈ j(A). As another application of Lemma 2.1.7, we may see using 2.1.5 that if B is a C ∗ -cover of an approximately unital operator algebra A, and if π : B → B(H) is a ∗-representation, then π is nondegenerate if and only if its restriction π|A is nondegenerate. Lemma 2.1.9 Let A be a Banach algebra with cai (et )t , and let π : A → B(H) be a contractive homomorphism. We let p be the projection onto [π(A)H]. Then π(et ) → p in the w∗ -topology of B(H). Moreover, for a ∈ A we have π(a) = pπ(a)p,

a ∈ A.

(2.1)

Proof Let K = [π(A)H]. If a ∈ A then π(et )π(a) → π(a), and so π(et )∗ π(a) converges to π(a) by 2.1.6. Hence π(et )∗ → IK strongly on K. If ζ, η ∈ H, then π(et )ζ, η = pπ(et )ζ, η = ζ, π(et )∗ pη −→ ζ, pη = pζ, η. Thus the bounded net (π(et ))t converges to p in the WOT. Hence it also converges in w∗ -topology to p, by A.1.4. Using the last fact and the separate w ∗ continuity of the product in B(H) (see A.1.2), we obtain for a ∈ A that π(a)p = w∗ - lim π(a)π(et ) = lim π(aet ) = π(a). t

t

Since pπ(a) = π(a) by the deﬁnition of p, the claim (2.1) follows at once.

2

2.1.10 (Reducing to the nondegenerate case) The relation (2.1) should be interpreted as follows. Let H, K, π be as above. If we regard B(K) as a subalgebra of B(H) in the natural way (by identifying any T in B(K) with the map T ⊕ 0 in B(K ⊕ K ⊥ ) = B(H)), then the homomorphism π is valued in B(K). Since π is nondegenerate when regarded as valued in B(K), this yields a principle whereby to reduce a possibly degenerate homomorphism to a nondegenerate one. Applying this to completely isometric homomorphisms, we obtain that for any approximately unital operator algebra A, there exist a Hilbert space H and a nondegenerate completely isometric homomorphism π : A → B(H). 2.1.11 (The unitization) Often problems concerning an operator algebra A are solved by ﬁrst tackling the case where A is unital; and then in the general case considering the unitization A1 of A. It is this process of unitization which we now discuss. If A is a nonunital operator algebra, then a unitization of A which is also an operator algebra may be obtained by regarding A as a subalgebra of B(H) for some H, and then taking A1 = Span{A, IH }. We will prove in Corollary 2.1.15 that up to completely isometric isomorphism, this unitization does not depend

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on the embedding A ⊂ B(H). Consequently, A1 will be called the unitization of A, and we usually use it without any reference to a concrete embedding of A in B(H). We will ﬁrst establish the more general Theorem 2.1.13, which is an extremely useful extension principle. The following simple lemma is a special case: / A. Lemma 2.1.12 Let A ⊂ B(H) be an operator algebra, and assume that IH ∈ Then for any n ≥ 1 and any matrices a ∈ Mn (A) and λ ∈ Mn , we have λMn ≤ a + λ ⊗ IH Mn (B(H)) . Proof Consider the algebra A1 = Span{A, IH } ⊂ B(H). Since IH ∈ / A, we may deﬁne a functional φ on A1 by letting φ(a + λIH ) = λ for any a ∈ A and any λ ∈ C. It is clear that φ is a homomorphism. From the basic theory of characters from any text on Banach algebras, we have that φ is contractive. Hence by 1.2.6, φ is completely contractive. This proves the result. 2 Theorem 2.1.13 (Meyer) Let A ⊂ B(H) be an operator algebra, and assume / A. Let π : A → B(K) be a contractive (resp. completely contractive) that IH ∈ homomorphism, K being a Hilbert space. We let A1 = Span{A, IH } ⊂ B(H), and we extend π to π 0 : A1 → B(K) by letting π 0 (a + λIH ) = π(a) + λIK ,

a ∈ A, λ ∈ C.

Then π 0 is a contractive (resp. completely contractive) homomorphism. 2.1.14 Before launching into the proof of Theorem 2.1.13, we recall a few facts concerning the so-called Cayley transform of operators on a Hilbert space H. To those familiar with the basic theory of operator semigroups, the following facts will be transparent (e.g. see [405] sections III.8 and IV.4). In order to be self-contained, we will prove these facts using some basic results about the Riesz or analytic functional calculus (e.g. see [106] section 2.4). We let D and P denote respectively the open unit disk and right-hand open half plane in C. We let z−1 d : D → P be the conformal map z → 1+z 1−z , and let c(z) = z+1 be its inverse. If −1 is not in the spectrum σ(S) of an operator S on H (resp. 1 ∈ / σ(T )), then we deﬁne the Cayley transform (resp. inverse Cayley transform) by c(S) = (S −I)(S +I) −1 (resp. d(T ) = (I + T )(I − T )−1 ). It follows from the spectral mapping theorem (e.g. see [106, 2.4.4 (iv)]) that 1 ∈ / σ(c(S)) (resp. −1 ∈ / σ(d(T ))). Thus we see from the functional calculus that these transforms are indeed inverses of each other. For example, c(d(T )) = T providing that 1 ∈ / σ(T ). An operator T ∈ B(H) will be called a strict contraction if T < 1. In that case, the usual Neumann series trick shows that the operator I − T is invertible and (I −T )−1 = k≥0 T k belongs to the operator algebra generated by I and T . Next, an operator S ∈ B(H) will be called strictly accretive if Re(S) = 21 (S + S ∗ ) is positive and invertible. That is, there is a scalar > 0 with Re(S) ≥ I. 1 After pre- and post-multiplying by Re(S)− 2 , it is easy to see that such an S is

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invertible. Similarly, for any z ∈ C with Re(z) ≤ 0, it is clear that S − zI is also strictly accretive, and therefore invertible in B(H). Thus the spectrum of S is contained in P. We claim that (S + I)−1 belongs to the operator algebra B generated by I and S. To see this, by the basic theory of Banach algebras it is enough to show that χ(S + I) = 0 for every character of B. However by the Hahn–Banach theorem any such χ extends to a state ϕ on B(H) (see A.4.2), and ϕ(S + I) = ϕ(S) + 1 = 0 (indeed Re(ϕ(S)) = ϕ(Re(S)) ≥ 0, by (A.11) say). Using the formula 4Re(S)ζ, ζ = (S + I)ζ2 − (S − I)ζ2 ,

ζ ∈ H,

(2.2)

a simple computation shows that: (1) S ∈ B(H) is strictly accretive if and only if −1 ∈ / σ(S) and the Cayley transform c(S) = (S − I)(S + I)−1 is a strict contraction. (2) T ∈ B(H) is a strict contraction if and only if 1 ∈ / σ(T ) and the inverse Cayley transform d(T ) = (I + T )(I − T )−1 is strictly accretive. Indeed (2) may be derived from (1) and the fact above that T = c(d(T )). Proof (Of Theorem 2.1.13) It is clear that π 0 is a homomorphism. We will show that if the nth ampliﬁcation πn is contractive, then so is πn0 . Fix T in Mn (A1 ) ⊂ B(2n (H)), with T < 1. We will show that πn0 (T ) < 1. We may write T uniquely as T = a + λ ⊗ IH , with a ∈ Mn (A) and λ ∈ Mn . By Lemma 2.1.12, we have λ < 1. To simplify the notation, we shall identify λ and λ ⊗ IH in the sequel. Let us apply the assertion (2) of 2.1.14 to λ. We ﬁnd two selfadjoint matrices α, β ∈ Mn such that (I + λ)(I − λ)−1 = α + iβ, and such that α is positive and invertible. We may write the preceding identity as α−1/2 (I + λ)(I − λ)−1 α−1/2 − iα−1/2 βα−1/2 = I.

(2.3)

Let us now apply 2.1.14 (2) to T . We ﬁnd that (I + T )(I − T )−1 is strictly accretive, from which we deduce that θ = α−1/2 (I + T )(I − T )−1 α−1/2 − iα−1/2 βα−1/2

(2.4)

is strictly accretive. According to (2.3) we can write this operator as θ = I + α−1/2 (I + T )(I − T )−1 − (I + λ)(I − λ)−1 α−1/2 . Now observe that (I + T )(I − T )−1 − (I + λ)(I − λ)−1 may be rewritten as (I − T )−1 (I + T )(I − λ) − (I − T )(I + λ) (I − λ)−1 , which is simply 2(I − T )−1 a(I − λ)−1 . Since (I − T )−1 belongs to the operator algebra generated by I and T (see 2.1.14), hence to Mn (A1 ), and since Mn (A) is an ideal of Mn (A1 ), we obtain that (I + T )(I − T )−1 − (I + λ)(I − λ)−1 ∈ Mn (A). Therefore θ − I belongs to Mn (A). Moreover by a fact in 2.1.14 again, (θ + I)−1 belongs to the operator algebra generated by I and θ, and hence also to M n (A1 ). Hence the Cayley transform (θ − I)(θ + I)−1 belongs to Mn (A).

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Since πn0 is a unital homomorphism and θ + I is invertible, πn0 (θ) + I is −1 = πn0 (θ + I)−1 , and hence invertible as well. Also πn0 (θ) + I −1 . πn0 (θ − I)(θ + I)−1 = πn0 (θ) − I πn0 (θ) + I By the assertion (1) of 2.1.14, the operator (θ −I)(θ +I)−1 is a strict contraction. Since it belongs to Mn (A), and since πn is contractive, we deduce that 0 πn (θ) − I πn0 (θ) + I −1 = πn (θ − I)(θ + I)−1 < 1. By 2.1.14 (2) we deduce that πn0 (θ), which is the inverse Cayley transform of the −1 operator πn0 (θ) − I πn0 (θ) + I , is strictly accretive. By (2.4), πn0 (I + T )(I − T )−1 = πn0 α1/2 θα1/2 + iβ = α1/2 πn0 (θ)α1/2 + iβ is strictly accretive. Since πn0 is a homomorphism, we have as above that −1 πn0 (I + T )(I − T )−1 = I + πn0 (T ) I − πn0 (T ) . By 2.1.14 (1) again, we have πn0 (T ) < 1, which concludes the proof.

2

Corollary 2.1.15 (Meyer) Let A ⊂ B(H) be a nonunital operator algebra and let π : A → B(K) be an isometric (resp. completely isometric) homomorphism. Then the unital homomorphism from Span{A, IH } into B(K) extending π is an isometry (resp. complete isometry). Proof By assumption, IH ∈ / A. Let B = π(A) and regard π as valued in B. Since A is isomorphic to B as algebras, B is nonunital. Hence IK ∈ / B. Using the notation from Theorem 2.1.13, we deduce that π 0 : Span{A, IH } → Span{B, IK } is a bijection, with (π 0 )−1 = (π −1 )0 . That theorem shows that π 0 and (π −1 )0 are both (completely) contractive. Thus π 0 is a (complete) isometry. 2 2.1.16 (Unitization of a subalgebra) It is clear from 2.1.15 that if C is a unital operator algebra with unit denoted by 1C and if A ⊂ C is a nonunital subalgebra, then A1 may be taken to be Span{A, 1C } ⊂ C. In particular if A, B are nonunital operator algebras with A ⊂ B, then the units of A1 and B 1 may be identiﬁed and A1 may be viewed as a unital-subalgebra of B 1 . 2.1.17 (Unitization of an approximately unital algebra) Although the unitization A1 is now deﬁned unambiguously, it is in general very diﬃcult to describe its norm (or its matrix norms) intrinsically. However if A is an approximately unital operator algebra, and if A ⊂ B(H) nondegenerately (see 2.1.10), then for any a ∈ A and any λ ∈ C, we claim that a + λIH = sup ac + λc : c ∈ A, c ≤ 1 . (We note in passing, although we do not need this here, that the right-hand side above is the ‘unitization norm’ from A.4.3, and that another equivalent

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description of it is given there.) To prove the claim, let (et )t be a cai for A. Since et → IH strongly by 2.1.5, a + λIH is the strong limit of aet + λet . Hence a + λIH ≤ sup aet + λet . t

The other direction is easier, since for any c ∈ A with c ≤ 1, ac + λc = (a + λIH )c ≤ a + λIH . This proves the result. With the same proof we see that more generally, for any integer n ≥ 1 and for any matrices [aij ] ∈ Mn (A) and [λij ] ∈ Mn , we have [aij + λij IH ] = sup [aij c + λij c] : c ∈ A, c ≤ 1 . (2.5) If A is an already unital operator algebra then Meyer’s result shows that there is an essentially unique unital operator algebra containing A completely isometrically as a codimension 1 ideal. Again we write this strictly larger algebra as A1 . In fact this ‘unitization’ is just the ‘∞-direct sum’ A ⊕∞ C (see 1.2.17). Proposition 2.1.18 Let A be an approximately unital operator algebra with a cai (et )t and denote the identity of A1 by 1. (1) If ψ : A1 → C is a functional on A1 , then limt ψ(et ) = ψ(1) if and only if ψ = ψ|A . (2) Let ϕ : A → C be any functional on A. Then ϕ uniquely extends to a functional of the same norm on A1 . Proof We ﬁrst prove (1). Suppose that ψ : A1 → C satisﬁes limt ψ(et ) = ψ(1). For a ∈ A and λ ∈ C, we obtain limt ψ(aet + λet ) = ψ(a + λ1), and so ψ(a + λ1) ≤ supψ(aet + λet ). t

Consequently, ψ(a + λ1) ≤ ψ|A sup aet + λet ≤ ψ|A a + λ1, t

which proves that ψ = ψ|A . Assume conversely that ψ = ψ|A . We may assume that this norm is equal to 1. By 1.2.8 there exist a Hilbert space H, a contractive unital homomorphism π : A1 → B(H) and two unit vectors ζ, η ∈ H such that ψ(x) = π(x)ζ, η for any x ∈ A1 . Let K = [π(A)ζ] ⊂ H and let p be the projection onto K. For any a ∈ A, we have π(a)ζ, η = pπ(a)ζ, η, and so π(a)ζ, η = π(a)ζ, pη ≤ π(a)ζpη ≤ apη. This shows that ψ|A ≤ pη. Hence, by hypothesis, pη = η = 1. Thus pη = η, that is, η ∈ K. Now recall from Lemma 2.1.9 that π(et ) → p in the w∗ -topology, and thus

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ψ(et ) = π(et )ζ, η −→ pζ, η = ζ, η = ψ(1). We now deduce (2) from (1). Let ϕ ∈ A∗ . Again we may write ϕ = π(·)ζ, η; and Lemma 2.1.9 and the ‘ﬁrst half’ of the last centered equation shows that (ϕ(et ))t converges. Extend ϕ to a map ψ : A1 → C by deﬁning ψ(1) = limt ϕ(et ). The ‘only if’ part of (1) ensures that ψ = ϕ. On the other hand, the ‘if’ part of (1) shows that this norm preserving extension is unique. 2 2.1.19 (States) If A is unital then a state on A is a functional ϕ on A of norm 1 which satisﬁes ϕ(1) = 1 (see A.4.2). If A is not unital but has a cai (e t ), then we deﬁne a state on A to be a functional ϕ on A of norm 1 which satisﬁes limt ϕ(et ) = 1. By the last result, such ϕ extends uniquely to a state on A1 . 2.2 A FEW BASIC CONSTRUCTIONS The class of operator algebras is stable under many of the categorical constructions introduced in Chapter 1. We shall now review the simplest of these in this short section. 2.2.1 (Direct sums) Let I be a set and let (Ai )i∈I be a family of operator algebras. Let A = ⊕i∈I Ai be the ∞-direct sum of this family, as studied in 1.2.17. Besides being an operator space, A is a Banach algebra for the pointwise product deﬁned by setting the product of (ai )i and (bi )i equal to (ai bi )i . Here (ai )i and (bi )i are in A. If we represent each Ai ⊂ B(Hi ) as an operator algebra acting on some Hilbert space Hi , the resulting embedding A ⊂ ⊕i∈I B(Hi ) ⊂ B ⊕2i∈I Hi is a homomorphism. Hence A is an operator algebra. It is clear that A is unital if each Ai is unital. 2.2.2 (The minimal tensor product) To deﬁne this we use facts from 1.5.1 and 1.5.2. Let A ⊂ B(H) and B ⊂ B(K) be two operator algebras. Equip the algebraic tensor product A ⊗ B with the joint multiplication deﬁned by letting ai ⊗ b i cj ⊗ dj = ai cj ⊗ bi dj , (2.6) i

j

i,j

for any ﬁnite families (ai )i , (cj )j in A and (bi )i , (dj )j in B. It is clear that the natural embedding A ⊗ B ⊂ B(H ⊗2 K) is an algebra homomorphism. Passing to the completion, we see that A ⊗min B is an operator algebra. If A and B are unital with identities denoted by 1A and 1B , then A ⊗min B is unital as well, with identity equal to 1A ⊗ 1B . It is also worthwhile to observe that if A and B are approximately unital, then the same holds for A ⊗min B. Indeed if (et )t and (fs )s are cai’s for A and B respectively, then by a simple density argument, any y ∈ A ⊗min B satisﬁes lim lim (et ⊗ fs )y = lim lim y(et ⊗ fs ) = y. s

t

s

t

58

A few basic constructions

It follows from the ‘functoriality’ property in 1.5.1 that if π : A → B(H) and ρ : B → B(K) are completely contractive homomorphisms, then there is a unique completely contractive linear map π ⊗ ρ : A ⊗min B −→ B(H) ⊗min B(K) ⊂ B(H ⊗2 K) with (π ⊗ ρ)(a ⊗ b) = π(a) ⊗ ρ(b) for a ∈ A, b ∈ B. By an obvious density argument one sees that since π ⊗ ρ is a homomorphism on A ⊗ B, it is also a homomorphism on A ⊗min B. 2.2.3 (Matrix algebras) A special case of the minimal tensor product of particular interest, is the case when B = B(K), or equivalently when B = MI for a cardinal I. If n is a ﬁnite integer then Mn (A) ⊂ Mn (B(H)) = B(2n (H)) is an operator algebra on 2n (H), the product of two elements [aij ] and [bij ] of Mn (A) being given by n [aij ][bij ] = (2.7) aik bkj . k=1

We recall from (1.36) that Mn ⊗min A = Mn (A) as operator spaces, and this isomorphism is easily seen to be a homomorphism too. Thus they are ‘equal as operator algebras’. This result extends with a similar proof to the case when I is an inﬁnite cardinal or set. Indeed recall from 1.2.26 that KI (A) is deﬁned to be the closure of the subalgebra Mﬁn I (A) of the algebra MI (B(H)) = B(2I (H)). Thus KI (A) is an operator algebra on 2I (H) and KI ⊗min A = KI (A)

as operator algebras.

(2.8)

Note that if [aij ] and [bij ] are two (inﬁnite) matrices belonging to KI (A), then their product is equal to the matrix [cij ] whose entries are deﬁned by the expression cij = k∈I aik bkj ∈ A, the latter sum being norm convergent. The last fact follows from the discussion preceding (1.21). On the other hand, MI (A) need not be an algebra at all. However there are several operator algebras inside MI (A) which occasionally play a role. For example, the subsets of MI (A) corresponding to CIw (RI (A)), RIw (CI (A)), CI (RIw (A)) and RI (CIw (A)) are all operator algebras. We write RIw (CI (A)) as Mwr I (A); and we write CIw (RI (A)) as Mwc (A). These two operator algebras contain the other I two algebras in the last list as subalgebras. These facts will not play much of a wr role for us, and we leave the details as an exercise. Note that M wc I (A) ∩ MI (A) is also an operator algebra, and it contains the operator algebra A ⊗ min MI as may be seen by inspecting rank 1 tensors in the latter space. 2.2.4 (Operator algebra valued continuous functions) Let Ω be a compact space, and A an operator algebra. The operator space C(Ω; A) discussed in 1.2.18 is an operator algebra for the product deﬁned by (f g)(t) = f (t)g(t). Namely,

Basic theory of operator algebras

59

if A is a subalgebra of B(H) then C(Ω; A) is a subalgebra of the C ∗ -algebra C(Ω; B(H)). If A is unital, then C(Ω; A) is unital as well, the identity being the constant function equal to the identity of A. According to (1.39), we have a completely isometric identiﬁcation C(Ω) ⊗min A = C(Ω; A). We notice here that this identiﬁcation holds as well ‘as operator algebras’. This is equivalent to saying that the linear mapping taking a ﬁnitesum k gk ⊗ ak ∈ C(Ω) ⊗ A to the function f ∈ C(Ω; A) deﬁned by f (t) = k gk (t)ak is a homomorphism if C(Ω) ⊗min A and C(Ω; A) are equipped with the multiplications deﬁned in 2.2.2 and above respectively. Similarly if Ω is merely a locally compact space, then the operator space C0 (Ω; A) is an operator algebra for the pointwise product. As operator algebras, we have that C0 (Ω) ⊗min A = C0 (Ω; A). 2.2.5 (Uniform algebras) By deﬁnition, a (concrete) uniform algebra is a unitalsubalgebra of C(Ω), for some compact space Ω. In this book, we will consider any uniform algebra as endowed with its minimal operator space structure (described by the formula (1.3)). Then an (abstract) uniform algebra is a unital operator algebra which is completely isometrically isomorphic to a concrete uniform algebra. In this way we regard uniform algebras as a subclass of the operator algebras. A characterization of the algebras in this subclass will be given in 3.7.9 (see also the Notes to Section 4.6). More generally, we will use the term function algebra for an operator algebra A for which there exists a compact space Ω and a completely isometric homomorphism π : A → C(Ω). Any function algebra is a minimal operator space. 2.2.6 (Disc algebra) This fundamental example of a uniform algebra has two equivalent deﬁnitions. Let us denote by D and T the open unit disc of C and the unitary complex group T = {z ∈ C : |z| = 1} respectively. Then the disc algebra A(D) is the subalgebra of C(D) consisting of all continuous functions F : D → C, whose restriction to D is analytic. By the maximal modulus theorem, the restriction of functions in A(D) to the boundary T is an isometry. Hence we may alternatively regard A(D) ⊂ C(T) as a uniform algebra acting on T. In that representation, A(D) consists of all elements of C(T) whose harmonic extension to D given by the Poisson integral is analytic. Equivalently, given any f ∈ C(T), we associate Fourier coeﬃcients ! k ∈ Z. (2.9) f(k) = f (z)z −k dm(z), T Then A(D) ⊂ C(T) is the closed subalgebra of all f ∈ C(T) such that f (k) = 0 for every k < 0. The (analytic) polynomials form a dense subalgebra of A(D). 2.2.7 (Operator algebra valued analytic functions) Let X be an operator space. For any f ∈ C(T; X), we may deﬁne Fourier coeﬃcients f (k) ∈ X by means of (2.9). According to 2.2.6, we let A(D; X) ⊂ C(T; X) denote the subspace of all f such that f(k) = 0 for every k < 0. Equivalently, as in the scalar case, A(D; X)

60

A few basic constructions

may be deﬁned to be the subspace of C(D; X) consisting of all functions F whose restriction to D is analytic. We have A(D) ⊗min X = A(D; X) completely isometrically. Indeed both spaces may be regarded as the closure of A(D)⊗X in C(T; X). (The density of A(D)⊗X in A(D; X) follows from a routine argument using Fejer kernels.) If X = B is an operator algebra, it follows from either of these deﬁnitions that A(D; B) is an operator algebra which is both a subalgebra of C(T; B) and C(D; B). In that case the above identiﬁcation holds ‘as operator algebras’. 2.2.8 (Opposite and adjoint algebras) We will use notations from 1.2.25. If A is an operator algebra, then the operator space Aop from 1.2.25, is also an operator algebra for the reversed product ◦ deﬁned on A by a ◦ b = ba. Indeed, if A is a subalgebra of a C ∗ -algebra B, then Aop may be identiﬁed with the matching subalgebra of the C ∗ -algebra B op discussed in 1.2.25. We remark in passing that if A = Aop , and if A is approximately unital, then A is commutative. This follows quickly from one of our later ‘Banach-Stone theorems’ (e.g. see 4.5.13) applied to the identity map from A to Aop . Similarly, the adjoint operator space A (written also as A∗ when there is no possible confusion with the dual space) from 1.2.25, is an operator algebra, with product a∗ b∗ = (ba)∗ , for a, b ∈ A. Indeed if A is a subalgebra of a C ∗ -algebra B, then A may be identiﬁed with the subalgebra {a∗ : a ∈ A} of B. Note that if A has a cai (et )t , then (e∗t )t is a cai for A∗ . If π : A → B is a completely contractive homomorphism between operator algebras, then the canonical completely contractive mappings π op : Aop → B op and π : A → B from 1.2.25 also are homomorphisms. In particular, given a completely contractive representation π : A → B(H), we obtain a canonical completely contractive representation π : A → B(H). It easily follows from Lemma 2.1.9 that if A is approximately unital and π is nondegenerate, then π ∗ is nondegenerate as well. 2.2.9 Let X ⊂ B(H) be an operator space acting on some Hilbert space H. Then the operator space deﬁned inside B(H ⊕ H) = M2 (B(H)) as 0 x : x∈X 0 0 is completely isometric to X. It is also a subalgebra of B(H (2) ), with trivial (i.e. zero) product. Thus any operator space may be regarded as an operator algebra. 2.2.10 The last construction has a unital version which turns out to be much more useful. Again we consider an operator space X ⊂ B(H) acting on some Hilbert space H. We deﬁne λ1 x (2.10) : x ∈ X, λ1 , λ2 ∈ C ⊂ B(H ⊕ H), U(X) = 0 λ2

Basic theory of operator algebras

61

where λ1 and λ2 stand for the operators λ1 IH and λ2 IH respectively. By deﬁnition, U (X) may be regarded as a subspace of the Paulsen system S(X) deﬁned in 1.3.14. It therefore follows from 1.3.15 that as an operator space, U(X) only depends on X and not on the Hilbert space H on which it acts. For any x, y ∈ X and any λ1 , λ2 , µ1 , µ2 ∈ C, we have λ1 x µ1 y λ1 µ1 λ1 y + µ2 x = . 0 λ2 0 µ2 0 λ2 µ2 Hence U(X) is a subalgebra of B(H (2) ), and the multiplication on U(X) does not depend on H. Thus U(X) is a unital operator algebra whose deﬁnition only depends on X. The norm on U(X) also only depends on the norm on X (and not on the full operator space structure of X), as we shall see below. Proposition 2.2.11 Let X and Y be operator spaces, and let u : X → Y be a linear contraction (resp. complete contraction). Then the mapping θ u from U(X) to U(Y ) deﬁned by " # λ1 x λ1 u(x) θu = , x ∈ X, λ1 , λ2 ∈ C, 0 λ2 0 λ2 is a contractive (resp. completely contractive) homomorphism. Proof It is clear that θu is a homomorphism. The ‘complete contraction case’ follows immediately from 1.3.15. To see the ‘contraction case’, we explicitly compute the norm of a matrix in U(X). We have λ1 x 2 2 2 2 2 2 0 λ2 = sup λ1 ζ + xη + |λ2 | η : ζ + η ≤ 1 . Here ζ, η ∈ H, where H is a Hilbert space on which X is represented completely isometrically. It is easily seen that this last quantity equals 2

sup |λ1 | 1 − η2 + xη + |λ2 |2 η2 : η ≤ 1 , (the being obvious, and the other is obtainable by letting ζ equal one inequality xη eiθ 1 − η2 xη

, for real θ). In turn the last centered quantity equals 2

sup |λ1 | 1 − t2 + xη + |λ2 |2 t2 : t ∈ [0, 1], η = t 2

= sup |λ1 | 1 − t2 + xt +|λ2 |2 t2 : t ∈ [0, 1] . This last expression, for ﬁxed λ1 , λ2 , may be regarded as a function f (x). Since f is an increasing function, the desired result follows easily. 2 Corollary 2.2.12 If X and Y are two isometric (resp. completely isometric) operator spaces, then U(X) and U(Y ) are isometric (resp. completely isometric) operator algebras.

62

The abstract characterization of operator algebras

Proof We need only prove the isometric case. Given an isometric isomorphism J : X → Y , apply Lemma 2.2.11 to J and J −1 to obtain that uJ : U (X) → U(Y ) is an invertible isometric homomorphism. 2 2.2.13 (Ultraproducts) Let U be an ultraﬁlter on a set I, and consider a family of operator algebras (Ai )i∈I . We let A be their ultraproduct along U, as deﬁned in 1.2.31. Assume that each Ai is a subalgebra of some B(Hi ). Then it follows from the last few results in 1.2.31 that A is an operator algebra, a subalgebra of the C ∗ -algebra i∈I B(Hi )/U. Clearly A is unital if each Ai is unital. 2.3 THE ABSTRACT CHARACTERIZATION OF OPERATOR ALGEBRAS The main result of this section is Theorem 2.3.2, usually referred to as the BRS theorem. This fundamental result gives a criterion for a unital (or more generally an approximately unital) Banach algebra with an operator space structure, to be an operator algebra. Among other things, the BRS theorem allows one to check that abstract constructions with operator algebras remain operator algebras. We discuss some of the most important such constructions later in this section. 2.3.1 Recall from 2.2.3 that if A is an operator algebra, and if n ≥ 1 is an integer, then Mn (A) is an operator algebra with product given by (2.7). In particular it is a Banach algebra, so that the product on A is a completely contractive bilinear map in the sense described in 1.5.4. This is equivalent (since the Haagerup tensor product linearizes completely bounded bilinear maps—see 1.5.4) to saying that the multiplication mapping m : A ⊗ A → A on A extends to a completely contractive mapping on the Haagerup tensor product A ⊗h A. That is, m : A ⊗h A −→ A ≤ 1. (2.11) cb The BRS theorem asserts that this property characterizes operator algebras, at least for unital or approximately unital algebras. We will see in 5.3.6 that this characterization may fail without the approximately unital assumption. Theorem 2.3.2 (BRS theorem) Let A be an operator space which is also an approximately unital Banach algebra. Let m : A ⊗ A → A denote the multiplication on A. The following are equivalent: (i) The mapping m : A ⊗h A −→ A is completely contractive. (ii) For any n ≥ 1, Mn (A) is a Banach algebra. That is, n aik bkj k=1

Mn (A)

≤ [aij ]M

n (A)

[bij ] M

n (A)

for any [aij ] and [bij ] in Mn (A). (iii) A is an operator algebra, that is, there exist a Hilbert space H and a completely isometric homomorphism π : A → B(H).

Basic theory of operator algebras

63

Proof The equivalence between (i) and (ii), and that (iii) implies these, was mentioned in 2.3.1. To see that (i) and/or (ii) implies (iii), we let A be as in (i), and we will ﬁrst assume that A is unital, with identity e of norm 1. Fix a complete isometry u0 : A → B(H0 ), for some Hilbert space H0 , and apply Theorem 1.5.7 (1) to the mapping u0 m : A ⊗h A → B(H0 ). We obtain a Hilbert space H1 , and two completely contractive mappings u1 : A → B(H0 , H1 ) and v1 : A → B(H1 , H0 ), such that u0 m(b, a) = v1 (b)u1 (a) for any a, b ∈ A. Since mcb ≤ 1 and u1 cb ≤ 1, the mapping u1 m : A ⊗h A → B(H0 , H1 ) is completely contractive. By Theorem 1.5.7 (1) again, we may factor u 1 m as well as a product of two mappings deﬁned on A. Proceeding by induction, we obtain that for any k ≥ 1, there exist a Hilbert space Hk , and two completely contractive mappings uk : A → B(H0 , Hk ) and vk : A → B(Hk , Hk−1 ), such that uk m(b, a) = vk+1 (b) uk+1 (a)

k ≥ 0, a, b ∈ A.

(2.12)

For k ∈ N deﬁne a seminorm | · |k on the algebraic tensor product A ⊗ H0 by ai ⊗ ξi = uk (ai )ξi k

i

Hk

i

for any ﬁnite families (ai )i in A and (ξi )i in H0 . These seminorms satisfy the following key inequalities. For any integers k ≥ 0, n ≥ 1, m ≥ 1, and for any [bpq ] ∈ Mn (A), [aij ] ∈ Mn,m (A), [ξij ] ∈ Mn,m (H0 ), we have 2 2 2 m(bpq , aqi ) ⊗ ξqi ≤ [bpq ]Mn (A) aqi ⊗ ξqi p

k

q,i

q

.

(2.13)

k+1

i

Indeed according to the deﬁnition of | · |k , we have 2 2 m(bpq , aqi ) ⊗ ξqi = uk m(bpq , aqi )ξqi p

k

q,i

p

q,i

p

q,i

2 = vk+1 (bpq )uk+1 (aqi )ξqi

by (2.12)

2 2 ≤ [vk+1 (bpq )] uk+1 (aqi )ξqi q

i

2 2 = [vk+1 (bpq )] aqi ⊗ ξqi q

i

.

k+1

Since vk+1 is completely contractive, this yields (2.13). Applying (2.13) with n = 1 and b11 = e, we see that ai ⊗ ξi ≤ ai ⊗ ξi i

k

i

k+1

64

The abstract characterization of operator algebras

for any ﬁnite families (ai )i in A and (ξi )i in H0 . This monotonicity allows us to deﬁne a new seminorm | · |∞ on A ⊗ H0 by letting ai ⊗ ξi = lim ai ⊗ ξi = sup ai ⊗ ξi . ∞

i

k→∞

k

i

k

i

k

Passing to the limit in (2.13), we clearly obtain that 2 2 2 m(bpq , aqi ) ⊗ ξqi ≤ [bpq ]Mn (A) aqi ⊗ ξqi p

∞

q,i

q

i

∞

(2.14)

for any [bpq ] ∈ Mn (A), [aqi ] ∈ Mn,m (A), and [ξqi ] ∈ Mn,m (H0 ). Now observe that | · |∞ is a prehilbertian seminorm. Indeed each Hk is a Hilbert space, hence |·|k is prehilbertian for any k ≥ 0. Thus each |·|k satisﬁes the parallelogram identity. Passing to the limit, we obtain that |·|∞ also satisﬁes this identity, which proves the observation. We may therefore deﬁne a Hilbert space H by ﬁrst taking the quotient of A⊗ H0 by the kernel N = {z ∈ A⊗ H0 : |z|∞ = 0} and then completing the resulting normed space (A ⊗ H0 )/N . Using (2.14) with n = 1, we see that for any b ∈ A, one may deﬁne a bounded linear mapping π(b) : H → H with π(b) ≤ b by letting ˙ ˙ ai ⊗ ξi +N m(b, ai ) ⊗ ξi +N, π(b) = ai ∈ A, ξi ∈ H0 . i

i

Then applying (2.14) with arbitrary n ≥ 1, we ﬁnd that the resulting linear mapping π : A → B(H) is actually a complete contraction. Since m is a multiplication our mapping π is clearly a homomorphism. That π is a complete isometry now reduces to showing that [bpq ] ≤ [π(bpq )] (2.15) for n ≥ 1 and [bpq ] ∈ Mn (A). Assume that [bpq ] > 1. Since u0 : A → B(H0 ) is a complete isometry, there exist ξ1 , . . . , ξn ∈ H0 such that 2 ξi 2 = 1 and u0 (bpq )ξq > 1. p

i

q

Since the sequence of seminorms | · |k is nondecreasing, for any p we have bpq ⊗ ξq ≥ bpq ⊗ ξq = u0 (bpq )ξq . ∞

q

q

0

q

˙ ) = b ⊗ ξ +N ˙ for b ∈ A and ξ ∈ H0 , we have Using the identity π(b)(e ⊗ ξ +N 2 ˙ π(bpq ) e ⊗ ξq +N > 1. p

q

Basic theory of operator algebras

65

On the other hand, since e = 1 and each uk is a contraction, we have e ⊗ ξq +N ˙ 2 = lim uk (e)ξq 2 ≤ ξq 2 = 1. k

q

q

q

This shows that [π(bpq )] > 1 and hence the inequality (2.15), which concludes our proof in the unital case. Finally, we consider the case that A is nonunital, but has a cai (et )t . Let A1 = A ⊕ C be the Banach algebraic unitization of A considered in A.4.3, with unit denoted by e. More generally, we deﬁne matrix norms on A1 by letting

[aij + λij e] [aij c + λij c] = sup : c ∈ A, c ≤ 1 (2.16) 1 M (A ) M (A) n

n

for any integer n ≥ 1 and for any matrices [aij ] ∈ Mn (A) and [λij ] ∈ Mn . By 1.2.16 these matrix norms satisfy Ruan’s axioms 1.2.12, and hence deﬁne an operator space structure on A1 . Moreover for any [aij ] ∈ Mn (A) we have [aij ] = lim [aij et ] ≤ sup [aij c] ≤ [aij ]. t

c ≤1

Thus we may regard A as a subalgebra of A1 completely isometrically. Given [aij ], [bij ] ∈ Mn (A) and [λij ], [µij ] ∈ Mn , note that by (2.16), [aij + λij e][bij + µij e] = sup [aij + λij e][bij c + µij c] : c ∈ Ball(A) , M (A) n

Moreover, given any c ∈ A we have [aij + λij e][bij c + µij c] = lim [aij et + λij et ][bij c + µij c]Mn (A) . Mn (A) t

If Mn (A) is a Banach algebra, we deduce that [aij et + λij et ][bij c + µij c] [aij + λij e][bij c + µij c] ≤ sup Mn (A) t ≤ [aij + λij e][bij c + µij c] by (2.16). Taking the supremum over all c ∈ A with c ≤ 1, shows that M n (A1 ) is a Banach algebra. By the ﬁrst part of the proof (the unital case), we see that 2 A1 is an operator algebra, and hence so too is its subalgebra A. 2.3.3 (Applications) We now apply the BRS theorem to show that various natural constructions produce operator algebras. As a ﬁrst example, we note that in analogy to 1.2.16, we may deﬁne an operator algebra seminorm structure on a unital algebra A. Namely, this is an operator seminorm structure in the sense of 1.2.16, such that 11 = 1, and such that the nth seminorm is submultiplicative for every n ∈ N. We let N be the nullspace of ·1 as usual. An argument similar to the one in 1.2.16, but using the BRS theorem instead of Ruan’s theorem, shows that (the completion of) A/N is a unital operator algebra.

66

The abstract characterization of operator algebras

We apply this principle to suprema and limsup’s of operator algebra structures, just as we did in 1.2.16. Suppose that A is a unital algebra, that Λ is a set, and that for each k ∈ Λ we have a unital operator algebra Bk and a unital homomorphism πk : A → Bk . We suppose that for all a ∈ A, the quantity sup{πk (a) : k ∈ Λ} is ﬁnite. We may then deﬁne an operator algebra seminorm structure on A by an = sup (πk )n (a)n k

for a ∈ Mn (A). By the last paragraph, (the completion of) a quotient of A is an operator algebra with these matrix norms. If Λ is a directed set, then replacing the sup by lim sup in the last equation yields another operator algebra. Throughout this book, a closed two-sided ideal of an operator algebra will simply be called an ideal. Long before the ﬁeld of operator spaces arose, it was known that if J is an ideal of an operator algebra A, then the quotient algebra A/J is isometrically isomorphic to an operator algebra (see the Notes section for references). This fact follows as an immediate corollary of the BRS theorem: Proposition 2.3.4 Let J be an ideal in an operator algebra A. Equip the quotient Banach algebra A/J with its quotient operator space structure (see 1.2.14). Then A/J is an operator algebra. That is, there exist a Hilbert space H and a completely isometric homomorphism π : A/J → B(H). Proof Assume that A is a subalgebra of B(K). Then J is also an ideal of the unital operator algebra A1 = Span{A, IK }; and the canonical embedding A/J ⊂ A1 /J is both a complete isometry and a homomorphism. Replacing A with A1 if necessary, we may therefore assume that A is unital. Then A/J is unital, and hence it suﬃces to show that A/J satisﬁes the assertion (ii) of Theorem 2.3.2. This is readily seen from the fact that A satisﬁes (ii), and from the deﬁnition of the quotient operator space structure. 2 2.3.5 (Factor theorem) As in 1.2.15, it follows that if π : A → B is a completely bounded homomorphism, and if J is an ideal in A contained in Ker(π), then π descends to a homomorphism A/J → B with a ‘cb-norm’ which is no larger than that of π. 2.3.6 (Interpolation) We shall apply Proposition 2.3.4 to the complex interpolation of operator algebras. We ﬁrst need a few facts concerning interpolation of Banach algebras. Consider a compatible couple of Banach spaces (A0 , A1 ) and recall from 1.2.30 the deﬁnition of the ‘interpolation space’ Aθ = [A0 , A1 ]θ for θ ∈ (0, 1). If A0 and A1 are Banach algebras, if A0 + A1 is a C-Banach algebra for some C ≥ 1, and if the embeddings A0 → A0 + A1 and A1 → A0 + A1 are both homomorphisms, then we say that (A0 , A1 ) is a compatible couple of Banach algebras. These conditions roughly mean that our two Banach algebras A0 and A1 have a ‘common multiplication’. In this case, this multiplication is both a contractive bilinear map from A0 × A0 into A0 and from A1 × A1 into A1 ; hence by interpolation, the multiplication extends to a contractive bilinear

Basic theory of operator algebras

67

map Aθ × Aθ → Aθ for any θ ∈ (0, 1). Each Aθ is therefore a Banach algebra for this common multiplication. The Banach algebra structure of Aθ has a useful alternative description in terms of quotients. Indeed, consider the Banach space F = F(A0 , A1 ) deﬁned in 1.2.30 (for A0 = X0 and A1 = X1 ), and observe that F is an algebra for the pointwise multiplication. Then the space F θ = {f ∈ F : f (θ) = 0} ⊂ F is an ideal of F and we may therefore consider the quotient algebra F/F θ . Then it is clear that the isometric identiﬁcation Aθ = F/F θ

(2.17)

provided by (1.23) holds at the algebraic level. Indeed if x = f (θ) and y = g(θ) are two arbitrary elements of Aθ , with f, g ∈ F, then xy = f (θ)g(θ) = (f g)(θ). If (A0 , A1 ) is a compatible couple of Banach algebras and if A0 and A1 are operator algebras, then we will say that (A0 , A1 ) is a compatible couple of operator algebras. In that case, Aθ = [A0 , A1 ]θ is both a Banach algebra (by the above discussion), and an operator space (by 1.2.30). Proposition 2.3.7 Let (A0 , A1 ) be a compatible couple of operator algebras. Then Aθ = [A0 , A1 ]θ is an operator algebra for any θ ∈ (0, 1). Proof We ﬁx some θ ∈ (0, 1) and apply 1.2.30 and 2.3.6. The embedding F ⊂ C0 (R; A0 ) ⊕∞ C0 (R; A1 ) is obviously a homomorphism; hence F is an operator algebra by 2.2.4. Thus Aθ is an operator algebra by (2.17) and 2.3.4. 2 2.3.8 (Direct limits) A similar construction to 2.3.3 yields the direct (or inverse) limit of operator algebras. There are several variants on this, for simplicity we sketch only the case of unital operator algebras and unital maps. Let Λ be a directed set, suppose that we have operator algebras An for each n ∈ Λ, and suppose that for each m ≥ n we have completely contractive unital homomorphisms πmn : An → Am with the property that πmn πnk = πmk . We suppose that πnn is the identity map, for all n. We consider the disjoint union of the An ’s, and take equivalence classes under the relation x ≡ y if and only if x ∈ An , y ∈ Am , and there exists a k ∈ Λ with πkn (x) = πkm (y). Of course addition and multiplication is well-deﬁned on equivalence classes. Deﬁne a matrix seminorm structure on the set of equivalence classes by [xij ] = lim supm {[πmn (xij )]}, where n is such that xij ∈ An for all i, j. We quotient by the nullspace of this seminorm, and denote the resulting space by lim An . Appealing to the BRS theorem in a similar → way to 2.3.3, we ﬁnd that this space is again an operator algebra. We write i n for the canonical completely contractive unital homomorphism An → lim An . We → leave as an exercise the fact that the algebra lim An has the appropriate ‘direct →

limit universal property’ (analoguous to that in [368] or Appendix L in [423]).

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2.3.9 (Matrix normed algebras) Let A be a Banach algebra which is also an operator space. We say that A is a matrix normed algebra if [bkl ] [aij bkl ] ≤ [aij ] Mmp (A)

Mn (A)

Mp (A)

for [aij ] ∈ Mn (A) and [bkl ] ∈ Mp (A). In this deﬁnition, the rows of [aij bkl ] are indexed by (i, k), whereas its columns are indexed by (j, l). From 1.5.11 the latter condition is equivalent to saying that the multiplication mapping m : A ⊗ A → A extends to a completely contractive mapping on A ⊗ A. That is, m : A ⊗ A −→ A ≤ 1. cb Since the operator space projective tensor norm dominates the Haagerup tensor norm (1.5.13), any operator algebra is a matrix normed algebra. The converse however is far from being true. Indeed note that if A is a Banach algebra, then Max(A) is a matrix normed algebra, as is clear from the identity 1.5.12 (1). Thus any Banach algebra may be regarded as a matrix normed algebra. However of course most Banach algebras are not isomorphic to an operator algebra (e.g. see 5.1.5 for more on this issue). A good example of a matrix normed algebra is CB(X) for an operator space X. With its canonical matrix norms (1.6) it is fairly obvious that CB(X) is a matrix normed algebra. However we shall see in 5.1.9 that if CB(X) is an operator algebra then X must be a Hilbert space. 2.4 UNIVERSAL CONSTRUCTIONS OF OPERATOR ALGEBRAS Various classical universal constructions in the C ∗ -theory have natural extensions to the nonselfadjoint framework. In this section we will present some of these, and show as an application of the BRS theorem that they may be characterized ‘internally’ in terms of certain ‘factorization formulae’. For example, following [67], we will consider the universal operator algebra of a semigroup, and the free product of operator algebras. Some other universal constructions will appear later on in the book (for example, the maximal tensor product of operator algebras will be discussed in Chapter 6). 2.4.1 (Ordering and isomorphism of C ∗ -covers) Let A be an operator algebra. If (B, j) and (B , j ) are C ∗ -covers of A (see 2.1.1), we then declare (B, j) ≤ (B , j ) if and only if there is a ∗-homomorphism π : B → B such that π ◦ j = j. By A.5.8, such a ∗-homomorphism π is automatically surjective and unique (this is because the ∗-algebra generated by A is dense in both B and B , and π is uniquely determined there, and is surjective). We say that (B, j) is A-isomorphic to (B , j ) if a π exists as above which is also one-to-one (and therefore a ∗-isomorphism by A.5.8). This is an equivalence relation, and we deﬁne C(A) to be the set of equivalence classes of C ∗ -covers of A, with the ordering above. In this ordering, we will see next that there is a largest element of C(A). This C ∗ -cover we think

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of as the noncommutative maximal ideal space of A. It is much more diﬃcult (see Section 4.3) to see that there is a smallest element of C(A) in this ordering, known as the C ∗ -envelope of A. Proposition 2.4.2 Let A be an operator algebra. Then there exists a C ∗ -cover ∗ (A), j) of A with the following universal property: if π : A → D is any (Cmax completely contractive homomorphism into a C ∗ -algebra D, then there exists a ∗ (A) → D such that π ˜ ◦ j = π. (necessarily unique) ∗-homomorphism π ˜ : Cmax Proof It suﬃces to assume that D = B(H), for a Hilbert space H. Suppose that the cardinality of A is less than or equal to a cardinal β that we choose such that β ℵ0 = β. Deﬁne F to be the set of completely contractive representations π : A → B(2J ) where J varies over the cardinals which are less than or equal to β. We write Hπ = 2J . Deﬁne j = ⊕ {π : π ∈ F}, that is, j(a) = ⊕π∈F π(a) for all a ∈ A. This is a completely contractive representation of A on a Hilbert space Hmax = ⊕π∈F Hπ . In fact j is also completely isometric, as may be seen by standard arguments. The projection of Hmax onto its ‘πth coordinate’ will be ∗ (A), or C ∗ (A) for brevity, to be the C ∗ -algebra written as Pπ . We deﬁne Cmax inside B(Hmax ) generated by j(A). Note that if θ : A → B(H) is any completely contractive representation of A, with dim(H) ≤ β, then there is a unitary U on H such that ρ = U ∗ θ(·)U ∈ F. Deﬁne ρ˜ : C ∗ (A) → B(Hρ ) to be ρ˜(T ) = Pρ T|Hρ . Then ρ˜ is a ∗-homomorphism deﬁned on C ∗ (A), and ρ˜ ◦ j = ρ. Then θ˜ = U ρ˜(·)U ∗ is a ∗-homomorphism C ∗ (A) → B(H), and θ˜ ◦ j = θ. Clearly θ˜ is the unique such ∗-homomorphism. Thus we have shown that C ∗ (A) has the desired universal property, but for extending completely contractive representations of A on Hilbert spaces of dimension ≤ β. From this fact it is not hard to show that C ∗ (A) has this universal property when H has arbitrary dimension. We brieﬂy sketch the argument. For π : A → B(H) as above, we let J0 be a set whose cardinality is equal to dim(H). Then we consider the set G of pairs (J, {Kj }j∈J ), where J is a subset of J0 , the Kj ’s are mutually orthogonal nonzero subspaces of H with dim(Kj ) ≤ β for any j ∈ J, and each Kj is reducing for π(A). Recall that this means that Kj is invariant for both π(A) and π(A)∗ . The set G has a natural order, namely (J, {Kj }j∈J ) ≤ (J , {Kj }j∈J ) if J ⊂ J and Kj = Kj for any j ∈ J. Applying a routine Zorn’s lemma argument, we see that G has a maximal element (J, {Kj }j∈J ). Since we supposed that Card(A) ≤ β and β ℵ0 = β, it is not hard to check that ⊕j Kj = H. Indeed if x ∈ H is orthogonal to each Kj , and if C ⊂ B(H) is the C ∗ -algebra generated by π(A), then [Cx] is reducing for π(A) and has dimension less than or equal to β. This is a contradiction. We leave the details to the reader, Thus π may be written as a direct sum of a family of completely contractive representations on Hilbert spaces of dimension ≤ β. Now it is clear that π has the desired extension π ˜. 2 ∗ (A) in 2.4.2 is called the 2.4.3 (The maximal C ∗ -algebra) The algebra Cmax ∗ maximal or universal C -algebra of A. Algebraically, the universal property in

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∗ 2.4.2 is saying that the functor A → Cmax (A) is the left adjoint to the forgetful ∗ functor from the category of C -algebras to the category of operator algebras. ∗ (A), j) is unital and it is easy to check that it is If A is unital, then (Cmax also characterized by the following universal property: if π : A → D is any unital completely contractive homomorphism into a unital C ∗ -algebra D, then there ∗ exists a (necessarily unique and unital) ∗-representation π ˆ : Cmax (A) → D such that π ˆ ◦ j = π. ∗ (A1 ) is the maximal It is also worth noticing that if A is not unital and if Cmax ∗ ∗ C -algebra of its unitization, then Cmax (A) coincides with the C ∗ -subalgebra of ∗ Cmax (A1 ) generated by A. Indeed let C be the latter C ∗ -algebra and let j be ∗ (A1 ). We consider a completely contractive the canonical embedding A1 → Cmax representation π : A → B(H). By Meyer’s theorem 2.1.13 there is a completely contractive unital homomorphism π 0 : A1 → B(H) extending π. By the universal ∗ ∗ (A1 ), there is a ∗-homomorphism ρ : Cmax (A1 ) → B(H) such property of Cmax 0 that ρ ◦ j = π . The restriction of ρ to C is the desired extension of π.

2.4.4 (The universal representation) We deﬁne the universal representation πu of a possibly nonselfadjoint operator algebra A to be the restriction of the ∗ (A) to A. universal representation (see A.5.5) of Cmax 2.4.5 (An example) Consider A = T2 , the upper triangular 2 × 2 matrices. According to Corollary 2.2.12, A = U(C) is completely isometrically isomorphic to the subalgebra of M2 (C([0, 1])) deﬁned as % $ √ λ1 µ · : λ1 , λ2 , µ ∈ C ⊂ M2 (C[0, 1]), 0 λ2 √ where · is the square root function on [0, 1]. We henceforth identify A with this ∗ (A) = {f ∈ M2 (C([0, 1])) : f (0) is a diagonal matrix}. subalgebra. Claim: Cmax We will prove this claim by showing that the last C ∗ -algebra has the property of Proposition 2.4.2. First we observe that the ∗-algebra E generated by A is dense in the space of matrices of the form √ b√ 1 b2 · b3 · b4 with bi ∈ C([0, 1]), and the latter space is dense in the algebra of functions f ∈ M2 (C([0, 1])) such that f (0) is a diagonal matrix. Then we consider a nondegenerate (completely) contractive representation π : A → B(L) on some Hilbert space L. This immediately gives a decomposition of L as a Hilbert space sum H ⊕ K say, and operator T : K → H, namely T = π(E12 ), so a contractive √ λ1 µ · λ1 IH µT that π maps any in A to . It is easily checked that 0 λ2 0 λ2 IK √ b1 (T T ∗ ) b2 (T T ∗ )T b√ 1 b2 · , bi ∈ C([0, 1]), −→ θ : b3 · b4 T ∗ b3 (T T ∗ ) b4 (T ∗ T )

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is a ∗-homomorphism from E into B(H ⊕ K) = B(L). Also, θ is bounded; for √ 1 example, b2 (T T ∗ )T = b2 (T T ∗ )(T T ∗ ) 2 ≤ b2 ·C([0,1]) by spectral theory. Hence θ extends to a ∗-homomorphism on the containing C ∗ -algebra. 2.4.6 (The enveloping operator algebra of a Banach algebra) This is an analogue of the well-known enveloping C ∗ -algebra of an involutive Banach algebra. We shall only consider unital algebras for the sake of simplicity. Let B be a unital Banach algebra which is also an operator space. We let I be the collection of all unital completely contractive homomorphisms π : B → B(H), H = Hπ being a Hilbert space. For any integer n ≥ 1 and any matrix b ∈ Mn (B), we deﬁne

|||b|||n = sup πn (b)Mn (B(H)) : π ∈ I . It is easy to see that each ||| · |||n is a seminorm on Mn (B), and that they deﬁne an ‘operator algebra seminorm structure’ on B in the sense of 2.3.3. We let O(B) denote the resulting operator algebra obtained by the procedure of 2.3.3 (namely, the completion of the quotient of B by the nullspace of |||·|||1 ). We will call O(B) the enveloping operator algebra of B. 2.4.7 (Universal property of O(B)) For B as above, there is a canonical completely contractive and unital homomorphism i : B → O(B), whose range is dense, and (O(B), i) has the following property: for any unital operator algebra A and for any unital completely contractive homomorphism π : B → A, there exists a (necessarily unique) unital completely contractive homomorphism π ˜ : O(B) → A such that π ˜ ◦i = π. It is easy to see that this property characterizes O(B) uniquely, up to appropriate isomorphism. Next we use the BRS theorem to give an alternative description of the enveloping operator algebra O(B) in terms of factorization matrix norms. We will then see some surprising applications of this. Proposition 2.4.8 Let B be a unital Banach algebra which is also an operator space. For n ∈ N and b ∈ Mn (B), we have (with the notation from 2.4.6) (2.18) |||b|||n = inf b1 b2 · · · bN , the inﬁmum over all integers N ≥ 1 and all possible factorizations of b in the form b = b1 b2 · · · bN , with b1 ∈ Mn,k1 (B), b2 ∈ Mk1 ,k2 (B), . . . , bN ∈ MkN −1 ,n (B) for some positive integers k1 , k2 , . . . , kN −1 . Proof For any N ≥ 1, let ZN = B ⊗ · · · ⊗ B be the N -fold tensor product of N copies of B. Let mN : ZN → B be the N -fold multiplication deﬁned by letting mN (b1 ⊗ · · · ⊗ bN ) = b1 · · · bN for any b1 , . . . , bN ∈ B, and then extending by linearity. We write m1 for the identity map on B. Then it follows from (1.40) that the right-hand side of (2.18) is equal to

|b|n = inf inf zh : z ∈ Mn (ZN ), IMn ⊗ mN (z) = b . N ≥1

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Fix b, b ∈ Mn (B), for some n ≥ 1, and let ε > 0 be an arbitrary positive number. For some integers N, N ≥ 1, there exist elements z in Mn (ZN ) and z in Mn (ZN ) such that IMn ⊗ mN (z) = b, IMn ⊗ mN (z ) = b , zh < ε+ |b|n , z z as an element of Mn (ZN +N ), we have and z h < ε + |b |n . Regarding IMn ⊗ mN +N (z z ) = bb . Thus |bb |n ≤ z z h ≤ zhz h ≤ ε + |b|n ε + |b |n . Since ε > 0 was arbitrary, we have proved that |bb |n ≤ |b|n |b |n ,

b, b ∈ Mn (B), n ≥ 1.

(2.19)

By adding on identity matrices decomposition of z (or z ), we can in the tensor assume that N = N . Then IMn ⊗ mN (z + z ) = b + b , and so |b + b |n ≤ z + z h ≤ zh + z h ≤ 2ε + |b|n + |b |n . Since ε is arbitrary, | · |n is subadditive. Similar arguments show that the quantities | · |n deﬁne an operator algebra seminorm structure in the sense of 2.3.3, simply because the Haagerup matrix norms on each ZN satisfy Ruan’s axioms (see 1.2.12), and because of (2.19). Thus, by 2.3.3, the completion of the quotient of B by the nullspace of the seminorm, is an operator algebra A. We shall now prove (2.18). First note that the canonical map i : B → A is a unital completely contractive homomorphism. Thus the universal property of O(B) ensures that ||| · |||n dominates | · |n , for each n ≥ 1. On the other hand, suppose that b ∈ Mn (B), and let π : B → B(H) be a unital completely contractive homomorphism. If b = IMn ⊗ mN (z) for some z ∈ Mn (ZN ), then b ∈ Mn (B) and πn (b) = IMn ⊗πmN (z). Since πmN : B ⊗· · ·⊗B → B(H) is the linear mapping which takes b1 ⊗· · ·⊗bN to π(b1 ) · · · π(bN ) for any b1 , . . . , bN ∈ B, by the ﬁrst paragraph in 1.5.8, we have that πmN is completely contractive on B ⊗h · · · ⊗h B. Hence πn (b) ≤ zh. It therefore follows from the deﬁnitions 2 of ||| · |||n and | · |n , that we have |||b|||n ≤ |b|n . 2.4.9 (The semigroup operator algebra) Let G be a discrete semigroup. The space 1G of summable families indexed by G, is a unital Banach algebra with the convolution product (e.g. see [106]). We deﬁne the semigroup operator algebra O(G) of G to be the enveloping operator algebra O(Max( 1G )). There is an obvious bijective correspondence between the contractive representations σ : G → B(H), the unital contractive homomorphisms π : 1G → B(H), and the unital completely contractive homomorphisms π : Max(1G ) → B(H). According to these correspondences, the matrix norms on O(G) may be described as follows. Let (eg )g∈G denote the canonical basis of 1G , and let C[G] = Span{eg : g ∈ G} be the semigroup algebra of G. Then for b = g λg ⊗ eg ∈ Mn ⊗ C[G], with λg ∈ Mn , we have

Basic theory of operator algebras |||b|||n = sup λg ⊗ σ(g) g

73

Mn (B(H))

where the supremum runs over all contractive representations σ : G → B(H). According to 2.4.7, the operator algebra O(G) is characterized by the following universal property. There exists a contractive representation i : G → O(G) whose range generates O(G) as an operator algebra, such that: for any contractive representation σ : G → A into a unital operator algebra A, there exists a (necessarily unique) unital completely contractive homomorphism σ ˜ : O(B) → A such that σ ˜ ◦ i = σ. 2.4.10 If G is a group, then 1G has a natural involution b → b∗ . Regarding G ⊂ 1G via the map g → eg above, then this involution on 1G is simply the unique ∗-linear extension to 1G of the map g → g −1 on G. Suppose that π is a contractive group homomorphism G → B(H). Since π(g)π(g −1 ) = π(g −1 )π(g) = IH for any g ∈ G, it follows from the last assertion in 2.1.3 that π(g) is unitary, with π(g −1 ) = π(g)∗ . By the basic theory of representations, π extends to a contractive ∗-homomorphism 1G → B(H), and extends further to a ∗-homomorphism from the full group C ∗ -algebra of G, into B(H). By 1.2.4 this homomorphism is completely contractive. By the universal property in 2.4.9, it follows that O(G) coincides with the full group C ∗ -algebra of G. The next proposition is a strengthening of Proposition 2.4.8, in the case that B = Max(1G ) for a semigroup G. Since its proof is similar to that of Proposition 2.4.8, we omit it. Proposition 2.4.11 Let G be a discrete semigroup and let b ∈ Mn ⊗ C[G] for some n ≥ 1. Then |||b|||n = inf α0 α1 · · · αN , where the inﬁmum runs over all possible product factorizations of b of the form b = α0 b1 α1 b2 · · · bN αN , where α0 ∈ Mn,k1 , α1 ∈ Mk1 ,k2 , . . . , αN ∈ MkN ,n are scalar matrices for some k1 , . . . , kN ∈ N, and b1 ∈ Mk1 (B), . . . , bN ∈ MkN (B) are diagonal matrices with entries in the set {eg : g ∈ G}. The latter factorization result is especially interesting in the case when G is either N0 (the additive semigroup of nonnegative integers), or N20 . Before coming to this, we need to review two important results from operator theory. 2.4.12 (Nagy’s dilation theorem) Let A(D) be the disc algebra from 2.2.6, and let P ⊂ A(D) denote the dense subalgebra of polynomials. Assume that T is a contraction on some Hilbert space H. Nagy’s dilation theorem [404, 405] asserts that there exists a Hilbert space K, an isometry J : H → K, and a unitary operator U ∈ B(K), such that T k = J ∗ U k J for any k ∈ N. Let π : C(T) → B(K) be the ∗-representation obtained by applying the spectral mapping theorem to U . That is, π(f ) = f (U ) for any f ∈ C(T). Then f (T ) = J ∗ π(f )J for any polynomial f . This shows that the mapping f ∈ P → f (T ) extends to a unital completely contractive homomorphism uT : A(D) → B(H).

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Any matrix [fij ] ∈ Mn (P) can be viewed as a matrix valued function, namely as the function f taking z to [fij (z)]. Moreover it follows from 2.2.5 that its norm as an element of Mn (A(D)) is equal to sup f (z)Mn : z ∈ D . Any such function f will be called a ‘matrix valued polynomial’ (in one variable). If f = k≥0 f (k) z k is the (ﬁnite) expansion of f , then uT (f ) = k≥0 f (k) ⊗ T k . Thus we have

≤ f Mn (A(D)) = sup : z∈D . f (k) ⊗ T k f (k) z k k≥0

Mn (B(H))

Mn

k≥0

This is a matricial version of von Neumann’s inequality (which is the case n = 1). 2.4.13 (Ando’s dilation theorem) Let A(D2 ) be the bidisc algebra, which may be deﬁned to be the closure of the set P 2 = P ⊗P of polynomials in two variables, in C(T2 ). Equivalently (see 2.2.7), we have A(D2 ) = A(D) ⊗min A(D) = A(D; A(D)). Ando’s dilation theorem (see [9, 405]) extends Nagy’s theorem as follows. If T, S are two commuting contractions on a Hilbert space H, then there exist a Hilbert space K, an isometry J : H → K, and two commuting unitary operators U, V on K, such that T k S l = J ∗ U k V l J for any pair of nonnegative integers k and l. By a similar argument to that of 2.4.12, we deduce that the mapping f ∈ P 2 → f (T, S) extends to a unital completely contractive homomorphism uT,S : A(D2 ) → B(H). In particular, this yields a version of von Neumann’s inequality for pairs of commuting contractions. 2.4.14 (The semigroup operator algebras of N0 and N20 ) If σ : N0 → B(H) is a contractive representation of G = N0 on a Hilbert space H, then T = σ(1) is a contraction. Conversely, any contraction T ∈ B(H) gives rise to a contractive representation σ : N0 → B(H), deﬁned by letting σ(k) = T k for any k ≥ 0. If we identify the algebra C[N0 ] with P in the usual way (ek = z k for any k ≥ 0), we deduce from this correspondence, 2.4.12, and the the last centered formula in 2.4.9 that O(N0 ) coincides with A(D) completely isometrically. Similarly, since contractive representations of N20 correspond to pairs of commuting contractions, it follows from 2.4.13 that O(N 20 ) = A(D2 ). In the following result, matrices are assumed to have (ﬁnite) sizes for which the matrix products make sense. Theorem 2.4.15 (1) For n ∈ N let f ∈ Mn (P) be a matrix valued polynomial. Then f (z)Mn < 1 for all z ∈ D if and only if there exists a factorization f (z) = α0 b1 (z)α1 b2 (z) · · · bN (z)αN ,

z ∈ C,

where α0 , α1 , . . . , αN are scalar matrices of norm < 1 and b1 , . . . , bN are diagonal matrices whose entries are of the form z k for (varying) k ∈ N.

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(2) Let n ∈ N and f ∈ Mn (P 2 ) a matrix valued polynomial in two variables z and w. Then f (z, w)Mn < 1 for every (z, w) ∈ D2 , if and only if there exist matrix valued polynomials of one variable f1 , . . . , fN , g1 , . . . ∈ gN , with f (z, w) = f1 (z)g1 (w)f2 (z) · · · gN −1 (w)fN (z)gN (w),

(z, w) ∈ C2 ,

with fi (z) < 1 and gi (w) < 1 for any 1 ≤ i ≤ N , and any w, z ∈ D. Proof Taking 2.4.14 into account, the ﬁrst assertion is a straightforward application of Proposition 2.4.11. For the second assertion, we only need to prove the ‘only if’ part. We let f ∈ Mn (P 2 ) be a matrix valued polynomial such that f (z, w)Mn < 1 for any (z, w) ∈ D2 . Applying Proposition 2.4.11 again, and the second paragraph of 2.4.14, we obtain a factorization of f of the form f (z, w) = α0 b1 (z, w)α1 b2 (z, w) · · · bN (z, w)αN ,

z, w ∈ C,

where α0 , α1 , . . . , αN are scalar matrices of norm < 1 and each bi is a diagonal matrix of the form bi (z, w) = Diag z n1 wm1 , z n2 wm2 , . . . , z nk wmk for some nonnegative integers n1 , m1 , n2 , . . . , mk . We write bi (z, w) = bi (z)bi (w), with bi (z) = Diag z n1 , z n2 , . . . , z nk and bi (w) = Diag wm1 , wm2 , . . . , wmk . Then f has the desired decomposition, with f1 = α0 b1 , fi = bi for any 2 ≤ i ≤ N , and gi = bi αi for any 1 ≤ i ≤ N . 2 2.4.16 (Free product of operator algebras) We now turn to another universal construction, the unital free product. We refer, for example, to [26, 368, 418] for some background on free products, in particular free products of unital algebras and free products of unital C ∗ -algebras. Let A and B be unital operator algebras, and let F(A, B) denote their unital algebra free product. This unital algebra is characterized by the following universal property. There exist unital one-to-one homomorphisms jA : A → F(A, B) and jB : B → F(A, B) such that jA (A) and jB (B) generate F(A, B) as an algebra, and such that: for any pair (π, ρ) of unital homomorphisms π : A → C and ρ : B → C into a unital algebra C, there exists a (necessarily unique) homomorphism π ∗ ρ : F(A, B) → C such that (π ∗ ρ) ◦ jA = π and (π ∗ ρ) ◦ jB = ρ. Let I be the collection of all pairs i = (π, ρ) of unital completely contractive homomorphisms π : A → B(H) and ρ : B → B(H), H = Hi being a Hilbert space. Then for any integer n ≥ 1 and any matrix y ∈ Mn (F(A, B)), we deﬁne

|||y|||n = sup IMn ⊗ (π ∗ ρ) (y)Mn (B(Hi )) : i = (π, ρ) ∈ I . It is easy to see that each ||| · |||n is a seminorm on Mn (F(A, B)). It turns out that ||| · |||n is actually a norm. This follows from [26] in the case when A and B are C ∗ -algebras. In the general case, let C and D be unital C ∗ -algebras such that A ⊂ C and B ⊂ D are unital-subalgebras and observe that by [26, Proposition

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2.1], the canonical embedding F(A, B) ⊂ F(C, D) is one-to-one. For any pair of ∗-representations π: C → and ρ : D → B(H) and any y ∈ Mn (F(A, B)), B(H) we have IMn ⊗ (π ∗ ρ) (y)Mn (B(H)) ≤ |||y|||n . Taking the supremum over all such possible pairs (π, ρ), we deduce from the above mentioned result in the C ∗ -algebra case, that y = 0 if |||y|||n = 0. We let A ∗ B be the completion of F(A, B) in the norm ||| · |||1 . Arguing as in 2.4.6, we see that A ∗ B is a unital operator algebra for the matrix norms (induced by) ||| · |||n . We call A ∗ B the unital free product of A and B. If A and B are unital C ∗ -algebras then A ∗ B coincides with the usual full amalgamated free product C ∗ -algebra, as may be seen using the above and A.5.8. 2.4.17 If A and B are two unital operator algebras, then there exists a pair of unital completely isometric homomorphisms π : A → B(H) and ρ : B → B(H), for a common Hilbert space H. Indeed if A and B are represented as unitalsubalgebras of B(H1 ) and B(H2 ) respectively, then take H = H1 ⊗2 H2 , and deﬁne π and ρ by letting π(a) = a⊗IH2 , and ρ(b) = IH1 ⊗b, for any a ∈ A, b ∈ B. It now follows from the centered equation in 2.4.16, that the embeddings jA and jB considered in 2.4.16 are unital completely isometric homomorphisms jA : A → A ∗ B

and

jB : B → A ∗ B.

From now on (except in the next item), we will suppress mention of jA and jB , and simply regard A and B as unital-subalgebras of A ∗ B. 2.4.18 (Universal property of the free product) It is clear that the unital free product operator algebra A ∗ B of A and B is characterized by the following property. There exist unital completely isometric homomorphisms jA : A → A∗B and jB : B → A∗B, whose ranges together generate A∗B as an operator algebra, such that: for any Hilbert space H and any pair (π, ρ) of unital completely contractive homomorphisms π : A → B(H) and ρ : B → B(H), there exists a (necessarily unique) unital completely contractive homomorphism π ∗ ρ from A ∗ B to B(H) such that (π ∗ ρ) ◦ jA = π and (π ∗ ρ) ◦ jB = ρ. One may use the BRS theorem to give the following description of the matrix norms on unital free products. The proof is similar to that of Proposition 2.4.8 (and Proposition 2.4.11), and is therefore omitted. Proposition 2.4.19 Let A and B be unital operator algebras, let F(A, B) be their unital algebra free product, and let y ∈ Mn (F (A, B)) for some n ≥ 1. Then zMn (A∗B) = inf a1 b1 a2 · · · aN bN , where the inﬁmum runs over all N ∈ N and all possible factorizations of y of the form z = a1 b1 a2 · · · aN bN , with a1 ∈ Mn,k1 (A), b1 ∈ Mk1 ,l1 (B), a2 ∈ Ml1 ,k2 (A), . . . , aN ∈ MlN −1 ,kN (A), bN ∈ MkN ,n (B) , where k1 , l1 , k2 , . . . , lN −1 , kN ∈ N. Our next objective is Theorem 2.4.21, for which we will need the following: Proposition 2.4.20 Let A be a unital operator algebra.

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(1) If u : A → B(H) is completely contractive then there exist a Hilbert space K, contractions V : H → K, W : K → H, and a unital completely contractive homomorphism π : A → B(K), such that u(a) = W π(a)V for a ∈ A. (2) Let B be a second unital operator algebra, and let u : A ⊗ h B → B(H) be a completely contractive map. Then there exist a Hilbert space K, two linear contractions V : H → K, W : K → H, and two unital completely contractive homomorphisms π : A → B(K) and ρ : B → B(K), such that we have u(a ⊗ b) = W π(a)ρ(b)V for any a ∈ A and b ∈ B. Proof Part (1) is an obvious consequence of 1.2.8, so that we only need to prove (2). If u : A ⊗h B → B(H) is completely contractive, then Theorem 1.5.7 (2) ensures that we can ﬁnd Hilbert spaces K1 , K2 , and contractive linear operators V : H → K2 , R : K2 → K1 and W : K1 → H, as well as unital completely contractive homomorphism π : A → B(K1 ) and ρ : B → B(K2 ), such that u(a ⊗ b) = W π(a)Rρ(b)V,

a ∈ A, b ∈ B.

(2.20)

We shall now modify this factorization into the desired form. Note that in (2.20), without loss of generality we may replace K 1 and K2 with 2I (K1 ) and 2I (K2 ), for any set I. Indeed it suﬃces to replace W ∈ B(K1 , H) with the row matrix [W 0 0 · · · ] ∈ B(2I (K1 ), H), replace V ∈ B(H, K2 ) with the column [V 0 0 · · · ]t ∈ B(H, 2I (K2 )), and to replace π(a), R, and ρ(b), with their corresponding multiple ρ(a) ⊗ I2I , etc. Choosing a suﬃciently large I, we may assume that K1 and K2 have the same dimension. Thus there exists a unitary U : K2 → K1 . Using this unitary we can write u(a⊗b) = W U U ∗ π(a)U U ∗ Rρ(b)V for a ∈ A and b ∈ B. Replacing W by W U , R by U ∗ R, and π by U ∗ π( )U , we therefore obtain that (2.20) holds true with K = K1 = K2 . Since R ∈ B(K) is a contraction, the operators I − RR ∗ and I − R∗ R are nonnegative. We may deﬁne a unitary Γ ∈ B(K ⊕ K) by the formula R (I − RR∗ )1/2 Γ= . −(I − R∗ R)1/2 R∗ Moreover for a ∈ A and b ∈ B, we can write π(a) 0 ρ(b) 0 V u(a ⊗ b) = W 0 Γ 0 π(a) 0 ρ(b) 0 π(a) 0 ρ(b) 0 V Γ . = W 0 Γ Γ∗ 0 π(a) 0 ρ(b) 0 V Replacing K by K ⊕ K, V by , ρ by ρ ⊕ ρ, π by Γ∗ π ⊕ π Γ, and W by 0 [ W 0 ]Γ, we see that (2.20) may be achieved with R = IK . 2 Besides A and B, the simplest subspace of A ∗ B is the linear span of all the products ab, for a ∈ A and b ∈ B. More speciﬁcally, note that the map (a, b) → ab from A × B to A ∗ B is bilinear, and therefore yields a canonical linearization E : A ⊗ B → A ∗ B. The range of this map has an attractive description:

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The second dual algebra

Theorem 2.4.21 Let A and B be two unital operator algebras. Then the canonical map E : A⊗B → A∗B above extends to a complete isometry A⊗ h B → A∗B. Proof Fix y = k ak ⊗ bk ∈ A ⊗ B, with ak ∈ A and bk ∈ B. Then a b ∈ A ∗ B. Let θ : A ∗ B → B(H) be a completely isometric E(y) = k k k homomorphism for some suitable H, and let π = θ|A and ρ = θ|B be the corresponding (completely isometric) homomorphisms on A and B. Then E(y)A∗B = θ(E(y)) = π(ak )ρ(bk ) ≤ yA⊗h B k

by the fact at the end of the ﬁrst paragraph in 1.5.4. Conversely, let u : A ⊗h B → B(H) be a completely isometric mapping for some suitable H. According to the second part of Proposition 2.4.20, we have a factorization of the form u(a ⊗ b) = W π(a)ρ(b)V for some contractions V ∈ B(H, K) and W ∈ B(K, H), and for some unital completely contractive homomorphisms π : A → B(K) and ρ : B → B(K). Then W π(ak )ρ(bk )V yA⊗h B = u(z) = k ≤ π(ak )ρ(bk ) ≤ E(y)A∗B k

by the universal property 2.4.18. This shows that A ⊗h B ⊂ A ∗ B isometrically. 2 A similar argument shows that A ⊗h B ⊂ A ∗ B completely isometrically. 2.5 THE SECOND DUAL ALGEBRA Passing to the second dual is a commonly used trick in C ∗ -algebra theory, as it allows one to work with von Neumann algebras and their various weak topologies. For similar purposes, it is important to understand the properties of the second dual A∗∗ of a nonselfadjoint operator algebra A, and thus we devote a section to this topic. We will not take for granted any facts about the second dual of a C ∗ -algebra not proved in this book, and will prove such facts as we go along. We begin with some background and notation on second duals of Banach algebras. 2.5.1 (Arens product) Let A be a Banach algebra. Then we may equip its second dual A∗∗ with two natural products as follows. Consider a ∈ A, ϕ ∈ A∗ , and η ∈ A∗∗ . We let aϕ and ϕa be the elements of A∗ deﬁned by aϕ, b = ϕ, ba

and

ϕa, b = ϕ, ab

for any b ∈ A. Then we let ηϕ and ϕη be the elements of A∗ deﬁned by ηϕ, b = η, ϕb

and

ϕη, b = η, bϕ

for any b ∈ A. By deﬁnition, the left and right Arens products ·λ and ·µ on A∗∗ are given by the following formulae, for η, ν ∈ A∗∗ and ϕ ∈ A∗ :

Basic theory of operator algebras η·λ ν, ϕ = η, νϕ

79 η·µ ν, ϕ = ν, ϕη.

and

These two products are called the Arens products. They have useful equivalent formulations in terms of double limits. Namely, let η, ν ∈ A∗∗ , and according to Goldstine’s lemma (see A.2.1), let (aα )α and (bβ )β be two nets in A converging to η and ν in the w∗ -topology of A∗∗ . From the last centered formula we have η·λ ν, ϕ = η, νϕ = limνϕ, aα = limν, ϕaα = lim lim ϕ, aα bβ . α

α

α

β

A similar calculation works for the other product, and thus: η·λ ν, ϕ = lim lim ϕ, aα bβ α

∗

β

and

η·µ ν, ϕ = lim lim ϕ, aα bβ (2.21) β

α

∗∗

for any ϕ ∈ A . It is easy to check that A is a Banach algebra with either product ·λ or ·µ , and that these products both extend the product of A. 2.5.2 (Arens regularity) A Banach algebra A is called Arens regular if the left and right Arens products coincide on A∗∗ . In that case, we will speak of the product ην of two elements of A∗∗ without any ambiguity, and we drop the notation ·λ or ·µ . It is easy to see using (2.21) say, that a subalgebra A of an Arens regular Banach algebra C, is again Arens regular, and that in this case A∗∗ becomes a subalgebra of C ∗∗ . 2.5.3 (A characterization of the Arens product) If a Banach algebra A is Arens regular, then the Arens product on A∗∗ is separately w∗ -continuous. To see this, ﬁx ϕ ∈ A∗ and η ∈ A∗∗ , and consider the functional ν → ην, ϕ on A∗∗ . By the formulae above, this functional coincides with ϕη ∈ A∗ . Hence it is w∗ -continuous. Similarly for any ν ∈ A∗∗ and any ϕ ∈ A∗ , the functional η → ην, ϕ coincides with νϕ ∈ A∗ , and hence is w∗ -continuous. Conversely, if A∗∗ is equipped with a product which extends that of A and is separately w∗ -continuous, then A is Arens regular and the product on A∗∗ is indeed the Arens product. This clearly follows from (2.21). From this last fact and A.5.7, it is clear that any C ∗ -algebra A is Arens regular and that the product on the W ∗ -algebra A∗∗ is the Arens product. Corollary 2.5.4 Any operator algebra is Arens regular. Proof We just saw that any C ∗ -algebra is Arens regular. The result then follows by the remarks at the end of 2.5.2. 2 2.5.5 (W ∗ -continuous extension of homomorphisms) Let A be an operator algebra (or more generally an Arens regular matrix normed algebra), and let π : A → B(H) be a completely contractive homomorphism. According to 1.4.8, we let π ˜ : A∗∗ → B(H) be the unique w∗ -continuous mapping extending π, and we recall that π ˜ also is completely contractive. We observe that moreover, π & is a homomorphism when A∗∗ is equipped with its Arens product. Indeed, this follows immediately from the separate w ∗ -continuity of the products on B(H) and A∗∗ (see A.1.2 and 2.5.3). If π above maps into a w ∗ -closed subalgebra M of B(H) then it is evident that π ˜ maps into M too (by w ∗ -continuity).

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The second dual algebra

Corollary 2.5.6 If A is an operator algebra, then A∗∗ is also an operator algebra. Indeed, there exist a Hilbert space H, and a completely isometric homomor˜ : A∗∗ → B(H) phism π : A → B(H) whose (unique) w ∗ -continuous extension π w∗ is a completely isometric homomorphism. In this case, A∗∗ ∼ &(A∗∗ ) = π(A) . =π Proof Let C be a C ∗ -algebra containing A as a subalgebra. Dualizing the embedding map A → C twice, we obtain by 1.4.3 a completely isometric embedding A∗∗ → C ∗∗ . By the remark at the end of 2.5.2, A∗∗ is a subalgebra of C ∗∗ . Since the second dual operator space structure on C ∗∗ coincides with the one inherited from its C ∗ -structure (see 1.4.10), we see that A∗∗ is an operator algebra. Consider the following commutative diagram of operator algebra embeddings: A ⊂ C ∩ ∩ A∗∗ ⊂ C ∗∗ Let πu : C → B(H) be the universal representation of C. According to 1.4.10, u : C ∗∗ → B(H) is a πu satisﬁes the conclusion of our proposition. That is, π ˜ is completely isometric homomorphism. If π is the restriction of πu to A, then π ˜ is a complete isometry. Then it follows by the restriction of π u to A∗∗ , and so π the Krein–Smulian theorem, as in the last few lines of the proof of A.5.6, that π(A)

w∗

= π &(A∗∗ ).

2

2.5.7 It is clear from the above proof and 1.4.10 that for an operator algebra A, the completely isometric identiﬁcation Mn (A)∗∗ ∼ = Mn (A∗∗ ) from Theorem 1.4.11, is also an isomorphism of algebras, for any n ≥ 1. We now turn to approximately unital algebras and mention a well-known result concerning Arens regular Banach algebras. Proposition 2.5.8 Let A be a Banach algebra and assume that A is Arens regular. Then A has a cai (et )t if and only if A∗∗ has an identity e of norm 1. Similarly, A has a right cai (et )t if and only if A∗∗ has a right identity e of norm 1. If A is an operator algebra, and (et )t is a cai or right cai for A, and if e is as above, then et → e in the w∗ -topology of A∗∗ . If A has both a left and a right cai, then A has a (two-sided) cai. Proof Assume that A admits a cai (et )t , and let e ∈ A∗∗ be a cluster point of this net for the w∗ -topology. Then for any a ∈ A, ae is a cluster point of aet for the w∗ -topology. We are using the separate w ∗ -continuity of the product on A∗∗ (see 2.5.3). Thus ae = a. Thus, by A.2.1 and separate w ∗ -continuity again, ηe = η for any η ∈ A∗∗ . Similarly eη = η, and so e is an identity (of norm 1). Analoguous arguments give the ‘only if’ direction in the ‘right cai case’. Next we note that a right identity of norm 1 for an operator algebra is unique if it exists. To see this simply note that such a right identity is by A.1.1

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a (selfadjoint) projection. If p, q are projections with pq = p and qp = q, then by taking adjoints we see that p = q. If e is an identity or right identity of norm one for A∗∗ and if (et )t is as above, then the previous argument shows that e is the unique w ∗ -cluster point of (et )t . Thus et → e in the w∗ -topology. The last assertion of the proposition follows from the other assertions, and the analoguous results for left identities. Next suppose that A∗∗ has a right identity e of norm 1. By Goldstine’s lemma (A.2.1) we may choose a net (et )t in Ball(A) converging to e in the w ∗ -topology of A∗∗ . Then for each a ∈ A it is clear that aet −→ a weakly in A.

(2.22)

Let F be the set of ﬁnite subsets of A. For any F = {a1 , a2 , . . . , an } ∈ F, deﬁne PF = {(a1 u − a1 , a2 u − a2 , . . . , an u − an ) : u ∈ Ball(A)}, a subset of A⊕∞ A⊕∞ · · ·⊕∞ A. By (2.22), 0 lies in the weak closure of PF . Since PF is convex, 0 also lies in its norm closure, by Mazur’s theorem from basic functional analysis. Let Λ = F × N, which we consider as a directed set with respect to the product ordering. Given λ = (F, m) for some F = {a1 , a2 , . . . , an } ∈ F and m ∈ N, there exists a uλ ∈ Ball(A) such that ak uλ − ak < 1/m for all 2 1 ≤ k ≤ n. Then (uλ ) is evidently a right approximate identity for A. Corollary 2.5.9 Suppose that A is an approximately unital operator algebra (or Arens regular matrix normed algebra), and that π : A → B(H) is a completely contractive homomorphism. If π ˜ : A∗∗ → B(H) is its w∗ -continuous extension (see 2.5.5), then π is nondegenerate if and only if π(et ) → IH in the w∗ -topology, ˜ being unital. for any cai (et )t of A. This is also equivalent to π Proof The ﬁrst ‘if and only if’ follows from Lemma 2.1.9. For the other, note ˜ (eA∗∗ ) (in the w∗ -topology). 2 that et → eA∗∗ by 2.5.8, so that π(et ) → π As another application of the ‘second dual’ we give the following principle, which is surprisingly useful for nonselfadjoint algebras. Theorem 2.5.10 Suppose that A and B are (norm closed) subalgebras of an operator algebra C. Suppose further that each of A and B are approximately unital; and that AB = B and BA = A. Then A = B as subsets of C. Proof We may assume that C is selfadjoint. By symmetry it is enough to show B ⊂ A. We shall consider second duals and apply Proposition 2.5.8. Since A ⊂ C and B ⊂ C are subalgebras, we may regard A∗∗ ⊂ C ∗∗ and B ∗∗ ⊂ C ∗∗ as w∗ -closed subalgebras. Recall from basic functional analysis (see A.2.3 (4)) that A = A∗∗ ∩ C. Thus we only need to show that B ⊂ A∗∗ . Let (et )t be a cai for A and let eA be the unit of A∗∗ , so that et → eA in the ∗ w -topology of C ∗∗ . Also et b → b for any b ∈ B by the equality AB = B. Since the product on C ∗∗ is separately w∗ -continuous, we obtain that eA b = b for any b ∈ B. Hence, again by separate w ∗ -continuity, we have eA η = η for any η ∈ B ∗∗ .

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In particular, if eB denotes the unit of B ∗∗ , then eA eB = eB . By symmetry, we also have eB eA = eA . Since eA and eB are projections in C ∗∗ , we deduce that eA = eB by taking adjoints. We can now conclude as follows: for any b ∈ B, we have b = beB = beA = w∗ -limt bet . Since BA ⊂ A we see that b belongs to the 2 w∗ -closure of A, and hence to A∗∗ . There is an analoguous version of the last result when AB = A and BA = B. It is interesting that these results are not true for general Banach algebras. 2.6 MULTIPLIER ALGEBRAS AND CORNERS 2.6.1 (Left multipliers) If A is simply an algebra then, according to the algebraists, a left multiplier of A is a right A-module map u : A → A. The left multiplier algebra LM (A) is the unital algebra of left multipliers of A. Let λ be the left regular representation of A on itself. Namely, λ(a)(b) = ab for a, b ∈ A; this is a homomorphism from A into the left multiplier algebra of A. If A is a Banach algebra which has a cai, then it follows from A.6.3 that any left multiplier is bounded. Thus Banach algebraists usually deﬁne the left multiplier algebra LM (A) of an approximately unital Banach algebra A to be BA (A), the unital Banach algebra of bounded right A-module maps on A. If, further, A is an operator algebra or a matrix normed algebra in the sense of 2.3.9, with cai (e t )t , then it follows immediately from the relation u(a) = limt u(et )a, which clearly holds for all u ∈ BA (A) and a ∈ A, that BA (A) = CBA (A) isometrically. Here CBA (A) is the set of completely bounded right A-module maps. Thus we deﬁne the left multiplier algebra LM (A) of such A to be the pair (CBA (A), λ), where λ : A → CBA (A) is the left regular representation of A mentioned above. Since LM (A) is a subalgebra of CB(A) (completely isometrically), it is clearly a matrix normed algebra. Note that the usual matrix norms on LM (A) = CBA (A) as deﬁned by (1.6), have a simple description in this case. Namely, due to the relation u(a) = limt u(et )a above, it is fairly evident that (2.23) [uij ]Mn (CBA (A)) = sup [uij (a)]Mn (A) : a ∈ Ball(A) . Equivalently, if we regard Mn (A) as a right A-module in the obvious way, we have Mn (LM (A)) = CBA (A, Mn (A)) = BA (A, Mn (A)) isometrically. Henceforth in this section A is an approximately unital operator algebra. One would wish the left multiplier algebra of A to be a unital operator algebra, and fortunately it turns out that CBA (A) with its usual matrix norms discussed in the last paragraph, is an abstract operator algebra. This may be seen by the BRS theorem, or by the Theorem 2.6.2 below. Theorem 2.6.2 Let A be an approximately unital operator algebra. Then the following algebras are all completely isometrically isomorphic: (1) {η ∈ A∗∗ : ηA ⊂ A}, (2) {T ∈ B(H) : T π(A) ⊂ π(A)}, for any nondegenerate completely isometric representation π of A on a Hilbert space H,

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83

(3) CBA (A). In particular, CBA (A) is a unital operator algebra. Proof Let LM be the algebra deﬁned in (1). According to 2.5.6, LM is an operator algebra. Any η ∈ A∗∗ with the property that ηA ⊂ A may be clearly regarded as an element of CBA (A) hence there is a canonical homomorphism θ : LM → CBA (A), which may easily be seen to be completely contractive using (2.23). Let (et )t be a cai for A. If θ(η) = 0 then ηet = 0. Using the separate w∗ -continuity of the product in A∗∗ (see 2.5.3), and Proposition 2.5.8, it follows that η = 0. Thus θ is one-to-one. Given v ∈ CBA (A), let η be a w∗ -cluster point of v(et ) in A∗∗ . Clearly η ≤ v. For a ∈ A, we have v(a) = lim v(et a) = lim v(et )a = ηa, t

t

where again we have used the w ∗ -continuity of the product. Hence θ(η) = v. Thus θ is an isometric surjection, and a similar proof shows that it is a complete isometry. This proves the completely isometric isomorphism between (1) and (3), and also shows that CBA (A) is a unital operator algebra. If we write LM (π) for the algebra in (2), then there is a canonical homomorphism ρ : LM (π) → CBA (A), namely ρ(T )(a) = π −1 (T π(a)) for a ∈ A. If [Tij ] ∈ Mn (LM (π)), then by (2.23) we have (2.24) [ρ(Tij )]n = sup [ρ(Tij )a]n = sup [Tij π(a)]n ≤ [Tij ]n , where the supremum is taken over a ∈ Ball(A). Thus ρ is completely contractive. To see that ρ is completely isometric, we take ε > 0, and choose a vector ζ in Ball(2n (H)) such that [Tij ]n ≤ [Tij ]ζ + ε. Then [Tij ]n ≤ lim [Tij π(et )]ζ + ε ≤ sup [Tij π(et )] + ε ≤ [ρ(Tij )]n + ε, t

t

the last inequality by (2.24). Thus ρ is completely isometric. To see that ρ is onto, suppose that v ∈ BA (A). Let η = θ−1 (v) ∈ LM, and let π ˜ : A∗∗ → B(H) be the w∗ -continuous homomorphism extending π as discussed in 2.5.5 say. Then for any a ∈ A, π ˜ (η)π(a) = π ˜ (ηa) = π(ηa). This shows that π ˜ (η) ∈ LM (π) and ρ(˜ π (η)) = θ(η) = v. Thus ρ is onto.

2

2.6.3 (Left multiplier operator algebras) More generally, for A as above, we consider pairs (D, µ) consisting of a unital operator algebra D and a completely isometric homomorphism µ : A → D, such that Dµ(A) ⊂ µ(A). Sometimes we write µ as µA to indicate the dependence on A. We say that two such pairs (D, µ) and (D , µ ) are completely isometrically A-isomorphic if there exists a completely isometric surjective homomorphism θ : D → D such that θ ◦ µ = µ . This is an equivalence relation.

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Multiplier algebras and corners

One should think of each of the three equivalent algebras in Theorem 2.6.2 as a pair (D, µA ) as in the previous paragraph. We spell out what the map µA is in each case. In (1), it is the usual inclusion iA : A → A∗∗ . In (2), it is π, and in (3) it is the map λ mentioned above the theorem. We deﬁne a left multiplier operator algebra of A, to be any pair (D, µ) as above which is completely isometrically A-isomorphic to (CBA (A), λ). We also write any such pair as LM (A). The following is clear from the last proof. Corollary 2.6.4 Each of the operator algebras in 2.6.2, together with its associated map µA discussed above, is a left multiplier operator algebra of A. That is, they are each completely isometrically A-isomorphic to (CB A (A), λ). 2.6.5 If A is unital, then it is easy to see that LM (A) = A. We also remark that if π : A → B(H) is as in 2.6.2 (2), then we may view LM (A) as a subset of the second commutant π(A) . Indeed if T ∈ B(H) with T π(A) ⊂ π(A), and if S ∈ π(A) and a ∈ A, then ST π(a) = T π(a)S = T Sπ(a). Since π is nondegenerate we have ST = T S. Thus T ∈ π(A) . Corollary 2.6.6 Let A be an approximately unital operator algebra, and ﬁx u in Mn (CBA (A)). Write Lu (resp Lu ) for the operator on Cn (A) (resp. on Mn (A)) given by the usual formula for multiplication of matrices. Then (1) uMn (CBA (A)) = Lu cb = Lu cb . (2) Mn (CBA (A)) ∼ = CBMn (A) (Mn (A)) ∼ = CBMn (A) (Cn (A)) ∼ = BA (A, Mn (A)) isometrically. (3) LM (Mn (A)) ∼ = Mn (LM (A)) completely isometrically. Proof We observed after (2.23) that BA (A, Mn (A)) ∼ = Mn (CBA (A)). We let LM(A) denote the algebra in (1) of Theorem 2.6.2. Clearly Mn (LM(A)) is isomorphic to LM(Mn (A)) as operator algebras, using the fact from 2.5.7 that Mn (A)∗∗ = Mn (A∗∗ ). This yields (3), by the way. Thus by 2.6.2 we have Mn (CBA (A)) ∼ = CBMn (A) (Mn (A)) isometrically. It is clear by inspection that Lu cb = Lu cb for u as above, so that CBMn (A) (Mn (A)) ∼ = CBA (Cn (A)) isometrically. This proves (1) and (2). 2 2.6.7 (Right and two-sided multiplier algebras) We may deﬁne the right (resp. two-sided) multiplier operator algebra RM (A) (resp. M (A)) in a similar way, and one obtains similar results. For example, these algebras may be taken to be RM (A) = {η ∈ A∗∗ : Aη ⊂ A} (resp. M (A) = {η ∈ A∗∗ : ηA ⊂ A, Aη ⊂ A}). However there are one or two pitfalls to be wary of. First, the canonical map RM (A) → A CB(A) into the algebra of completely bounded left A-module maps is a completely isometric anti-homomorphism. Thus RM (A) should be identiﬁed with A CB(A) with the reverse of the usual multiplication on the latter. This ‘twist’ is related to the reason why algebraists often write operators on the right of the variable. Second, the canonical embedding LM (A) → CBA (A) does not restrict, as one might ﬁrst guess, to an embedding of M (A) into the A-bimodule maps on A. The traditional way for Banach algebraists to circumvent this last

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diﬃculty is to consider two-sided multipliers as a pair of maps in the double centralizer algebra (see the Notes on Section 2.6). Instead, we take the following approach. We consider pairs (D, µ) consisting of a unital operator algebra D and a completely isometric homomorphism µ : A → D such that Dµ(A) ⊂ µ(A) and µ(A)D ⊂ µ(A). We deﬁne two such pairs to be completely isometrically A-isomorphic just as we did in 2.6.3. We use the term ‘(two-sided) multiplier operator algebra of A’, and write M (A), for any pair (D, µ) as above which is completely isometrically A-isomorphic to {x ∈ A∗∗ : xA ⊂ A and Ax ⊂ A}. In fact these may be characterized quite nicely as subalgebras of LM (A). We leave the following as a (simple algebraic) exercise: Proposition 2.6.8 Suppose that A is an approximately unital operator algebra. If (D, µ) is a left multiplier operator algebra of A, then the closed subalgebra {d ∈ D : µ(A)d ⊂ µ(A)} of D, together with the map µ, is a (two-sided) multiplier operator algebra of A. Corollary 2.6.9 If A is a C ∗ -algebra then M (A) is the diagonal ∆(LM (A)). In particular, M (A) is a C ∗ -algebra. Proof If η ∈ LM (A) ⊂ A∗∗ then Aη ⊂ A

⇔

η∗ A ⊂ A

⇔

hence η ∈ M (A) if and only if η ∈ ∆(LM (A)).

η ∗ ∈ LM (A), 2

∗

2.6.10 (Essential ideals) If C is a C -algebra then an essential (two-sided) ideal of C is an ideal I of C for which the canonical homomorphism from C to B(I) associated to the product map C × I → I, is one-to-one. It is a pleasant ∗-algebraic exercise to show that this is equivalent to saying that K ∩ I = (0) for every nontrivial (two-sided closed) ideal K of C. Any C ∗ -algebra A is an essential ideal of M (A). Clearly, if C is a C ∗ -algebra containing A as an essential ideal, then the canonical homomorphism C → BA (A) = CBA (A) = LM (A) is completely contractive. Hence by the last paragraph in 2.1.2, for example, it yields a one-to-one ∗-homomorphism into ∆(LM (A)) = M (A) (see 2.6.9). Thus M (A) is the largest C ∗ -algebra with A as an essential ideal. If A is a w∗ -dense ideal in a W ∗ -algebra M , then M = LM (A) = M (A). Indeed, the complete contraction above from M to CBA (A), is one-to-one by the w∗ -density of A. That it is completely isometric and surjective is easily seen by considering, for any T ∈ CBA (A), a w∗ -limit point of (T (et )) in M , where (et ) is the approximate identity for A. The other assertions are now easy. In fact, this argument works more generally if M is a matrix normed algebra which is a dual space with separately w∗ -continuous product, and if A has a cai. 2.6.11 (Multiplier-nondegenerate morphisms) Let A, B be approximately unital algebras. A completely contractive homomorphism π : A → LM (B) will be called a left multiplier-nondegenerate morphism, if B is a nondegenerate left

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module with respect to the natural left module action of A on B via π. This is equivalent, by Cohen’s factorization theorem A.6.2, to saying that for any left cai (et )t of A, we have π(et )b → b for all b ∈ B; or to saying that any b ∈ B may be written b = π(a)b for some a ∈ A, b ∈ B. Similarly for right multiplier-nondegenerate morphism; and a (two-sided) multiplier-nondegenerate morphism is a map π : A → M (B) that has both of these properties. Thus if π actually maps into B, then π is multiplier-nondegenerate if and only if (π(et ))t is a cai for B, for any cai (et )t for A. Note that the canonical map A → M (A) is multiplier-nondegenerate. Proposition 2.6.12 If A, B are approximately unital operator algebras, and if π : A → M (B) is a multiplier-nondegenerate morphism then π extends uniquely to a unital completely contractive homomorphism π ˆ : M (A) → M (B). Moreover π ˆ is completely isometric if and only if π is completely isometric. Proof By 2.6.7 we may regard M (A) and M (B) as subalgebras of A∗∗ and B ∗∗ respectively. Let π ˜ : A∗∗ → B ∗∗ be the (unique) w∗ -continuous homomorphism extending π (see 2.5.5). We will prove that π ˜ maps M (A) into M (B) and that π ˜ (·)|M(A) is the unique bounded homomorphism on M (A) extending π. As usual we let (et )t be a cai for A. Let η ∈ M (A) and let b ∈ B. Then ˜ (η)π(et )b → π ˜ (η)b. Since π ˜ is a homomorphism π(et )b → b by 2.6.11, hence π ˜ (η)b = limt π(ηet )b is the limit of a net of B, hence and ηet ∈ A, we ﬁnd that π belongs to B. Similarly, b˜ π (η) ∈ B for any b ∈ B, which shows that π ˜ (η) ∈ M (B). Assume now that π ˆ : M (A) → M (B) is a bounded homomorphism extending π, and let η ∈ M (A) and b ∈ B. Each ηet belongs to A, and this net converges to η in the w∗ -topology by Proposition 2.5.8. Hence π ˜ (η) = w∗ - lim π(ηet ). t

On the other hand, π(et )b → b, and hence π ˆ (η)b = lim π ˆ (η)π(et )b = lim π ˆ (ηet )b = lim π(ηet )b. t

t

t

This shows that π ˆ=π ˜|M(A) . It remains to prove the last assertion, so suppose that π is completely isometric. Using (2.23) we have for η ∈ M (A) that π (η)(π(a)b) = π(ηa)b ˆ π (η)cb ≥ ˆ if a ∈ Ball(A), b ∈ Ball(B). Taking the supremum over all b ∈ Ball(B) gives ˆ π (η)cb ≥ π(ηa) = ηa. Taking the supremum over all such a ∈ Ball(A), gives that ˆ π(η) cb ≥ ηcb. So π ˆ is isometric, and similarly it is completely isometric. 2 2.6.13 The analoguous result for ‘left multiplier-nondegenerate morphisms’, in which we replace M (·) by LM (·) in 2.6.12, holds with the same proof.

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As a corollary of the last result, and 2.1.7 (2), we note that if B is any C ∗ cover of an approximately unital operator algebra A, then LM (A) ⊂ LM (B), RM (A) ⊂ RM (B), and M (A) ⊂ M (B), as unital-subalgebras. As another corollary, suppose that J is a left ideal of an operator algebra A, that J has a cai, and that π : J → B(H) is a completely contractive nondegenerate representation. We claim that π extends uniquely to a completely contractive representation π ˜ of A on H. Indeed the uniqueness follows from the relation π ˜ (a)π(x)ζ = π ˜ (a)˜ π (x)ζ = π(ax)ζ, for a ∈ A, x ∈ J, ζ ∈ H. Viewing B(H) as the multiplier algebra of S ∞ (H), it is easy to see that π is a multipliernondegenerate morphism in the sense above. By 2.6.12, π extends to a completely contractive representation of LM (J) on H. Since LM (J) = CBJ (J), it is clear that there is a canonical completely contractive homomorphism from A into LM (J). Composing the last two homomorphisms gives the desired result. We end this section with some relations between multipliers and ‘corners’. 2.6.14 (Corners) If p is a projection in M (A) then we say that pAp is the 1-1-corner of A (with respect to p). In this case pAp is an operator algebra (indeed it is a C ∗ -algebra if A is one), as is the 2-2-corner (1 − p)A(1 − p). We say that pA(1 − p) is the 1-2-corner of A (with respect to p). Consider the subset B of M2 (A) consisting of all 2 × 2 matrices whose i-j entry belongs to the i-j-corner of A with respect to p. This set B is a norm closed subalgebra of M2 (A), which we claim is canonically completely isometrically isomorphic to A. To see this, ﬁrst suppose that A = B(H). Then A ∼ = B(K ⊕ K ⊥), and it is fairly clear that this is ∗-isomorphic to B in this case, via a ∗-isomorphism π say. By 1.2.4, π is completely isometric. For a general subalgebra A ⊂ B(H), the claim is established by considering the restriction of π to A. In fact, it is usually helpful to regard the 2 × 2 matrix operator algebra B as a subalgebra of the 2 × 2 matrix ∗-algebra corresponding to B(K ⊕ K ⊥ ). 2.6.15 (Corner-preserving maps) Suppose that A and B are approximately unital operator algebras, and that p and q are projections in M (A) and M (B) respectively. As in 2.6.14, this allows us to decompose A and B as 2 × 2 matrix operator algebras. We say that a map π : A → B is corner-preserving, or decoupled, if π maps each of the four corners of A into the corresponding corner of B. In this case we let πij be the restriction of π to the i-j-corner of A, viewed as a map from the i-j-corner of A into the i-j-corner of B. We call πij the i-j-corner of π, and sometimes write π = [πij ] to indicate this situation. Suppose that π : A → M (B) is a contractive multiplier-nondegenerate morphism (see 2.6.11) and that p is a projection in M (A). If we deﬁne q = π ˆ (p), where π ˆ is as in 2.6.12, then q is a contractive idempotent, and is consequently, by 2.1.3, a projection in M (B). Also, π is corner-preserving as a map into M (B) (with respect to the corners deﬁned via p and q = π ˆ (p)). To see this, note for example that π(pa(1 − p)) = π ˆ (pa(1 − p)) = qπ(a)(1 − q) for any a ∈ A. 2.6.16 (Corner-preserving completely positive maps) Suppose that Φ : A → B is a unital completely positive map between unital C ∗ -algebras. Suppose that

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p is a projection in A and that q = Φ(p) is a projection in B. Then a similar principle to 2.6.15 holds. Obviously Φ restricted to the linear span of p and 1 − p (which is a two-dimensional C ∗ -algebra) is a unital ∗-homomorphism. It follows by 1.3.12 that Φ(pa) = qΦ(a) and Φ(ap) = Φ(a)q for all a ∈ A. Hence Φ is again corner-preserving, and we may write Φ = [ϕij ] just as in 2.6.15. Since Φ is completely positive and hence ∗-linear it follows that ﬁrst, ϕ11 and ϕ22 are completely positive, and second, that ϕ12 (x∗ ) = ϕ21 (x)∗ for any x ∈ pA(1 − p). In other words, ϕ21 = ϕ12 in the sense of 1.2.25. In passing we observe that using 1.2.10 and 1.3.3, we may extend (the ﬁrst part of) these results to the case when Φ : A → B is a unital completely contractive homomorphism between unital operator algebras. 2.6.17 We may rephrase 2.6.16 in the language of bimodules. Note that the existence of projections p ∈ A and q ∈ B, is equivalent to saying that A and B each contain D2 = C ⊕ C as a unital C ∗ -subalgebra. That Φ may be written as a matrix of maps [ϕij ] with respect to these projections p, q, is equivalent to saying that Φ is a D2 -bimodule map. We can therefore extend the observation 2.6.16 to ∗ the case when A and B contain a copy of Dn = ∞ n as a unital C -subalgebra, and Φ : A → B is a unital completely contractive map. In this case, A and B decompose as n× n matrix algebras, with respect to the n canonical idempotents in Dn . Also, Φ is a Dn -bimodule map if and only if we may write Φ = [ϕij ]. In this case we say again that Φ is corner-preserving. Again 1.3.12 implies that if Φ is ‘the identity map’ on the copies of Dn , then Φ is corner-preserving. 2.7 DUAL OPERATOR ALGEBRAS We shall now consider dual objects in the category of operator algebras, both from the concrete and from the abstract point of view. By deﬁnition, a concrete dual operator algebra is a w ∗ -closed subalgebra of B(H), for some Hilbert space H. As a consequence of 1.4.7, any concrete dual operator algebra is a dual operator space. In the converse direction, let M be an operator algebra together with a given additional topology τ on M , and we suppose that τ is the w ∗ -topology given by some predual for M . We call such an algebra M , together with the topology τ , an (abstract) dual operator algebra, if there exist a Hilbert space H and a w∗ -continuous completely isometric homomorphism π : M → B(H). In this case, the range π(M ) is w∗ -closed by the Krein–Smulian theorem A.2.5, and π is a w∗ -homeomorphism onto its range. Hence π(M ) is a concrete dual operator algebra acting on H, which we may identify with M . If M is a dual operator algebra, then we say that the topology τ mentioned above, is a dual operator algebra topology on M . In the selfadjoint situation, we use the term ‘von Neumann algebra’ for a concrete selfadjoint dual operator algebra, and the term ‘W ∗ -algebra’ for an abstract selfadjoint dual operator algebra (see A.5.1 and A.5.3). By the famous theorem of Sakai, a W ∗ -algebra is exactly the same as a C ∗ -algebra with a Banach space predual; and in this case the Banach space predual is unique (see

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[380, Sections 1.13 and 1.16]). We will see that in the nonselfadjoint context, some new complications may occur. For example, in general the predual of an operator algebra need not be unique. That is, an operator algebra may possess more than one dual operator algebra topology (see Corollary 2.7.8 and Proposition 2.7.16). However we will establish an abstract characterization of dual operator algebras (see Theorem 2.7.9) which is close to Sakai’s result. We often identify any two dual operator algebras M and N which are w ∗ homeomorphically and completely isometrically isomorphic. In this case we sometimes say that M ∼ = N as dual operator algebras. 2.7.1 If M is a dual operator algebra, then we will reserve the notation M ∗ for any operator space predual of M which induces the given dual operator algebra topology on M . If M ⊂ B(H) is represented as a w ∗ -closed subalgebra (by a map π as above), then by 1.4.6 we may take M∗ = S 1 (H)/M⊥ . 2.7.2 Unital dual operator algebras play a particularly prominent role in the sequel. A slight modiﬁcation of the argument in 2.1.4 shows that any unital dual operator algebra may be represented as a w ∗ -closed subalgebra of some B(H) containing IH . (Note that the map a → a|K in 2.1.4 is a w∗ -continuous complete isometry into B(K), and is thus a w∗ -homeomorphism by A.2.5.) 2.7.3 As a consequence of Corollary 2.5.6, the second dual of any operator algebra is a dual operator algebra, as one might hope. The next result collects together several superﬁcial facts about dual algebras: Proposition 2.7.4 Let M be a (possibly nonunital) dual operator algebra. (1) The product on M is separately w ∗ -continuous. (2) For any a ∈ M and any ϕ ∈ M∗ , the functionals ϕa and aϕ (using notation from 2.5.1) both belong to M∗ . (3) If M is approximately unital, then M is actually unital. (4) The w∗ -closure of a subalgebra of M is a dual operator algebra. (5) The unitization (see Section 2.1 above) of M is also a dual operator algebra. (6) If M is nonunital, and if π : M → B(H) is a w ∗ -continuous contractive homomorphism, then the canonical unital extension of π to the unitization of M (see 2.1.13) is w∗ -continuous. Proof Item (1) is clear from the separate w ∗ -continuity of the product on B(H) (see A.1.2). Item (2) is a restatement of (1). For (3), assume that (e t )t is a cai for M , and let e ∈ M be a w∗ -cluster point of this net. For any a ∈ M and any ϕ ∈ M∗ , and passing to a subnet, we have ϕ, et a = aϕ, et −→ aϕ, e = ϕ, ea by (2). Since et a → a, we deduce that ϕ, ea = ϕ, a. Similarly, ϕ, ae = ϕ, a. Since ϕ ∈ M∗ was arbitrary, this shows that e is a unit (of norm 1) for M . For (4), let x, y belong to the w∗ -closure of the subalgebra A in M , and let (aα )α and (bβ )β be two nets of A converging to x and y respectively in

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the w∗ -topology. By (1), xbβ = w∗ -limα aα bβ . Hence xbβ belongs to A ∗

w∗

w∗

. Then

xy = w -limβ xbβ , and so xy ∈ A . Finally, suppose that M is a w∗ -closed nonunital subalgebra of B(K), and write I for IK . Suppose that (xt )t and (λt )t are nets in M and C respectively, with (xt + λt I) converging in the w∗ -topology. By the Hahn–Banach theorem, it is easy to see that (λt )t converges in C. It follows that (xt )t converges in M , in 2 the w∗ -topology. From this, (5) and (6) follow easily. 2.7.5 (Examples) The following completes the list considered in Section 2.2. We refer the reader to the extensive literature on dual algebras for deeper and more interesting examples (see the ﬁrst paragraph of the Notes section for this chapter for a few references). (1) As in 2.2.1, the ∞-direct sum of dual operator algebras is again a dual operator algebra. We leave this fact as an exercise using the last facts in 1.4.13. In particular, if M is a dual operator algebra and I is a set, then ∞ I (M ) is a dual operator algebra. (2) Let M and N be two dual operator algebras. Then it follows from 2.2.2 and 2.7.4 (4), that their normal spatial tensor product M ⊗ N (deﬁned in 1.6.5) is a dual operator algebra. We leave it as an exercise that if π, θ are w ∗ -continuous completely bounded representations of M and N , then the extension of π ⊗ θ to M ⊗ N is again a w∗ -continuous completely bounded homomorphism. (3) The result in 2.2.4 extends as follows. Let (Ω, µ) be a σ-ﬁnite measure space, let M be a dual operator algebra with separable predual, and recall from 1.6.6 the w∗ -continuous completely isometric isomorphism jM from L∞ (Ω) ⊗ M onto the dual operator space L∞ (Ω; M ) of w∗ -measurable essentially bounded functions f : Ω → M . If f, g ∈ L∞ (Ω; M ) then the pointwise product function (f g)(t) = f (t)g(t) is w ∗ -measurable and hence belongs to L∞ (Ω; M ). Moreover the resulting multiplication on L∞ (Ω; M ) is separately w∗ -continuous. Indeed these results are established in [380, Theorem 1.22.13] in the case when M is a W ∗ -algebra and the proofs given there extend almost verbatim to the nonselfadjoint case. Now consider L∞ (Ω; M ) as equipped with this pointwise product. Then the last lines of 1.6.6 show that the restriction of jM to L∞ (Ω) ⊗ M is a homomorphism. Since the products on L∞ (Ω) ⊗ M and L∞ (Ω; M ) are both separately w ∗ -continuous, we deduce that jM is a homomorphism. Thus L∞ (Ω; M ) is a dual operator algebra for the pointwise multiplication and L∞ (Ω) ⊗ M = L∞ (Ω; M ) ‘as dual operator algebras’. These results extend the well-known fact (e.g. see [380, Theorem 1.22.13]) that L ∞ (Ω; M ) is a W ∗ -algebra if M is a W ∗ -algebra. (4) The discussion in 2.2.6 and 2.2.7 can be continued as follows. Let H ∞ (D) ⊂ L∞ (T) be the Hardy space of all f ∈ L∞ (T) whose negative Fourier coeﬃcients vanish. Then H ∞ (D) is a w∗ -closed subalgebra of L∞ (T), and hence it is a unital dual operator algebra. Using Poisson integrals, H ∞ (D) can also be described as the algebra of all bounded analytic functions from D into C. More generally, if M is a dual operator algebra with separable predual, then the space

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H ∞ (D; M ) of all functions f ∈ L∞ (T; M ) whose negative Fourier coeﬃcients vanish is a dual operator algebra. Moreover H ∞ (D; M ) can also be described as the algebra of all bounded analytic functions from D into M and we have an identiﬁcation H ∞ (D; M ) = H ∞ (D) ⊗ M ‘as dual operator algebras’. We omit the proofs of these assertions, as they diverge from our main concerns. (5) When M = MI , for a cardinal I, then (2) yields a dual operator algebra structure on MI ⊗ N . On the other hand, if N ⊂ B(H) is represented as a dual 2 operator algebra on H, then MI (N ) is the w∗ -closure of Mﬁn I (N ) in B(I (H)) by Corollary 1.6.3 (3), hence is a dual operator algebra by 2.7.4 (4). From this it is not hard to see that the identiﬁcation MI ⊗ N = MI (N ) given by (1.54) holds ‘as dual operator algebras’. This is analoguous to (2.8) and is related to the latter by the following proposition. Proposition 2.7.6 Let A be an operator algebra, and I any cardinal. Then KI (A)∗∗ ∼ = MI (A∗∗ ) as dual operator algebras. Proof The canonical embedding A ⊂ A∗∗ induces a completely isometric homomorphism π : KI (A) → KI (A∗∗ ) ⊂ MI (A∗∗ ). By (1.62) the unique w∗ -continuous extension π ˜ of π to KI (A)∗∗ is a completely isometric isomorphism. However by 2.5.5, it is a homomorphism too. 2 We now return to the algebra U(X) constructed in 2.2.10. Lemma 2.7.7 Let X be an operator space. (1) If U(X) is a dual operator algebra, then X ⊂ U(X) is w ∗ -closed. Hence X is a dual operator space. (2) If X is a dual operator space, then U(X) is a dual operator algebra. Indeed U(X) has an operator space predual for which the canonical embedding from X into U(X) is w∗ -continuous. Proof Assume that U(X) is a dual operator algebra. Let p be the projection 10 in U (X) deﬁned by p = . Then p U(X)(1 − p) is the subspace of U(X) 00 corresponding to X. It is a fairly obvious general principle that a corner pM (1−p) of a dual algebra M is w∗ -closed. Indeed by 2.7.4 (1) and the fact that p is a projection, it follows that if (yt ) is a net in pM (1 − p) converging to b ∈ M say, then yt = pyt (1 − p) converges to pb(1 − p). Thus b = pb(1 − p) ∈ pU(X)(1 − p). Assume conversely that X is a dual operator space. By 1.4.7 we may assume that X ⊂ B(H) is a w∗ -closed subspace of B(H) for some Hilbert space H. Then U (X), as deﬁned by (2.10), is easily seen to be a w ∗ -closed subalgebra of B(H (2) ) and the result follows at once. 2 Corollary 2.7.8 There exist a unital operator algebra A with two distinct dual operator algebra topologies. Equivalently, there exist two completely isometric homomorphisms π1 : A → B(H1 ) and π2 : A → B(H2 ) such that M1 = π1 (A) and M2 = π2 (A) are w∗ -closed but π1∗ (M1 ∗ ) = π2∗ (M2 ∗ ).

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Proof Let E be a Banach space with two distinct preduals Z1 and Z2 . More precisely, we think of these preduals as two distinct subspaces Z1 ⊂ E ∗ and Z2 ⊂ E ∗ . We consider the operator space X = Min(E), and the unital operator algebra A = U (X) of 2.2.10. Let J : X → U (X) be the canonical embedding. Since X = Max(Z1 )∗ = Max(Z2 )∗ , the second part of Lemma 2.7.7 ensures that there exist two completely isometric homomorphisms π1 : A → B(H1 ) and π2 : A → B(H2 ) respectively, such that M1 = π1 (A) and M2 = π2 (A) are w∗ closed, and such that π1 ◦ J : X → M1 and π2 ◦ J : X → M2 are w∗ -continuous. In the last statement X is regarded as the dual of Z1 and of Z2 respectively; that is, (π1 ◦ J)∗ (M1∗ ) = Z1 and (π2 ◦ J)∗ (M2∗ ) = Z2 . Thus if π1∗ (M1∗ ) and π2∗ (M2∗ ) 2 were equal, we would have the contradiction that Z1 = Z2 . We have noticed that a dual operator algebra is a dual operator space. The main result of this section is the following converse, which may be regarded as a dual version of the BRS theorem, and also as a nonselfadjoint version of Sakai’s theorem mentioned at the start of this section. Theorem 2.7.9 Let M be an operator algebra which is a dual operator space. Then the product on M is separately w ∗ -continuous, and M is a dual operator algebra. That is, there exists a Hilbert space H and a w ∗ -continuous completely isometric homomorphism π : M → B(H). The ﬁrst assertion saying that the product on M is separately w ∗ -continuous will only be proved in Chapter 4, in 4.7.2. The rest of the proof will require several intermediate steps of independent interest, and will be completed in 2.7.13 below. We start with a version of Proposition 2.4.20 (1), for unital dual operator algebras. Theorem 2.7.10 Let M be a unital operator algebra which is also a dual operator space. Let u : M → B(H) be a w ∗ -continuous completely contractive map, H being a Hilbert space. Then there exist a Hilbert space K, two linear contractions V : H → K, W : K → H, and a unital w∗ -continuous completely contractive homomorphism π : M → B(K), such that u(a) = W π(a)V for any a ∈ M . Proof We ﬁrst apply Proposition 2.4.20 (1) to u. We obtain a Hilbert space G, two linear contractions T : H → G and S : G → H, and a unital completely contractive homomorphism θ : M → B(G) such that u(a) = Sθ(a)T,

a ∈ M.

(2.25)

Moreover we may assume that G = [θ(M )T (H)]. Let K = [θ(M )∗ S ∗ (H)] ⊂ G, and let p = PK be the projection onto K. If a, b, c ∈ M and ζ, η ∈ H then u(acb)ζ, η = pθ(b)T ζ, θ(c)∗ θ(a)∗ S ∗ η = pθ(c)pθ(b)T ζ, θ(a)∗ S ∗ η. Thus u(acb) = Sθ(a)θ(c)pθ(b)T = Sθ(a)pθ(c)pθ(b)T.

(2.26)

Setting a = b = 1 in (2.26) yields u = W π(·)V , where W = S|K , V = pT and π = pθ(·)|K . A well-known general principle states that if θ : A → B(H)

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is a homomorphism and if K is a subspace of H such that θ(A)∗ K ⊂ K (or equivalently, such that θ(A)K ⊥ ⊂ K ⊥ , then π = PK θ(·)|K is a homomorphism. To see this, note that for any a, b ∈ A and any ζ, η ∈ K, we have π(ab)ζ, η = θ(ab)ζ, η = θ(a)θ(b)ζ, η = θ(b)ζ, θ(a)∗ η = π(b)ζ, θ(a)∗ η. On the other hand, π(a)π(b)ζ, η = θ(a)π(b)ζ, η = π(b)ζ, θ(a)∗ η. Thus π is a unital completely contractive homomorphism. To show that π is w∗ -continuous, we ﬁx a, b ∈ M and ζ, η ∈ H. We noticed before Theorem 2.7.10 that the product on M is separately w ∗ -continuous (by 4.7.2). Since u is w∗ continuous, the mapping c → u(acb) is therefore w ∗ -continuous from M into B(H). Also observe that for any c ∈ M , we have using (2.26) that u(acb)ζ, η = pθ(c)pθ(b)T ζ, θ(a)∗ S ∗ η = π(c)pθ(b)T ζ, θ(a)∗ S ∗ η. If (ct )t is a bounded net in M converging in the w ∗ -topology to c ∈ M , then π(ct )k1 , k2 → π(c)k1 , k2

(2.27)

for k1 = pθ(b)T ζ, k2 = θ(a)∗ S ∗ η. The span of the set of such k1 is dense in K, and similarly for the span of the set of such k2 . Therefore, by a simple norm density argument, (2.27) holds for all k1 , k2 ∈ K. That is, π(ct ) → π(c) in the WOT. Since this is a bounded net, it converges by A.1.4 in the w ∗ -topology. Thus π is w∗ -continuous by A.2.5 (2). 2 We next give a dual version of Proposition 2.3.4. Recall from 1.4.4, that if X = Z ∗ is a dual operator space and if Y ⊂ X is w ∗ -closed, then the quotient space X/Y is the dual operator space of Y⊥ ⊂ Z. Proposition 2.7.11 Let M be a dual operator algebra and let J ⊂ M be a w∗ -closed (two-sided) ideal. Then M/J is a dual operator algebra. Proof We may assume that M ⊂ B(H) is a w ∗ -closed subalgebra of some B(H). Then by 2.7.4 (5) its unitization M 1 = Span{M, IH } is also a w∗ -closed subalgebra, any ideal of M is still an ideal of M 1 , and the canonical embedding M/J ⊂ M 1 /J is w∗ -continuous. Replacing M by M 1 if necessary, we may therefore assume that M is unital. If M is unital, then N = M/J is unital, hence N is both a dual operator space and a unital operator algebra (by 2.3.4). By 1.4.7, there exists a w ∗ -continuous linear complete isometry u : N → B(H), for some Hilbert space H. Choose K, π, V and W as in Theorem 2.7.10, with u = W π(·)V . For any n ≥ 1 and any matrix [aij ] ∈ Mn (N ), we have [aij ] = [u(aij )] ≤ W [π(aij )] V ≤ [π(aij )] ≤ [aij ] . n n n n n Hence π is a complete isometry, so that N is a dual operator algebra.

2

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Lemma 2.7.12 Let M be an Arens regular Banach algebra. Assume moreover that M = X ∗ is a dual Banach space, and that the product on M is separately w∗ -continuous. Let iX : X → X ∗∗ be the canonical embedding. Then the adjoint mapping Q = i∗X : M ∗∗ → M is a homomorphism. Proof Take η and ν in M ∗∗ and consider two nets (at )t and (bs )s in M converging respectively to η and ν in the w ∗ -topology. Then it follows immediately from the separate w∗ -continuity that Q(ην) = w∗ - lim lim Q(at bs ) = w∗ - lim lim at bs = w∗ - lim at Q(ν) = Q(η)Q(ν). t

s

t

s

t

2

Thus Q is a homomorphism.

2.7.13 (Proof of Theorem 2.7.9) Assume that M is an operator algebra which is also a dual operator space. We already emphasized that the product on M is separately w∗ -continuous (by 4.7.2). Let Q : M ∗∗ → M be deﬁned as in the above lemma. Combining the latter with 2.5.4, we obtain that Q is a homomorphism. Since iX is a complete isometry, Q also is a w ∗ -continuous complete quotient map, by 1.4.3. Thus Ker(Q) is a w∗ -closed ideal of M ∗∗ . Factoring through its kernel, Q induces a w∗ -continuous (by A.2.4) and completely isometric homomorphism from M ∗∗ /Ker(Q) onto M . In fact this map is a w ∗ -homeomorphism, by A.2.5. Since M ∗∗ is a dual operator algebra, as we remarked in 2.7.3, Proposition 2.7.11 shows that M ∗∗ /Ker(Q) is a dual operator algebra. Thus M is a dual operator algebra. Corollary 2.7.14 Suppose that A is an operator algebra, and that m is any product on A∗∗ extending the product on A, for which A∗∗ is (completely isometrically) an operator algebra. Then m must be the Arens product. Proof By Theorem 2.7.9, m is automatically w ∗ -continuous. By 2.5.3, m is the Arens product. 2 In the light of Sakai’s theorem (which we described at the start of this section), one might imagine that Theorem 2.7.9 remains valid if we replace the assumption that M is a dual operator space by the weaker assumption that M is merely a dual Banach space. In fact this is not true, and our next goal is to show this. We will use the following subtle result which clariﬁes some diﬀerences between Banach space and operator space duality. Lemma 2.7.15 There exists an operator space X with a unique Banach space predual Z, such that X possesses no operator space predual. Moreover, whenever Z is equipped with an operator space structure, the canonical mapping between the dual operator space Z ∗ and X is not a complete isomorphism. Proof Let B be an operator space that will be speciﬁed later on. From 1.4.13, 1 (B ∗ )∗ = ∞ (B ∗∗ )

and

c0 (B)∗ = 1 (B ∗ )

(2.28)

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completely isometrically. Thus c0 (B)∗∗ = ∞ (B ∗∗ ). If η ∈ ∞ (B ∗∗ ), let η˙ denote its class in the quotient space ∞ ((B ∗∗ )op )/c0 ((B ∗∗ )op ) (see 1.2.25 for this notation), and write η¯ for the pair (η, η). ˙ Consider the operator space ∞ ∗∗ ∞ ∗∗ X = η¯ : η ∈ (B ) ⊂ (B ) ⊕∞ ∞ ((B ∗∗ )op )/c0 ((B ∗∗ )op ) . (2.29) The map κ : X → ∞ (B ∗∗ ) taking η¯ to η is a complete contraction, that is, ηMn (X) ηMn (∞ (B ∗∗ )) ≤ ¯ ∞

(2.30)

∗∗

for any n ≥ 1 and any η ∈ Mn ⊗ (B ). Moreover since η → η˙ is a contraction, κ is an isometric isomorphism. The main point to be kept in mind is the operator space structure on X induced by the right-hand side of (2.29). Obviously, η˙ = 0 for any η ∈ c0 (B). Thus if η ∈ Mn ⊗ c0 (B) then η Mn (X) . ηMn (c0 (B)) = ¯ That is, the canonical map ι : c0 (B) → X is a complete isometry. Fix z ∈ Mn ⊗ 1 (B ∗ ) for some arbitrary integer n ≥ 1, and let u be the associated linear mapping from ∞ (B ∗∗ ) to Mn (as in (1.27), and using (2.28)). Note that u is w∗ -continuous, and therefore it is the unique w ∗ -continuous extension to ∞ (B ∗∗ ) of the map u|c0 (B) : c0 (B) → Mn . Hence according to 1.4.8, and also using (1.27) and (2.28), we have u : ∞ (B ∗∗ ) −→ Mn = u|c (B) : c0 (B) −→ Mn = zM (1 (B ∗ )) . 0 n cb cb It follows from the last equations, and the fact that κ and ι are completely contractive, and that u ◦ κ ◦ ι = u|c0 (B) , that we also have (2.31) zMn (1 (B ∗ )) = u ◦ κ : X −→ Mn cb . Being isometric to ∞ (B ∗∗ ), X is a dual Banach space by (2.28). We shall now assume that B is a C ∗ -algebra. Then ∞ (B ∗∗ ) is a W ∗ -algebra, and so 1 (B ∗ ) is the unique Banach space predual of X. Equation (2.31) may be rephrased as the fact that if we view 1 (B ∗ ) ⊂ X ∗ in the natural way, then the operator space structure induced on 1 (B ∗ ) from X ∗ , coincides with its usual one from 1.4.13. Let Z denote 1 (B ∗ ) equipped with another operator space structure for which X = Z ∗ completely isomorphically via the canonical map. Then Z = 1 (B ∗ ) completely isomorphically by our rephrased version of (2.31). Passing to operator space duals, we ﬁnd that X = ∞ (B ∗∗ ) completely isomorphically as well. This means that there is a constant c > 0 such that η ˙ Mn (∞ ((B ∗∗ )op )/c0 ((B ∗∗ )op )) ≤ cηMn (∞ (B ∗∗ )) for any n ≥ 1 and any η ∈ Mn (∞ (B ∗∗ )). Restricting to matrices η whose entries are constant sequences valued in B, we deduce that ηMn (B op ) ≤ cηMn (B) ,

n ≥ 1, η ∈ Mn (B).

Hence we obtain a contradiction by choosing our B so that the last inequality is 2 false (take for example B to be B(H) or S ∞ (H), with dim(H) = ∞).

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An example which also satisﬁes the last lemma, but which looks much simpler, is obtained by deﬁning X to be the subspace of B(H) ⊕∞ (B(H)/S ∞ (H))op consisting of pairs (T, T˙ ) for T ∈ B(H). The arguments of the lemma go through verbatim almost to the end. However the last three lines of the proof (in the case that c > 1) then require nontrivial facts about the Calkin algebra B(H)/S ∞ (H). Corollary 2.7.16 There exists a unital operator algebra A with a Banach space predual for which the product is separately w ∗ -continuous but for which there is no complete isometry J : A → B(H) which is w ∗ -continuous, or whose range is w∗ -closed. Such an algebra A has no operator space predual. Proof Let X be an operator space satisfying Lemma 2.7.15, and let Z denote its Banach space predual. Consider the unital operator algebra A = U(X) deﬁned by 2.2.10, and we will prove that it satisﬁes the requirements above. First note that Min(X) = Max(Z)∗ is a dual operator space (see 1.4.12). Thus U(Min(X)) is a dual operator algebra by Lemma 2.7.7 (2). Moreover A is isometrically isomorphic to U (Min(X)) by Proposition 2.2.12. This proves that A has a Banach space predual for which the product is separately w ∗ -continuous. Second, note that if A were a dual operator space, then it would be a dual operator algebra by Theorem 2.7.9. By Lemma 2.7.7 (1), this would imply the contradiction that X ⊂ A is a dual operator space. If there existed a w ∗ -continuous complete isometry J : A → B(H) then the range of J is w ∗ -closed by A.2.5, so that A would be a dual operator space (using 1.4.6). 2 2.8 NOTES AND HISTORICAL REMARKS Nonselfadjoint operator algebras were ﬁrst studied by Kadison and Singer [215], and a little later by Ringrose (see references in [364]). Shortly after this study commenced, many major and foundational papers on the subject were written by Arveson—for example, see [20–23, 25]. It was Arveson who deﬁned complete contractions, and complete isometries, and recognized their importance to operator algebras and operator spaces. Around that time, motivated in large part by operator theory and the invariant subspace problem, dual operator algebras began to be studied extensively using Banach space techniques. We will not attempt the long list of names that might be mentioned here, we would be sure to omit some. For some citations see [23, 32, 84, 108, 240, 337, 354], and references therein. Operator algebras occur quite naturally in mathematics, for example as the ‘classifying object’ for many problems. In the 1970s also, Varopoulos initiated another direction in the study of operator algebras as Banach algebras. For example, see [417,116,124,123,411,82]; amongst other things, these authors gave some characterizations, and produced many surprising examples of Banach algebras which are/are not bicontinuously isomorphic to operator algebras. We say a little more on this work (which greatly inﬂuenced the subject of our book) in Chapter 5. We have included some results on Banach algebras in the Appendix; others may be found in the standard texts, for example [74, 106, 297, 298].

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Again, we will not attempt to describe the huge literature on nonselfadjoint operator algebras from the 1980s and later. The deepest examples and special classes of operator algebras are out of the range of our book. We hope that at some future point we might see a text entitled ‘Operator algebras by example’ to follow [108,110]! General results, which apply to all operator algebras, seem to be rather scarce in the literature. Our book of course only treats topics connected with operator spaces, and even then is incomplete due to limitations of space. We have omitted to address a few major topics which are extensively covered in other books [108, 314, 335, 337], such as dilation theory. 2.1: The result 2.1.7 is from [46]. The unitization of approximately unital (matrix normed) operator algebras may ﬁrst appear in [344]. Meyer’s results [277] suggest that perhaps other questions about the ball can be transformed into questions involving positivity. The diagonal ∆(A) has played a role since the beginning of the subject. Arazy and Solel show in [14] that a surjective linear isometry T : A → B between unital operator algebras takes the diagonal onto the diagonal. It then follows from a result of Kadison [212] that T (1) is a unitary v in ∆(B), so that v ∗ T (·) is a unital isometry of A onto B. They also show that for unital surjective isometries, T (xy + yx) = T (x)T (y) + T (y)T (x) for all x, y ∈ A. See also [196]. If A is a unital C ∗ -algebra, then a = sup |ϕ(a)| for a ∈ A+ , the supremum over states ϕ of A. There is a result resembling this which is true if A is an operator algebra. Namely, deﬁne a matrix state of A to be a completely contractive unital ϕ : A → Mn . Then a = sup ϕ(a) for any a ∈ A, where the supremum is taken over matrix states of A. A similar formula holds for a ∈ M n (A). Indeed, suppose that A ⊂ B(H), that a ∈ Mn (A), and consider a ﬁnite subset of vectors in H on which a ‘achieves its norm’ within . Let K be the span of these vectors, let PK be the projection onto K, and let ϕ(x) = PK x|K . Since B(K) ∼ = Mm for some m, we may view ϕ as a matrix state on A. The result is now easy (see [69] and [394]). If A is an operator algebra which is not approximately unital, we say that a homomorphism π : A → B(H) is ∗-nondegenerate if the span of terms of the form c1 c2 · · · cn ζ, for ζ ∈ H and ci ∈ π(A) ∪ π(A)∗ , is dense in H. The universal representation deﬁned in 2.4.4 is a good example of a ∗-nondegenerate representation. For operator algebras with cai, one can show that ∗-nondegeneracy coincides with our earlier notion of nondegeneracy. Indeed, it is not hard to see using A.1.5 that an operator algebra A possesses a cai if and only if whenever π : A → B(H) is a ∗-nondegenerate completely contractive homomorphism, and whenever x ∈ H, then x ∈ [π(A)x] (see also [72]). Our book focuses on approximately unital operator algebras for the most part, however some analoguous theory of operator algebras with a one-sided cai may be found in [55] and [60]. 2.2: The constructions in this section are well-known, with the exception of a few of the facts mentioned in 2.2.3. An early study of the spatial tensor product of nonselfadjoint algebras may be found in [316]. The constructions in 2.2.9 and

98

Notes and historical remarks

2.2.10 we have heard attributed to Arveson. The norm relation in 2.2.11 is wellknown (e.g. see Chapter IV Section 2 in [159] for a generalization); its usage in 2.2.11 is taken from [279]. A frequently used variant of the algebra in 2.2.10, is its subalgebra with repeated diagonal entries. Simple modiﬁcations to 2.2.1 show that the ‘c0 -sum’ (see 1.2.17) of operator algebras is an operator algebra. Many other general operator algebra constructions have been considered by others. As just one example, we mention the work of Gilfeather and Smith (see [170, 385], and references therein; in the former paper the automorphisms of certain operator algebras are parametrized and studied using the weak* Haagerup tensor product of 1.6.9). 2.3: The BRS theorem is due to Blecher, Ruan, and Sinclair [69]. The nonunital case appeared ﬁrst in [373]. The proof given here is due to Le Merdy [246]; we have presented this proof in part because of its capacity to generalize. In fact one need not assume associativity of the product in the BRS theorem, this is automatic as proved in the original paper [69]. That the quotient Banach algebra of an operator algebra A by an ideal is isometrically isomorphic to an operator algebra, was proved by Cole when A is a function algebra [424], and then extended to general operator algebras by Bernard [34] and Lumer [259] in the early 1970s. A direct proof of this may also be found in [337], and it is used there to give an alternative proof of the BRS theorem. It is an interesting open question as to whether every unital Banach algebra for which von Neumann’s inequality holds, is an operator algebra (e.g. see [125]). Interpolation of operator algebras considered as Banach algebras was brieﬂy studied in [415]. See [44, 62] for interpolation in the sense considered here. Nonselfadjoint direct limit operator algebras have been well studied; Power used the BRS theorem to construct such algebras in [350]. In [241] there is an explicit identiﬁcation of the ‘C ∗ -envelope’ (see Section 4.3) of an interesting class of direct limit algebras. See also [111, 112, 351] for example. Matrix normed algebras were ﬁrst mentioned in [42]; it was observed there that the Fourier algebras of a locally compact group were examples of these. They were used extensively in the middle 1990s by Eﬀros, Ruan, and Kraus, under the name completely contractive Banach algebra, in order to study Banach algebras which have an operator space structure of speciﬁc interest. For example, see [377] or Chapter 16 of [149], and references therein. 2.4: The name ‘C ∗ -cover’ is due to Muhly we believe, however the concept ∗ itself was developed by Arveson. The maximal universal C ∗ -algebra Cmax (A) of an operator algebra A was mentioned in [67, 64], and it (and the associated ‘universal representation of A’) played a crucial role in [50, 52, 72]. The latter ∗ (A), and paper treated the nonunital case ﬁrst. Another construction of Cmax Example 2.4.5, may be found in [50]. Duncan has established some functorial ∗ (·), and computed it for certain interesting classes of algebras properties of Cmax (e.g. see [129]). Note that saying that one C ∗ -cover (B, j) is dominated by another (B , j ) in the ordering deﬁned in 2.4.1, is the same as saying that if one forms any poly-

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nomial p in variables coming from A and the set {a∗ : a ∈ A} of formal adjoints, then pB ≤ pB . In this case, the kernel I of the canonical ∗-homomorphism from B to B is called a boundary ideal of B for A. Note that then B /I ∼ = B as C ∗ -algebras, and indeed also as C ∗ -covers of A. We show in Theorem 4.3.1 that a minimal C ∗ -cover exists in this ordering. We will take this fact for granted in the following discussion. Let A be an operator algebra, and consider the family of all C ∗ -covers of A. This has two extremal elements, namely the maximal one ∗ (A) from 2.4.3, and a minimal one whose existence we are taking on faith, Cmax and which we denote by Ce∗ (A). For any C ∗ -cover (B, i) of A, we deﬁne KB to ∗ be the kernel of the ∗-homomorphism Cmax (A) → B provided by 2.4.2. It is easy to see from the universal property in 2.4.2 that (B, i) ≤ (B , i ) if and only if KB ⊂ KB . In particular (B, i) and (B , i ) are equivalent with respect to the canonical equivalence relation on C ∗ -covers, if and only if KB = KB . Next consider the ideal J deﬁned to be KB in the particular case when B = Ce∗ (A). Then by the above universal property of these covers, it is easy to see that K B ⊂ J for any C ∗ -cover B of A. Hence such KB is an ideal of J. Conversely, for any ∗ closed ideal K of J, it is easy to see that Cmax (A)/K is a C ∗ -cover of A. Putting all the above together, we have that the correspondence B → K B is a bijective order-reversing correspondence between the partially ordered set C(A) of equivalence classes of C ∗ -covers of A, and the set of closed ideals of J. The last set is a lattice. Indeed by [122] 3.2.2, this lattice is lattice isomorphic to the set of open sets in a topological space, namely the open sets in the spectrum Jˆ of the C ∗ -algebra J. We recall that the spectrum Jˆ is the set of the equivalence classes of irreducible representations of J. Thus we have shown that if A is an operator algebra, then C(A) is a complete lattice, which is lattice anti-isomorphic to a certain topology. More precisely, C(A) is lattice anti-isomorphic to the set of open sets in the generally ˆ In this canonical way, one can associate a non-Hausdorﬀ topological space J. topological space with any operator algebra A. It is an interesting exercise to compute this topological space explicitly in the case that A = T 2 , using 2.4.5. The matrix factorization idea and several of the key examples in this section are from [67]. Some remarks on [67], and a list of other examples that may be treated in this framework, including the enveloping operator algebra discussed here, may be found in [45]. In connection with 2.4.14, it should be noticed that since von Neumann’s inequality does not extend to the case of three commuting contractions [416], the unital operator algebra O(N30 ) cannot be isometrically identiﬁed with the tridisc algebra A(D3 ) ⊂ C(T3 ). Pisier put the matrix factorization idea to excellent use in his notion of similarity degree—see [337] and references therein. He also has various extensions and complements to this circle of ideas in [337]. For example, the notion of the ‘universal operator algebra’ O(E) of an operator space E is thoroughly developed there, together with its relations to the topics above. See [250] for some versions appropriate to the w ∗ -topology. Paulsen has also further exploited such factorization formulae in a series of recent papers. For example, he cleverly uses such formulae to recover and extend

100

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a result of Agler related to interpolation. See [314, Chapter 18], [312, 313], and references therein. One may deﬁne the noncommutative maximal ideal space of any Banach al∗ (O(B)). This gebra which is also an operator space, to be the C ∗ -algebra Cmax is easily seen to have the following universal property: if π : B → D is any completely contractive homomorphism into a C ∗ -algebra D, then there exists a ∗ (O(B)) → D such that π ˜ ◦ j = π. (necessarily unique) ∗-homomorphism π ˜ : Cmax A variant of 2.4.20 (2) and 2.4.21, in the C ∗ -algebra case, was ﬁrst proved by Christensen, Eﬀros, and Sinclair [92]. The version presented here is due to Pisier [329]. The ‘unital free product operator algebra’ discussed in this section is ‘amalgamated over the scalars’. If one does not amalgamate, one obtains another free product, or ‘coproduct’, of operator algebras. See [337] and references therein. We have also not discussed ‘reduced free products’ here (e.g. see [418]). It is also shown in [67] that the free product operator algebra construction is injective. That is, if Ai is a unital-subalgebra of Bi for i = 1, 2, then A1 ∗ A2 ⊂ B1 ∗ B2 completely isometrically. It is fairly clear that if G1 and G2 are two discrete semigroups, then the universal properties in 2.4.9 and 2.4.18 show that O(G1 ) ∗ O(G2 ) = O(G1 ∗ G2 ). 2.5: Most Banach algebras are not Arens regular; for example, see [106] and [130]. The latter is a good survey of the Arens product and the basic features of Arens regularity. For instance if G is an inﬁnite locally compact Hausdorﬀ group, then the group algebra L1 (G) is not Arens regular [438]. See [107] (and references therein) for more on this theme. A version of the result 2.5.6 appeared in [144] (one also needs 1.4.11 to make the results match precisely). The fact 2.5.8 has perhaps been known since [97]. The principle 2.5.10 is from [65], with a diﬀerent proof. It is easy to see that if A is an Arens regular matrix normed algebra, then A∗∗ is a matrix normed algebra too. 2.6: The two-sided multiplier algebra of an operator algebra was ﬁrst studied by Poon and Ruan [344] in terms of the double centralizer algebra. This object may be deﬁned just as in the Banach algebra case (see [297, 106]). Poon and Ruan check that with natural matrix norms, the double centralizer algebra is a unital operator algebra containing A completely isometrically, and that it is isomorphic to the appropriate subalgebras of A∗∗ and B(H) (for nondegenerate completely isometric representations of A on H). One-sided and two-sided multiplier algebras of operator algebras were developed from a diﬀerent angle around the same period by Blecher, Muhly, and Paulsen as a necessary tool for the theory of nonselfadjoint operator algebra Morita equivalence (see [65, 46, 309]). Proposition 2.6.12 is taken from [46]. Kraus and Ruan discuss multiplier algebras for a matrix normed algebra in [236]. Item 2.6.17 is well known. We leave it as an exercise (using 2.1.6, say) that if A is an approximately unital operator algebra, then its diagonal (2.1.2) satisﬁes ∆(A) = A ∩ ∆(M (A)) = A ∩ ∆(LM (A)). As another exercise, the reader might check that for a unital operator algebra A

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and a cardinal I, we have LM (KI (A)) = Mwr I (A) (see 2.2.3 for this notation). wc Note that this implies that M (KI (A)) = Mwr I (A) ∩ MI (A) for such A. If A is nonunital then these do not hold necessarily; in this case we can add to the list of inﬁnite matrix operator algebras in 2.2.3, the algebras LM (KI (A)), RM (KI (A)), and also the subalgebras CIw (A)RI (A) and CI (A)RIw (A). 2.7: These results are due to Le Merdy [248, 249], with the following exceptions. The separate w∗ -continuity assertion in Theorem 2.7.9 follows immediately from the notion of operator space multipliers and Magajna’s result 4.7.1. It had been ﬁrst established in the unital case by Blecher in [54]. Corollary 2.7.16 is also from the latter paper. The proof of Theorem 2.7.9 shows that an operator algebra which is a dual Banach space is isometrically w ∗ -homeomorphic to a dual operator algebra if and only if the product is separately w ∗ -continuous. Ruan ﬁrst used the canonical projection from M ∗∗ to M to study dual operator algebras [372, Lemma 2.2]. The fact that the predual of a dual operator algebra need not be unique is probably quite old (we heard it ﬁrst mentioned by Westwood, using the U (X) trick too), and was further investigated by Ruan [372]. Godefroy’s work on unique preduals (e.g. see the survey [171]) may also be combined with the U(X) trick. The ﬁrst example of an operator space which is a dual Banach space but not a dual operator space was found by Le Merdy in [244]. The later example described in Lemma 2.7.15 is a variation on examples found independently by Peters and Wittstock [323], and Eﬀros, Ozawa, and Ruan [141]. We are grateful to Neufang for communicating the example of Peters and Wittstock to us. Proposition 2.7.11 was independently obtained by Arias and Popescu [16]. Observation 2.7.14 is new, and is related to [172, Theorem II.1]. By an almost identical argument to that of 2.4.2, if M is a dual operator algebra then one may deﬁne a maximal W ∗ -algebra W ∗ (M ). There is a w∗ continuous completely isometric homomorphism j : M → W ∗ (M ) whose range generates W ∗ (M ) as a W ∗ -algebra, and which possesses the following universal property: given any w∗ -continuous representation π : M → B(H), there exists a w∗ -continuous ∗-representation of W ∗ (M ) on H extending π. See [72]. General dual function algebras are discussed in [257,443,83], for example. Indeed by Theorem 2.7.9, the objects called uniform dual algebras by those authors, are exactly the function algebras with a Banach space predual. For a sample of other recent results on general dual operator algebras the reader might consult [54], [55] (for dual algebras with a one-sided identity), [113] (and references therein), or [352, Theorem 4]. The ideas in the proof of 2.7.9 break down for general Banach algebras. Indeed, the related question for Banach algebras may be phrased: Does a unital Arens regular Banach algebra with a predual, necessarily have a separately w∗ -continuous product? We are grateful to Lau for showing us a counterexample.

3 Basic theory of operator modules

3.1 INTRODUCTION TO OPERATOR MODULES Operator modules are one of the main themes of our book. One is presented immediately with operator modules when one ﬁrst meets operator algebras, in the form of the Hilbert spaces on which they act. Then as one gets deeper into the theory of operator algebras one ﬁnds that they arise naturally in many places, and provide an important tool for the analysis of operator algebras. Indeed they form a class that is large enough to include the Hilbert modules just mentioned, the operator algebras themselves, and important subclasses such as the C ∗ -modules; and yet is not so large that one loses too much contact with the underlying Hilbert space geometry. For example, we shall see in Chapter 8 that most basic C ∗ -module constructions, such as their usual tensor products, are in essence operator module constructions, but are not Banach module constructions. In this chapter we treat the basics of the ‘completely isometric’ theory of modules. Thus in this chapter if X and Y are (left, say) A-modules over an algebra A which are also operator spaces, then we consider these modules to be ‘the same’ if X and Y are completely isometrically isomorphic via an A-module map. In this case we say that X and Y are completely isometrically A-isomorphic, and we write X ∼ = Y completely A-isometrically. A similar terminology will be used for bimodules. We shall not discuss the ‘completely isomorphic’ theory of modules here; this will be touched on in Section 5.2. Unless stated otherwise, throughout the ﬁrst section of this chapter, the symbols A, B are reserved for algebras which are at least Banach algebras with an operator space structure. They do not necessarily have any kind of identity or approximate identity. This class includes of course all operator algebras. 3.1.1 (Operator modules) A concrete left operator A-module is a linear subspace X ⊂ B(K, H), which we take to be norm closed as always, together with a completely contractive homomorphism θ : A → B(H) for which θ(A)X ⊂ X. Such an X is a left A-module via θ. An (abstract) left operator A-module is an operator space X which is also a left A-module, such that X is completely isometrically A-isomorphic to a concrete left operator A-module. This is equivalent to saying that there is a complete isometry Φ : X → B(K, H), for some Hilbert spaces H, K, and a completely contractive homomorphism θ : A → B(H) such that Φ(ax) = θ(a)Φ(x) for any a ∈ A and x ∈ X.

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Similar deﬁnitions hold for right modules and bimodules. Thus a concrete right operator B-module is a subspace X ⊂ B(K, H) with Xπ(B) ⊂ X for a completely contractive homomorphism π : B → B(K). For a concrete operator A-B-bimodule we have both θ(A)X ⊂ X and Xπ(B) ⊂ X. In fact it is often convenient to treat one-sided modules as bimodules. This is done by simply deﬁning another action of the scalars C on X in the obvious way. 3.1.2 (Examples of operator modules) (1) If H, K are Hilbert spaces, and if θ : A → B(H) and π : B → B(K) are completely contractive homomorphisms, then B(K, H) is a concrete operator A-B-bimodule (with the canonical module actions). (2) Submodules of operator modules are clearly operator modules. (3) Any operator space X is an operator C-C-bimodule. (4) Any operator algebra A is an operator A-A-bimodule. (5) If A is an approximately unital operator algebra and if p, q are projections in A (or projections in the multiplier algebra M (A)), then qAp is an operator (qAq)-(pAp)-bimodule. If q = 1 − p, then we say as in 2.6.14, that qAp is a corner of A. We shall in fact see in 3.3.4 that these are essentially the only examples of operator bimodules over approximately unital operator algebras. If A is a C ∗ -algebra, then the operator bimodule qAp is an example of a ‘right C ∗ -module’ over pAp (see Chapter 8), and conversely we shall see in 8.1.19 that every right C ∗ -module is a corner of a C ∗ -algebra. (6) A closed left or right ideal J in a C ∗ -algebra A is an operator bimodule over JJ and J J. Here J denotes the adjoint space of J (see 1.2.25). More generally, let Z be a norm closed linear subspace of B(H), or of B(K, H) for two Hilbert spaces H and K, such that ZZ Z ⊂ Z. We say that such Z is a ternary ring of operators, or TRO. Such a space Z is an operator bimodule over the C ∗ -algebras ZZ and Z Z. This example will be important in Sections 4.4 and 8.3. 3.1.3 (h-modules) Let X be an operator space which is also a left A-module. We say that X is a (left) h-module over A if the module action on X, viewed as a linear mapping A ⊗ X → X, extends to a complete contraction A ⊗h X −→ X .

(3.1)

Since the Haagerup tensor product linearizes completely bounded bilinear maps (see 1.5.4), the last line is also equivalent to saying that the spaces M n (X) are left Banach Mn (A)-modules in the canonical way, for every n ∈ N. That is, axn ≤ an xn ,

n ≥ 1, a ∈ Mn (A), x ∈ Mn (X).

(3.2)

Similar deﬁnitions hold for right modules. Then an h-bimodule over A and B is by deﬁnition an A-B-bimodule which is both a left h-module over A and a right h-module over B. Any left operator A-module is an h-module over A. Indeed suppose that X is a θ(A)-submodule of B(K, H), for a completely contractive θ as in 3.1.1. Then for

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a ∈ Mn (A) and x ∈ Mn (X), we have θn (a)xn ≤ θn (a)n xn ≤ an xn . Thus we have veriﬁed (3.2). Of course a similar result holds for right modules, and hence for bimodules. In Section 3.3 and in 4.6.7, we will prove that conversely, if A and B are approximately unital and if X is a nondegenerate A-B-bimodule, then X is an operator A-B-bimodule provided that the actions A ⊗h X → X and X ⊗h B → X are both completely contractive. 3.1.4 (Matrix normed modules) It is natural in view of the discussion in A.6.1 to consider the A-modules that correspond to completely contractive homomorphisms A → CB(X), for a operator space X. We shall call these matrix normed modules. It is easy to see that these are exactly the left A-modules X, such that X is also an operator space, and such that also [aij xkl ]nm ≤ [aij ]n [xkl ]m

(3.3)

for all [aij ] ∈ Mn (A), [xkl ] ∈ Mm (X), and all m, n ∈ N. By 1.5.11, a reformulation of (3.3) is that the module action on X extends to a complete contraction

A ⊗ X −→ X.

(3.4)

Similar deﬁnitions hold for right matrix normed modules. Then a matrix normed A-B-bimodule is by deﬁnition an A-B-bimodule which is both a (left) matrix normed A-module and a (right) matrix normed B-module. 3.1.5 (Examples of matrix normed modules) (1) Every h-module, and hence every operator module is a matrix normed module. To see this we simply compare (3.1) and (3.4), and we apply 1.5.13. Similar assertions hold for bimodules. (2) If X is a matrix normed A-B-bimodule, then X ∗ is a matrix normed B-A-bimodule for the dual actions deﬁned by bϕ, x = xb, ϕ

ϕa, x = ϕ, ax

and

for a ∈ A, b ∈ B, x ∈ X, ϕ ∈ X ∗ . That this is a matrix normed bimodule is a simple exercise using the deﬁnitions (see 1.2.20). (3) Any matrix normed algebra A (see 2.3.9) is a matrix normed A-module over itself. More generally, if X is an operator space, then A ⊗ X is a left matrix normed A-module. To see this we appeal to (3.4). That the obvious left A-module action on A ⊗ X satisﬁes (3.4) follows from the ‘associativity’ and ‘functoriality’ of the operator space projective tensor product (see 1.5.11):

m⊗IX A ⊗ (A ⊗ X) ∼ = (A ⊗ A) ⊗ X −→ A ⊗ X

where m is the product mapping on A. A similar argument shows that if Y is any left matrix normed A-module, then Y ⊗ X is a left matrix normed A-module. In the case that X = Sn1 , the dual of Mn , then we write Sn1 [A] for A ⊗ Sn1 . This

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is a basic ‘building block’ in the category of matrix normed modules over A, as we will see in 3.4.7. (4) Suppose that X is a Banach A-module over a Banach algebra A. In 2.3.9 we showed that Max(A) is a matrix normed algebra. By a similar argument (using 1.5.12 (2) and (3.4)) it is easy to see that Max(X) is a matrix normed Max(A)-module. Thus the class of matrix normed modules contains the class of Banach A-modules in some sense. This shows that one usually cannot expect very strong general theorems to hold for all matrix normed modules. 3.1.6 (Hilbert modules) These constitute one of the simplest and most important classes of operator modules. We deﬁne a Hilbert A-module to be a Hilbert space H which is a left A-module whose associated homomorphism θ : A → B(H) is completely contractive. If A is a C ∗ -algebra then the last condition is equivalent to saying that θ is a ∗-homomorphism, as we noted in 1.2.4. We sometimes write such a Hilbert module as Hθ . Some authors call this a completely contractive Hilbert module, and are also interested in the case when θ is merely bounded or completely bounded. We will be more restrictive. Unless otherwise stated we will assume that our Hilbert modules are nondegenerate, which by 2.1.5 is equivalent to saying that θ is nondegenerate. Moreover a Hilbert module H will always be viewed as an operator space by assigning it the ‘column Hilbert space structure’ H c discussed in 1.2.23. It is easy to see that Hilbert modules are exactly the left operator modules from Example 3.1.2 (1) in the case that K = C. An alternative, and often useful, identiﬁcation of Hilbert modules with certain concrete operator modules proceeds as follows. Let θ : A → B(H) be a completely contractive homomorphism, let η be a ﬁxed unit vector in a Hilbert space K, and let H ⊗ η be deﬁned as in 1.2.23, namely, the set of rank one operators {ζ ⊗ η ∈ B(K, H) : ζ ∈ H}. Being left invariant under θ(A), H ⊗ η is a concrete left operator A-module. By 1.2.23 this module is completely isometric to H c , and it is clear that this isometry is also an A-isomorphism. The next result is a useful characterization of Hilbert modules. Proposition 3.1.7 Let H be a Hilbert space, let θ : A → B(H) be a nondegenerate bounded homomorphism, and consider H c as a left A-module in the canonical way (via θ). The following are equivalent: (i) θ is completely contractive, (ii) H c is a left operator A-module, (iii) H c is a left matrix normed A-module. Proof That (i) implies (ii) follows from Example 3.1.2 (1) as we noted in 3.1.6. That (ii) implies (iii) follows from 3.1.5 (1). That (iii) implies (i) follows from the fact that B(H) = CB(H c ) completely isometrically (see (1.14)), and from the ﬁrst few lines of 3.1.4. 2 3.1.8 (The ∞-direct sum of operator modules) For a family {Xi : i ∈ I} of operator A-B-bimodules, the ∞-direct sum ⊕i Xi discussed in 1.2.17 is an operator A-B-bimodule. This is easily seen by an argument analoguous to the

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one in 2.2.1, using, for example, a property of the ∞-sum mentioned in the last paragraph of 1.4.13. Similarly, ⊕i Xi is an operator bimodule over ⊕i Ai and ⊕i Bi , if each Xi is an operator Ai -Bi -bimodule. 3.1.9 (The 1-direct sum of matrix normed modules) Let {Xi : i ∈ I} be a family of left matrix normed A-modules say. The 1-sum ⊕1i Xi deﬁned in 1.4.13, with the obvious left A-module action, is a left matrix normed A-module. One way to see this is to note that the canonical complete contractions A ⊗ Xi → Xi give, via the universal property of the 1-direct sum (see 1.4.13), a complete contraction ⊕1i (A ⊗ Xi ) → ⊕1i Xi . Putting this together with (1.52) yields a canonical complete contraction

A ⊗ (⊕1i Xi ) −→ ⊕1i Xi . Appealing to (3.4) we see that ⊕1i Xi is a left matrix normed A-module. 3.1.10 (Quotient modules) Let X be an operator space, and suppose that X is also an A-B-bimodule. If Y is an A-B-submodule of X (as always, assumed to be closed), then the quotient operator space (see 1.2.14) X/Y is an A-B-bimodule in the canonical way. This is simple algebra, indeed if q : X → X/Y is the canonical quotient map then q is an A-B-bimodule map too. It is also easy to see that if in addition A and B act nondegenerately on X, then they also act nondegenerately on X/Y . This follows from the simple fact that if V is a dense subspace of X then q(V ) is dense in X/Y . If X is a matrix normed A-B-bimodule, then we claim that X/Y is a matrix normed A-B-bimodule too. To see this suppose that z ∈ Mm (X/Y ) and a ∈ Mn (A), both with norm < 1. Then there is a matrix x ∈ Mm (X) of norm < 1 which is mapped to z by the quotient map. Then [aij xkl ] has norm < 1 in Mnm (X), so that [aij zkl ] = qnm ([aij xkl ]) has norm < 1 in Mnm (X/Y ). Thus X/Y is a matrix normed A-module, and similarly it is a matrix normed B-module. A similar proof shows that the quotient of an h-bimodule is an h-bimodule. 3.1.11 (Multiplier actions) There is a useful way to make a nondegenerate module over an approximately unital algebra, into a module over a unital algebra. For example, suppose that X is a nondegenerate left Banach module over an approximately unital Banach algebra A. As at the beginning of Section 2.6, we deﬁne LM (A) = BA (A), and we deﬁne a left action of LM (A) on X by setting ηx = η(a)x if η ∈ LM (A) and x = ax with a ∈ A, x ∈ X. Notice that ηx = η(a)x = lim η(et a)x = lim η(et )x, t

t

(3.5)

if (et )t is any cai for A. This together with Cohen’s factorization theorem A.6.2 shows that this action of LM (A) on X is well deﬁned. Indeed from (3.5) it is easy to check that X is a nondegenerate left Banach LM (A)-module. An almost identical argument shows that if X is a nondegenerate left matrix normed module

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over an approximately unital matrix normed algebra A, then the action described in (3.5) makes X into a nondegenerate left matrix normed LM (A)-module. Assume now that in addition X is a nondegenerate left operator A-module. We may assume that X is a θ(A)-submodule of B(K, H) for some Hilbert spaces K, H and a completely contractive homomorphism θ : A → B(H). By replacing H by its subspace [XK], which is θ(A)-invariant, we may assume that θ is nondegenerate. By deﬁnition, H c is a Hilbert A-module with A-module action deﬁned via θ. Hence H c is a matrix normed A-module by 3.1.7. By the last paragraph, H c is canonically a matrix normed LM (A)-module. Using 3.1.7 again, we see that we have extended θ to a completely contractive unital homomorphism θ˜: LM (A) → B(H). Thus B(K, H) becomes a left operator LM (A)-module. With notation as in (3.5) we have ˜ ˜ ˜ ηx = θ(η(a))x = θ(ηa)x = θ(η)θ(a)x = θ(η)x.

˜ Thus θ(LM (A))X ⊂ X and the resulting action coincides with the one deﬁned in (3.5). Thus X is actually a left operator LM (A)-submodule of B(K, H). In particular, a nondegenerate left operator A-module is also a left operator module over M (A) or over the unitization A1 , if A is an operator algebra, say. 3.1.12 (Prolongations) Let A, B be Banach algebras and assume that X is a left Banach A-module and that θ : B → A is a contractive homomorphism. If we deﬁne bx = θ(b)x for x ∈ X and b ∈ B, then X becomes a left Banach B-module. This B-module is called the prolongation of X by θ, and is sometimes written as θ X. It is easy to see that if A, B, X are also operator spaces and if θ is completely contractive, then θ X is an operator B-module (resp. matrix normed B-module) if X is an operator A-module (resp. matrix normed A-module). Assume now that A, B are approximately unital matrix normed algebras and that X is a nondegenerate left operator A-module. If θ : B → LM (A) is a completely contractive homomorphism, then using 3.1.11 and the considerations in the last paragraph, we see that X becomes a left operator B-module. If θ is a left multiplier-nondegenerate morphism (in the sense of 2.6.11) from B to LM (A), then X is easily seen to be a nondegenerate left operator B-module. A commonly encountered particular case of this, occurs when A is a left ideal in B. Since B is a matrix normed algebra, the canonical map from B into CBA (A) is a completely contractive homomorphism. Hence X becomes an operator module over B, which is nondegenerate if B is approximately unital. 3.1.13 (Matrix spaces) If X is a left operator module or matrix normed module, and if I, J are sets, then there are two obvious ways to attempt to make the matrix space MI,J (X) discussed in 1.2.26 into a module, namely via the formulae: a[xij ] = [axij ] or [aij ][xij ] = aik xkj , k

assuming that these are deﬁned. Similarly for right modules or bimodules. It is these actions that are referred to in the next result.

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Proposition 3.1.14 If X is an operator (resp. matrix normed) A-B-bimodule, and if I, J are sets, then KI,J (X) and MI,J (X) are operator (resp. matrix normed) A-B-bimodules. If A and B are operator algebras, and if X is an operator A-B-bimodule, then the following also hold: (1) KI,J (X) and MI,J (X) are operator KI (A)-KJ (B)-bimodules, wr (2) MI,J (X) is an operator Mwc I (A)-MJ (B)-bimodule (see 2.2.3). (3) KI,J (X) and MI,J (X) are operator MI -MJ -bimodules. Proof We treat only the left module action, the right being similar. We ﬁrst prove (2). Suppose that X is an operator A-B-bimodule. We may assume that X is a θ(A)-submodule of B(K, H), where θ : A → B(H) is a completely contractive homomorphism. Using (1.19), we will identify MI,J (B(K, H)) and MI (B(H)) with B(K (J) , H (I) ) and B(H (I) ) respectively. Thus we have a completely isometric embedding MI,J (X) ⊂ B(K (J) , H (I) ). Consider the ‘ampliﬁcation’ θI , a (I) ), this is completely contractive (see 1.2.26). It is map from Mwc I (A) to B(H an easy exercise to check that θI is a homomorphism. By (1.21) it is clear that (J) , H (I) ). We have now established (2). MI,J (X) is a Mwc I (A)-submodule of B(K The second assertion in (1) follows from (2), since, for example, KI (A) is a subalgebra of Mwc I (A). To see the other assertion in (1), note that the product of two ‘ﬁnitely supported’ matrices in MI (A) and MI,J (X) respectively, is clearly ﬁnitely supported, and thus is in KI,J (X). By density considerations we must have KI (A) KI,J (X) ⊂ KI,J (X). Taking A = B = C in (2) gives the second assertion in (3). One way to see the ﬁrst part of (3) is to ﬁrst note that KI,J (X) is an operator KI -KJ -bimodule, by the ﬁrst part of (1), and then to use 3.1.11. We ﬁnally prove the ﬁrst part of the proposition. Here we only assume that A, B are Banach algebras with an operator space structure. If X is an operator A-B-bimodule then a variant of the proof of (2) above shows that M I,J (X) is an operator A-B-bimodule, and KI,J (X) is clearly an A-B-submodule. Consider now the case that X is merely a matrix normed A-B-bimodule. If I and J are ﬁnite integers then it is quite obvious from a formula such as (1.5) that MI,J (X) also is a matrix normed A-B-bimodule. In the case that I, J are inﬁnite sets, we observe that since the norm in Mnm (MI,J (X)) ∼ = MI,J (Mnm (X)) is the supremum of the norm of ﬁnite submatrices, the calculation boils down to the previous case. Thus MI,J (X) is a matrix normed A-B-bimodule. As before, KI,J (X) is a submodule, and hence is also a matrix normed A-B-module. 2 Corollary 3.1.15 If A is an approximately unital operator algebra, then CI (A) is a nondegenerate operator KI (A)-A-bimodule, and RI (A) is a nondegenerate operator A-KI (A)-bimodule. 3.1.16 (Opposite and adjoint modules) These constructions allow one to make a left module into a right module, and vice versa. We will need the notation in 1.2.25 and 2.2.8. Assume for simplicity that A is an operator algebra. If X is a left operator A-module (resp. matrix normed A-module), then X op is canonically a right operator module (resp. matrix normed module) over Aop . Similarly, the

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adjoint operator space X is a right operator module (resp. matrix normed module) over the adjoint operator algebra A . We leave these assertions as exercises for the interested reader. 3.2 HILBERT MODULES We characterized Hilbert modules in 3.1.6 and the proposition following it. In this section we say a little more about their theory and applications. 3.2.1 (Submodules and quotients of Hilbert modules) It is clear that a submodule (as always, assumed to be closed) of a Hilbert A-module is a Hilbert A-module. Also, the quotient of a Hilbert A-module by a closed submodule is a Hilbert A-module too. This may be seen by combining 3.1.7 with the fact that the classes of nondegenerate matrix normed A-modules and Hilbert column spaces are both closed under quotients (see 3.1.10 and 1.2.23). However there is another way to see this fact which is more informative. Of course a submodule K of a Hilbert A-module H is nothing more than a θ(A)-invariant subspace, where θ : A → B(H) is the associated homomorphism. Let PK ⊥ denote the projection onto K ⊥ = H K. Then the completely contractive map π = PK ⊥ θ(·)|K ⊥ from A into B(H K) is a homomorphism. Indeed this follows from the ‘general principle’ proved in the lines following (2.26) in 2.7.10. This makes H K into a Hilbert A-module and it is easy to check that this A-module is completely A-isometrically isomorphic to the quotient module H/K discussed in 3.1.10. If θ is a representation of A on a Hilbert space H, then a closed subspace L of H is called a semi-invariant subspace for this representation, if PL θ(·)|L is a homomorphism from A into B(L). Here PL is the projection onto L. Proposition 3.2.2 (Sarason) Let θ : A → B(H) be a representation. A subspace L of H is a semi-invariant subspace for θ if and only if there exist two A-submodules H2 ⊂ H1 ⊂ H such that L = H1 H2 . Proof The ‘if part’ follows from the above discussion, with H and K replaced by H1 and H2 respectively. Conversely, if L is a semi-invariant subspace of H, let H1 be the closure of the span of L and θ(A)L, and let H2 = H1 L. Then L = H1 H2 . Let P be the projection of H onto L. For a, b ∈ A and ζ ∈ L, P θ(a)(I − P )θ(b)ζ = P θ(a)θ(b)ζ − P θ(a)P θ(b)ζ = 0 by hypothesis. Clearly P θ(a)(I − P )L = {0}. Thus by linearity and continuity, P θ(a)(I − P )H1 = {0}. Hence P θ(a)H2 = {0}, so that θ(A)H2 ⊂ L⊥ . Since H1 is θ(A)-invariant, we see now that H2 is θ(A)-invariant. 2 3.2.3 (Reducing submodules) Let H be a Hilbert A-module with associated homomorphism θ : A → B(H)), and let K be an A-submodule, that is, a closed θ(A)-invariant subspace. We say that K is a reducing submodule (or is A-reducing) if its orthogonal complement K ⊥ also is θ(A)-invariant. We also use the term summand for such a K. It is easy to see that K is reducing if and only

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if θ(a) commutes with PK ⊥ for any a ∈ A. In that case, the restriction of θ(·) to K ⊥ maps into K ⊥ , and the resulting homomorphism mapping into B(K ⊥ ) coincides with the homomorphism π = PK ⊥ θ(·)|K ⊥ discussed in 3.2.1. Hence the submodule K ⊥ coincides with the quotient module H/K. If A is a C ∗ -algebra, then so is θ(A) by A.5.8. Thus in this case, every closed A-submodule of H is A-reducing. 3.2.4 (Direct sums of Hilbert A-modules) It is clear that the usual Hilbert space direct sum ⊕i∈I Hi of a family {Hi : i ∈ I} of Hilbert A-modules, is again a Hilbert A-module. This simply corresponds to the usual ‘direct sum’ of the associated representations. If each Hi is a copy of a single Hilbert module H, then according to 1.1.4 we write H (γ) for ⊕i∈I Hi , if γ is the cardinal associated with I. We call this a multiple of H. If θ : A → B(H) is the associated representation, then we write θ(γ) for the associated representation of A on H (γ) , and call this a multiple of θ. We will need the simple fact that θ(γ) (A)

w∗

=

w ∗ (γ) θ(A) .

(3.6) w∗

Indeed, suppose that T is in the right-hand set, that is, there exists S ∈ θ(A) (γ) with S (γ) = T . If (St ) is a net in θ(A) converging to S, then St → T , and so (γ) T is in the left-hand set. Conversely, if (St ) is a net in θ(A) such that St → T , then to see that T is in the right-hand set, note that if Pi is the projection (γ) onto the ith copy of H, then Pi St Pj → Pi T Pj in the w∗ -topology, for all i, j. (γ) (γ) However Pi St Pj = 0 if i = j, and Pi St Pi = St . 3.2.5 (The category HM OD) By 1.2.23, linear maps T between Hilbert modules are bounded (resp. isometric) if and only if they are completely bounded (resp. completely isometric) with respect to the usual operator space structure on the Hilbert modules (that is, their ‘column space’ structure). Indeed for such T , we have T = T cb. We deﬁne A HM OD to be the category of Hilbert A-modules, with bounded A-module maps as the morphisms. Sometimes these morphisms are called ‘intertwining maps’, since they ‘intertwine’ the associated representations. We say that two Hilbert A-modules are spatially equivalent, and write H ∼ = K, if they are isometrically A-isomorphic, that is if there exists a unitary A-module map from H onto K. We say that representations π, θ of A are quasi-equivalent if there is a multiple of π which is spatially equivalent to a multiple of θ. Thus two Hilbert A-modules H and K are quasi-equivalent if and only if there are cardinals α and β (which by basic set theory we may assume to be equal) such that H (α) ∼ = K (β) . If A is a C ∗ -algebra this coincides with the usual deﬁnition of quasi-equivalence found in C ∗ -algebra texts (e.g. see [122, 5.3.1]). 3.2.6 (Hilbert modules and noncommutative ring theory) The transplanting of important ideas from the purely algebraic theory of rings and modules (eg. see [8, 368]), into the theory of operator algebras, usually happens in two main

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ways, at least from the standpoint of this book. In both, the ring is of course replaced by an operator algebra A. In the ﬁrst method, one takes the A-modules to be operator modules or matrix normed modules, including A itself. This is the main method the reader will encounter in this book. In the second method one takes the modules to be Hilbert modules. Although there are some strong relations between these two methods, they are usually quite distinct. For example, the notion of ‘projective module’ means quite diﬀerent things in the two categories. As examples of the importance of the second method we cite the work of Rieﬀel and Connes on Hilbert space modules, which developed into the theory of correspondences (discussed brieﬂy in Section 8.5 below), the work of Douglas and coauthors on Hilbert modules over function modules (see Douglas and Paulsen’s text [127], or [126] and references therein, for recent progress), and the work of Muhly and Solel on Hilbert modules over operator algebras (e.g. see [281,279]). We are unfortunately not able here to touch on the deep and important ideas developed by these authors. Instead we shall give a simpler illustration of the transplanting of ring theoretic ideas via Hilbert modules. Namely, we devote most of the rest of this section to a discussion of topics connected to the algebraic concept of a ‘generator of a category’, leading to a generalization of von Neumann’s double commutant theorem to nonselfadjoint operator algebras. Henceforth in this section we assume that A is an approximately unital operator algebra, and we recall that all Hilbert modules are assumed nondegenerate. This is simply to make things clearer, the general case of the topics below is discussed in [72]. 3.2.7 (Universal Hilbert modules and generators) A Hilbert A-module H (or equivalently, the associated nondegenerate representation of A on H) is said to be A-universal, if every Hilbert A-module K is isometrically A-isomorphic (that is, spatially equivalent) to an A-reducing submodule of a direct sum of copies of H. We say that a module H in A HM OD is a generator (resp. cogenerator) for A HM OD if for every nonzero morphism R : K → L of A HM OD, there exists a morphism T : H → K (resp. T : L → H) of A HM OD with RT = 0 (resp. T R = 0). As in pure algebra (e.g. see [8]), there is a host of equivalent formulation of the notions of generator and cogenerator (see [72]). 3.2.8 (Examples of generators) We shall not use this, but if A is a C ∗ -algebra, then it is easy to see that a Hilbert A-module H is A-universal if and only if H is a generator for A HM OD, and if and only if the unique w∗ -continuous map from A∗∗ to B(H) extending the representation of A on H, is one-to-one. These assertions are easy to deduce from, for example, the later results 3.2.11 (6), 3.2.12, 3.8.5, and 3.8.6 and the remark following it. The situation is a little diﬀerent for general nonselfadjoint algebras. One can see that A-universal representations are generically ‘very large’. However generators and cogenerators may be quite ‘small’, as we see next. 3.2.9 (A generator for T 2 ) To illustrate the deﬁnition, we show that the direct sum of the usual two-dimensional representation of T 2 , the upper triangular 2×2

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matrices, with the one-dimensional representation of ‘evaluation at the 1-1 entry’, is a generator. There are more sophisticated ways of obtaining this fact, but to see it directly from the deﬁnition given here, suppose that R : K → L is a nonzero T 2 -module map between Hilbert T 2 -modules. We may write K = K1 ⊕ K2 , with the action of T 2 on K given by the expression ab ζ aζ + bSη = , 0c η cη for a linear map S : K2 → K1 . Here a, b, c ∈ C, ζ ∈ K1 , η ∈ K2 . Since R = 0 we have R|K1 = 0 or R|K2 = 0. First suppose that R|K1 = 0. Then the action of T 2 on K1 is just multiplication by the scalar in the 1-1 entry. Choose ζ in K1 (K1 ∩Ker(R)), ζ = 0, and take a nonzero map T : C3 → Cζ which is zero except on the third coordinate. It is easy to see that T is a module map into K1 , and RT = 0. Thus we may assume that R|K1 = 0 and R(η0 ) = 0 for some η0 ∈ K2 . Deﬁne a map T : C3 → K1 ⊕ Cη0 taking e1 to Sη0 , e2 to η0 , and e3 to 0. One easily checks that T is a module map into K, and RT = 0. 3.2.10 (A connection with the C ∗ -theory) Let A be an approximately unital ∗ (A) be its ‘maximal C ∗ -algebra’ (see 2.4.3). From operator algebra, and let Cmax ∗ (A)-modules, 2.4.2 it follows that may regard Hilbert A-modules as Hilbert Cmax and vice versa, in a canonical way. Suppose that H, K are Hilbert A-modules, that θ and π are the representations of A on H and K respectively, and that T : H → K is a bounded A-module map. Then H, K are Hilbert modules over the adjoint operator algebra A as well (see the last part of 2.2.8). If a ∈ A then π(a)T = T θ(a), and so T ∗ π (a∗ ) = T ∗ π(a)∗ = θ(a)∗ T ∗ = θ (a∗ )T ∗ . Hence T ∗ : K → H is an A -module map. Conversely, by symmetry, if T ∗ is an A -module map then T is an A-module map. We call a bounded A-module map T between Hilbert A-modules adjointable if T ∗ is also an A-module map. From the last two paragraphs we deduce that ∗ T is adjointable if and only if T is a Cmax (A)-module map. We shall not need adjointable maps very much, except in the special case that i is an isometric map between Hilbert A-modules, such that i and i∗ are A-module maps. In this case ∗ it follows from the last paragraph that i and i∗ are Cmax (A)-module maps. In particular, we deduce that unitary morphisms, that is, unitary A-module maps, ∗ (A)-module maps. Thus two Hilbert A-modules are spatially equivalent are Cmax ∗ (A)-modules. as A-modules if and only if they are spatially equivalent as Cmax From the above, it also follows that the set of A-reducing submodules of a Hilbert ∗ (A)-submodules of H. Indeed A-module H is the same as the set of closed Cmax Hilbert A-module direct sums (resp. summands) of Hilbert A-modules are the ∗ (A)-module direct sums (resp. summands). same as Hilbert Cmax We collect together several simple deductions:

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Proposition 3.2.11 Let A be an approximately unital operator algebra. ∗ (1) A Hilbert A-module H is A-universal if and only if H is Cmax (A)-universal. (2) The universal representation of A considered in 2.4.4 is A-universal. (3) Two Hilbert A-modules are quasi-equivalent as Hilbert A-modules if and only ∗ if they are quasi-equivalent as Hilbert Cmax (A)-modules. (4) Any two A-universal representations are quasi-equivalent. (5) If θ is a representation of A which is quasi-equivalent to an A-universal representation, then θ is an A-universal representation. (6) Any A-universal module is a generator and a cogenerator for A HM OD. (7) H is a generator for A HM OD if and only if H cogenerates A HM OD. (8) If π and θ are quasi-equivalent representations of A, then there exists a (necessarily unique) w∗ -homeomorphic completely isometric isomorphism

ρ : π(A)

w∗

−→ θ(A)

w∗

such that ρ ◦ π = θ.

Proof (1) and (3) are obvious from the discussion above. Item (2) follows from the discussion above too, together with the fact that the universal representation of a C ∗ -algebra C is C-universal in the sense above (see the last fact in A.5.5). Item (5) is clear by the deﬁnitions, or is an easy exercise. Assertion (4) is a simple application of set theory, and the well-known ‘Eilenberg swindle’. If H and K are two A-universal representations, then there exist cardinals α and β, and Hilbert A-modules M and N , such that H ⊕ M ∼ = K (α) (β) and K ⊕ N ∼ = H . By adding on extra multiples of H or K to the last two equations, we may suppose that α = β. We may also assume (by replacing by a larger cardinal if necessary) that α · α equals α. Then K (α) ∼ = K (α) ⊕ K (α) ⊕ · · · ∼ = H ⊕ M ⊕ H ⊕ M ⊕ ··· ∼ = H ⊕ K (α) ⊕ K (α) ⊕ · · · ∼ = H ⊕ K (α) by associativity. Since α · α = α, a multiple of the last equation yields that K (α) ∼ = H (α) ⊕ K (α) . Similarly, H (α) ∼ = K (α) ⊕ H (α) , hence H (α) ∼ = K (α) . Item (7) follows from the deﬁnitions of generators and cogenerators, and the ﬁrst two paragraphs of the discussion in 3.2.10. To prove (6), suppose that H is A-universal, and that R : K → L is a nonzero bounded A-module map. There is a cardinal α such that K may be identiﬁed with an A-reducing submodule of H (α) . Let Q be the associated projection onto K from H (α) . Let i be the inclusion map of H into H (α) as its ith summand H. If every map R ◦ Q ◦ i is zero, then R ◦ Q and R are zero, which is a contradiction. Thus H is a generator. Using (1) we have that H is A -universal. By the above and (7), H is a cogenerator. If π and θ in (8), are spatially equivalent, then (8) is clear. However this case, together with (3.6), easily implies the general case. 2 Thus the A-universal representations comprise one entire equivalence class for the equivalence relation of quasi-equivalence.

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3.2.12 (A property of universal representations) If a nondegenerate representation π : A → B(H) is A-universal, then we claim that its w ∗ -continuous extension w∗ π ˜ : A∗∗ → B(H) (see 2.5.5) is a complete isometry. Hence π(A) ∼ = A∗∗ . Indeed, if π = πu is the universal representation of A (see 2.4.4), then the claim follows immediately from the proof of 2.5.6. For a general A-universal representation θ of A, we use (4) and (8) of 3.2.11. By those results, there exists a w ∗ -continuous w∗

w∗

completely isometric isomorphism ρ : πu (A) → θ(A) such that ρ ◦ πu = θ. ˜ Thus θ˜ is a complete isometry. By w∗ -continuity, we deduce that ρ ◦ π u = θ. 3.2.13 (The double commutant property) We say that a subspace Z of B(H) has the double commutant property if its double commutant equals its w ∗ -closure. w∗ W OT ⊂ Z . We say In this case, it also equals the WOT-closure, since Z ⊂ Z that a nondegenerate completely contractive representation π of an algebra A, or the associated Hilbert module Hπ , has the double commutant property if the w∗

latter property holds for π(A). That is, if π(A) = π(A) in B(H). We make several simple observations. If Z ⊂ B(H) then, by the w ∗ -continuity w∗

w∗

of the involution (see A.1.2), we have Z = (Z ) . It is easy algebra to check that (Z ) = (Z ) . Hence Z has the double commutant property if and only if Z has the double commutant property. If K is another Hilbert space, then it is also easy algebra to see that (Z ⊗ IK ) = Z ⊗ IK in B(H ⊗ K). In particular, and using also (3.6), if π : A → B(H) is a nondegenerate completely contractive representation and if α is a cardinal, then π has the double commutant property if and only if π (α) has the double commutant property. Theorem 3.2.14 Let A be an approximately unital operator algebra. If H is a generator or cogenerator for A HM OD, then H has the double commutant property. In particular, for any A-universal representation π we have π(A) = π(A)

w∗

∼ = A∗∗ ,

completely isometrically.

Proof The last part follows from the preceding assertions, Proposition 3.2.11 (6), and 3.2.12. The ‘cogenerator case’ follows from the ‘generator case’ by Proposition 3.2.11 (7) and an assertion in 3.2.13. Thus we may suppose that H is a generator for A HM OD, with associated nondegenerate representation π. Let L = H (∞) be the countably inﬁnite multiple of H and let θ = π (∞) be its associated representation. It is easy to see that L is a generator for A HM OD too. Fix a nonzero ζ ∈ L, and consider K = [θ(A)ζ] ⊂ L. By 2.1.5 we have ζ ∈ K. Suppose that T ∈ θ(A) . If V : L → K is a bounded A-module map then V (regarded as a map into L) is in θ(A) . Thus T V (L) = V T (L) ⊂ K. Consequently, T maps the range of V into K. Claim: T (K) ⊂ K. To see this, let W be the closure in K of the span of the ranges of all such V as above. Clearly W is an A-submodule of K, and by the above, T (W ) ⊂ K. If W = K, consider the quotient Hilbert A-module K/W , and the canonical quotient map q : K → K/W . Since L is a generator, there exists a bounded A-module map V : L → K such that qV = 0. This contradicts the deﬁnition of W , and proves our claim.

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For ζ ∈ L, the Claim above shows that T ζ ∈ [θ(A)ζ] ⊂ [θ(A) the set θ(A)

w∗

, which coincides by (3.6) with π(A)

these facts together, T ∈ θ(A) w∗

w∗

w∗

w∗

ζ]. By A.1.5,

⊗ I∞ , is ‘reﬂexive’. Putting

. Thus we have shown that θ(A) is contained

in θ(A) . Since the other inclusion trivially holds, θ has the double commutant property. By 3.2.13, π also has this property. 2 Thus we have a formal generalization of von Neumann’s double commutant theorem. Indeed assume that A is a C ∗ -algebra and let π : A → B(H) be a w∗

nondegenerate ∗-representation. To deduce the identity π(A) = π(A) from the above proof, just observe that the submodule K ⊂ L automatically reduces A, as we noted in the last line of 3.2.3. Hence the orthogonal projection V from L onto K is an A-module map. Thus T ζ = T V ζ = V T ζ ∈ K for ζ ∈ L, which is what is needed to obtain the desired conclusion. A similar proof yields an analogue of 3.2.14 above, for any unital dual operator algebra M . In particular, there exist w ∗ -continuous completely isometric representations π of M with π(M ) = π(M ). See [72] for details. 3.3 OPERATOR MODULES OVER OPERATOR ALGEBRAS The next theorem, the main result in this section, is a variation on a theorem of Christensen, Eﬀros, and Sinclair. We will refer to it as the ‘CES theorem’. It gives a complete characterization of nondegenerate h-bimodules over a pair of approximately unital operator algebras. A generalization to bimodules over Banach algebras which are also operator spaces will be given in 4.6.7. Recall from 3.1.3 that if A, B are operator algebras, say, then an A-B-bimodule X is called an h-bimodule, if the module actions extend to complete contractions A ⊗h X → X and X ⊗h B → X. Call these maps u and v respectively. Thus we obtain from the functoriality and associativity of the Haagerup tensor product (see 1.5.5) a sequence of complete contractions A ⊗h X ⊗h B

u⊗IB

−→

v

X ⊗h B −→ X.

The composition of these maps is a complete contraction A ⊗h X ⊗h B → X, taking a ⊗ x ⊗ b to axb for a ∈ A, b ∈ B and x ∈ X. It is often convenient to formulate the bimodule action as a single map from A ⊗h X ⊗h B to X. Theorem 3.3.1 (CES theorem) Let A and B be approximately unital operator algebras, and let X be an operator space which is a nondegenerate h-bimodule over A and B. Then there exist Hilbert spaces H and K, a completely isometric linear map Φ : X → B(K, H), and completely contractive nondegenerate representations θ of A on H, and π of B on K, such that θ(a)Φ(x) = Φ(ax)

and

Φ(x)π(b) = Φ(xb),

(3.7)

for all a ∈ A, b ∈ B and x ∈ X. Thus X is completely A-B-isometric to the concrete operator A-B-bimodule Φ(X). Moreover, Φ, θ, π may chosen to all be completely isometric, and such that H = K.

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If A = B then one may choose, in addition to all the above, π = θ. Proof Suppose that X ⊂ B(L), for a Hilbert space L, and apply 1.5.7 (2) (and the remark after it) to the map from A ⊗h X ⊗h B to X ⊂ B(L) described above 3.3.1. We obtain axb = Rθ1 (a)Ψ(x)π1 (b)S for a ∈ A, b ∈ B and x ∈ X. Here Ψ : X → B(K1 , H1 ) is a linear complete contraction, S : L → K1 and R : H1 → L are contractions; and θ1 and π1 are, respectively, the restrictions to A and B of ∗-representations of C ∗ -covers, on the Hilbert spaces H1 and K1 . Thus θ1 and π1 are completely contractive homomorphisms. We may assume, using 2.1.10 if necessary, that θ1 and π1 are nondegenerate representations. Let K be the subspace [π1 (B)SL] of K1 , let P be the projection of K1 onto K, and let π = π1 (·)|K = P π1 (·)|K . View π as a completely contractive nondegenerate representation of B on K. Let H be the subspace [θ1 (A)∗ R∗ L] of H1 , let Q be the projection of H1 onto H, and let θ = Q θ1 (·)|H . Note that θ1 (A)∗ H ⊂ H, so that H ⊥ is θ1 (A)-invariant. Hence by 3.2.1, θ is a completely contractive representation of A on H, and it is easy to see using 2.1.6 that θ is nondegenerate. Let Φ = QΨ(·)|K : X → B(K, H). For any a ∈ A, b ∈ B, π1 (b)S maps into K, whereas Rθ1 (a) = Rθ1 (a)Q. Hence for any x ∈ X, we have axb = Rθ1 (a)Φ(x)π1 (b)S.

(3.8)

One consequence of the last relation is that Φ(x) ≥ Rθ1 (et )Φ(x)π1 (fs )S = et xfs , where (et )t and (fs )s are cai’s for A and B respectively. From this it is clear that Φ is an isometry. Similarly, Φ is a complete isometry. Consider a, a2 ∈ A, b, b2 ∈ B, x ∈ X, and ζ, η ∈ L. By (3.8), (a2 axbb2 )ζ, η = Rθ1 (a2 )Φ(axb)π1 (b2 )Sζ, η = Φ(axb)π1 (b2 )Sζ, θ1 (a2 )∗ R∗ η. On the other hand, by (3.8) again this same quantity also equals Rθ1 (a2 a)Φ(x)π1 (bb2 )Sζ, η = θ1 (a)Φ(x)π1 (b)π1 (b2 )Sζ, θ1 (a2 )∗ R∗ η = θ(a)Φ(x)π(b)π1 (b2 )Sζ, θ1 (a2 )∗ R∗ η. Putting together, this shows that Φ(axb)π1 (b2 )Sζ, θ1 (a2 )∗ R∗ η = θ(a)Φ(x)π(b)π1 (b2 )Sζ, θ1 (a2 )∗ R∗ η. Now using linearity and density considerations, one sees that θ(a)Φ(x)π(b) = Φ(axb). If A and B are unital, we clearly have (3.7). In the contrary case, we note that a simpliﬁcation of the argument above can be used to prove each of the relations in (3.7) separately. This proves the ﬁrst assertion or two of the theorem.

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Next, we take a suﬃciently large multiple of all our maps. More speciﬁcally, for some cardinal γ we know that H (γ) = K (γ) unitarily. We have θ(a)γ Φ(x)γ π(b)γ = Φ(axb)γ . Take V : H (γ) → K (γ) to be a unitary, and deﬁne Φ = V Φ(·)γ , θ = V θ(·)γ V ∗ , and π = π(·)γ . Then it is easy to see that θ and π are nondegenerate completely contractive representations on K (γ) . Thus we may assume that H = K. By a similar argument we may also assume that there exist completely isometric nondegenerate representations ρ and σ, of A and B respectively, on H. More explicitly, take two more Hilbert spaces on which there do exist such representations and take large enough multiples of all our Hilbert spaces so that these multiples are unitarily equivalent. To adjust things so that all maps are completely isometric, we replace H by H (2) , and we modify θ, π, Φ by considering instead the representations θ ⊕ ρ, σ ⊕ π, and the 2 × 2 operator matrix which is zero except for a Φ(·) as the 1-2 entry. This gives the ‘Moreover’ assertion of the theorem. If A = B, take Φ, θ, π as in the ‘Moreover’ assertion of the theorem. We replace H by H (2) , and we modify θ, π, Φ by considering instead the representations θ ⊕ π, π ⊕ θ, and the 2 × 2 operator matrix which is zero except in the 1-2 entry; which entry is Φ(x). This gives the ﬁnal assertion. 2 3.3.2 (Quotients of operator modules) Suppose that X is a nondegenerate operator A-B-bimodule over a pair of approximately unital operator algebras. If Y ⊂ X is a closed A-B-submodule, then the quotient X/Y is nondegenerate and is an h-bimodule by the last sentence in 3.1.10. Hence the CES theorem ensures that X/Y is an operator A-B-bimodule. 3.3.3 (CES-representations) Let A and B be Banach algebras which are also operator spaces, and let X be an A-B-bimodule which is also an operator space. To say that X is an operator A-B-bimodule is equivalent to saying that X possesses a CES-representation. By the latter term we mean a triple (Φ, θ, π) consisting of a linear complete isometry Φ : X → B(K, H), and completely contractive representations θ of A on H, and π of B on K, satisfying (3.7). Indeed if there exists such a triple, then X is completely A-B-isometric to the concrete operator A-B-bimodule Φ(X). If A and B are operator algebras and if θ and π are completely isometric then we say that (Φ, θ, π) is a faithful CES-representation. Similar deﬁnitions pertain to one-sided modules, except that in this case the CES-representation will be a pair as opposed to a triple. If (Φ, θ, π) is a CES-representation of an A-B-bimodule X, with Φ(X) a subspace of B(K, H), then one may adjust the representation to obtain certain additional properties. For example, it is always possible to replace the CESrepresentation by one for which H = K, by the trick in the third last paragraph of the proof of 3.3.1. Or, if we replace H by H = [Φ(X)K], and θ by θ = θ(·)|H viewed as a map from A to B(H ), then we obtain a new CES-representation of X having the additional properties that Φ(X)K is now dense in H, and

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also that θ is a nondegenerate representation (even if θ was degenerate). Note however that θ need not be isometric even if the original θ was isometric. Finally, replace both H by H above, and K by [Φ(X) H]; and deﬁne Φ = Φ(·)|K , viewed as a map from X into B(K , H ), and deﬁne θ as above, and deﬁne π = PK π(·)|K , viewed as a map from B into B(K ). We claim that (Φ , θ , π ) is a CES representation too, but now θ and π are both nondegenerate. The proof of this follows from routine arguments of the type we have seen in 3.3.1, for example, so that we will only sketch part of it. For x ∈ X, ζ ∈ K, η ∈ H we have Φ(x)ζ, η = ζ, Φ(x)∗ η = PK ζ, Φ(x)∗ η = Φ (x)PK ζ, η. Thus Φ = Φ (·)PK , and it follows that the complete contraction Φ is a complete isometry. A similar argument shows that (3.7) holds for the adjusted maps. It is easy to see that θ is a nondegenerate homomorphism. To see that π is a homomorphism one may use the argument in 3.3.1 showing that θ was a homomorphism there. To see that π is nondegenerate, use 2.1.6 together with the fact that Φ(x)π(et ) = Φ(xet ) → Φ(x), for a cai (et )t for B, and x ∈ X. 3.3.4 (The algebra of a bimodule) We consider a ‘bimodule variant’ of the U(X) construction of 2.2.10. Suppose ﬁrst that X is an A-B-bimodule over algebras A and B. Consider the algebra D of 2 × 2 matrices a x 0 b for a ∈ A, b ∈ B, x ∈ X. The product here is the formal product of 2 × 2 matrices, implemented using the module actions and algebra multiplications. Assume that A and B areunital and that X is nondegenerate. Let p and q 1A 0 0 0 be the idempotents and . Then pDp ∼ = A and qDq ∼ = B as 0 0 0 1B algebras, and pDq ∼ = X as bimodules. Conversely, given any unital algebra D, and idempotents p, q ∈ D with p + q = 1D , then pDq is a pDp-qDq-bimodule. This characterizes nondegenerate bimodules in terms of unital algebras. For a nondegenerate operator A-B-bimodule X over approximately unital operator algebras A and B, we may form the 2 × 2 matrix algebra D as above. If (Φ, θ, π) is a faithful and nondegenerate CES-representation on Hilbert spaces H and K as provided by Theorem 3.3.1, then we may identify D with the concrete operator algebra C of matrices θ(a) Φ(x) , 0 π(b) which should be viewed as a nondegenerate subalgebra of B(H ⊕K). This algebra is approximately unital, and by elementary facts from Section 2.6, the canonical diagonal projections p = IH ⊕ 0 and q = 0 ⊕ IK are in M (C). Again we see that pCp ∼ = A and qCq ∼ = B as algebras, and pCq ∼ = X as bimodules, but now these

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isomorphisms are also completely isometric. The converse of this is true too, as we noted in 3.1.2 (5). This characterizes nondegenerate operator bimodules over approximately unital operator algebras as ‘corners’ pC(1 − p) of approximately unital operator algebras C (in the sense of 2.6.14). Lemma 3.3.5 Suppose that A, B, C are approximately unital operator algebras, that X is a nondegenerate operator A-B-bimodule, and that Y is a nondegenerate operator B-C-bimodule. Then there exists two faithful CES-representations (Φ, π, θ) and (Ψ, θ, σ) for X and Y respectively, where all representations are on a single Hilbert space H. Thus, up to appropriate identiﬁcations, we may view A, B, C as subalgebras of B(H), and X, Y as subbimodules of B(H). If B is unital, then we may assume in addition that θ is a unital map too. Proof By 3.3.1 there exist faithful CES-representations (Φ, π, θ) and (Ψ, ρ, σ) of the two bimodules. We may suppose that the representations in the ﬁrst triple are on a Hilbert space H1 , and that the representations in the second triple are on a Hilbert space H2 . Let H = H1 ⊕ H2 , and replace θ and ρ with θ(b) ⊕ ρ(b). Then replace Φ with Φ ⊕ 0, Ψ with 0 ⊕ Ψ, π with π ⊕ 0, and σ with 0 ⊕ σ. 2 3.4 TWO MODULE TENSOR PRODUCTS 3.4.1 (The algebraic module tensor product) Suppose that X and Y are, respectively, right and left modules over an algebra A. If Z is a vector space, then bilinear map u : X × Y → Z is said to be balanced (or A-balanced) if u(xa, y) = u(x, ay) for all x ∈ X, y ∈ Y, a ∈ A. Given such X and Y , we deﬁne the algebraic module tensor product X ⊗A Y to be the quotient of X ⊗ Y by the subspace spanned by terms of the form xa ⊗ y − x ⊗ ay, for x ∈ X, y ∈ Y, a ∈ A. It is clear that X ⊗A Y has the following universal property: given any vector space Z, and any balanced bilinear u : X × Y → Z, then there exists a unique linear u˜ : X ⊗A Y → Z mapping the class of x ⊗ y to u(x, y) for all x ∈ X, y ∈ Y . Thus the module tensor product ‘linearizes balanced bilinear maps’. 3.4.2 (Two operator module tensor products) We now introduce two operator space analogues of the algebraic module tensor product construction, the module Haagerup tensor product, and the module operator space projective tensor product. Let A be an algebra, and let X and Y be operator spaces which are, respectively, right and left A-modules. We deﬁne X ⊗hA Y (resp. X ⊗A Y ) to be the quotient in the sense of 1.2.14, of X ⊗h Y (resp. X ⊗ Y ) by the closure of the subspace spanned by terms of the form xa ⊗ y − x ⊗ ay. For the purpose of this paragraph and the next one only we write x ⊗A y for the equivalence class of x ⊗ y in either X ⊗hA Y or X ⊗A Y . Later on, as is customary in algebra, we will continue to write x ⊗ y for this equivalence class. Such expressions are the elementary tensors. We will also write ⊗A for the canonical bilinear completely contractive map X × Y → X ⊗hA Y (resp. jointly completely contractive map X × Y → X ⊗A Y ). In particular, the norm of x ⊗A y in either module tensor

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product is dominated by xy for x ∈ X, y ∈ Y ; and a similar assertion holds for matrices. The elements in the linear span of the range of ⊗A we will refer to as ﬁnite rank tensors, they are dense in X ⊗hA Y (resp. X ⊗A Y ). It is clear that the space W = X ⊗hA Y with the map µ = ⊗A , possesses the following universal property: Given any operator space Z and bilinear completely contractive balanced u : X × Y → Z, then there exists a (necessarily unique) linear completely contractive u ˜ : W → Z such that u ˜ ◦ µ = u. Thus X ⊗hA Y ‘linearizes’ balanced completely bounded maps. In fact this property characterizes X ⊗hA Y , analoguously to the fact at the end of 1.5.4. A similar universal property characterizes X ⊗A Y , namely the property of ‘linearizing’ balanced jointly completely bounded maps. 3.4.3 (An alternate construction) There is an alternate way of deﬁning the two module tensor products of 3.4.2, which provides some extra information quite easily. We treat the module Haagerup tensor product, the other is similar. Begin with a right A-module X and a left A-module Y as before and form their algebraic module tensor product X ⊗A Y (see 3.4.1). On X ⊗A Y we deﬁne a sequence of matrix seminorms { · n }∞ n=1 by the formula zn = inf{x y : z = x A y},

(3.9)

where x ranges over Mn,p (X), y ranges over Mp,n (Y ). Here p is variable, z is in Mn (X ⊗A Y ), and xA y is deﬁned to be [ k xik ⊗A ykj ]. Following verbatim the usual proof that the Haagerup tensor product is an operator space (see 1.5.4), one sees that the sequence of seminorms just deﬁned is an operator seminorm structure in the sense of 1.2.16. By 1.2.16, the completion of the quotient of X ⊗A Y by the null space N of · 1 is an operator space. It is straightforward to check that this completed quotient has the universal property described in 3.4.2, and hence is completely isometrically isomorphic to X ⊗hA Y . Note that ˙ in the quotient space, to the isomorphism takes an ‘elementary tensor’ x ⊗ y +N the equivalence class x ⊗A y in the quotient space of X ⊗h Y considered in our ﬁrst construction of the module Haagerup tensor product. Indeed the following diagram of canonical maps commutes: X⊗  Y −→ X ⊗A Y   ( ( X ⊗h Y −→ X ⊗hA Y One useful and immediate consequence of this is a useful norm formula. If z ∈ Mm,n (X ⊗ Y ), then the norm in Mm,n (X ⊗hA Y ) of its equivalence class is precisely the inﬁmum considered in (3.9), now viewing z ∈ Mm,n (X ⊗A Y ). There is a similar formula valid for tensors which are not ﬁnite rank:

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Lemma 3.4.4 Let A be an algebra, and let X and Y be operator spaces which are, respectively, right and left A-modules. If z ∈ X ⊗hA Y and if > 0 is given, R(X) and y = [y1 y2 · · · ]t ∈ C(X) such that then there exists x = [x1 x2 · · · ] ∈ ∞ z equals the norm convergent sum k=1 xk ⊗ yk in X ⊗hA Y . Moreover this can be done with xR(X) = yC(Y ) < z + . If z is ﬁnite rank, then the x and y above may be chosen in Rn (X) and Cn (X) respectively, for some n ∈ N. Proof The last assertion was mentioned in the last paragraph of 3.4.3. The rest follows immediately from 1.5.6 and the basic properties of a Banach space 2 quotient map (in this case, the canonical map X ⊗h Y → X ⊗hA Y ). Lemma 3.4.5 (Functoriality of the module tensor product) Let B be an algebra, let X1 and X2 be operator spaces which are right B-modules, and let Y1 and Y2 be operator spaces which are left B-modules. If u : X1 → X2 and v : Y1 → Y2 are completely bounded B-module maps, then the map u ⊗ v extends uniquely to a well deﬁned linear completely bounded map (which we also write as u ⊗ v) from X1 ⊗hB Y1 to X2 ⊗hB Y2 (resp. X1 ⊗B Y1 to X2 ⊗B Y2 ). Indeed, u ⊗ vcb ≤ ucbvcb . Proof We just prove this for the Haagerup tensor product; the other is identical. By the functoriality of the Haagerup tensor product 1.5.5, we obtain a completely bounded linear map X1 ⊗h Y1 → X2 ⊗h Y2 taking x ⊗ y to u(x) ⊗ v(y). Composing this map with the quotient map X2 ⊗h Y2 → X2 ⊗hB Y2 we obtain a map X1 ⊗h Y1 → X2 ⊗hB Y2 . It is easy to see that the kernel of the last map contains all terms of form xa ⊗ y − x ⊗ ay, with a ∈ B, x ∈ X1 , y ∈ Y1 , so that 2 we obtain a map X1 ⊗hB Y1 → X2 ⊗hB Y2 with the required properties. Lemma 3.4.6 Let A be an approximately unital Banach algebra, which is also an operator space. If X is a left h-module (resp. matrix normed module) over A, then the operator space A ⊗hA X (resp. A ⊗A Y ) is canonically completely isometrically isomorphic to the essential part of X (see A.6.4). Proof We prove the case involving A⊗hA X, the other is identical. By deﬁnition (see 3.1.3), the module action A × X → X is a completely contractive balanced bilinear map. By 3.4.2, it induces a completely contractive map m from A ⊗ hA X to X. Conversely, if (et )t is a cai for A, let φt : X → A ⊗hA X be the map x → et ⊗ x, which is easily seen to be completely contractive. We have φt (m(a ⊗ x)) = et a ⊗ x −→ a ⊗ x,

a ∈ A, x ∈ X.

Hence by linearity and density considerations, φt (m(z)) → z for all z ∈ A⊗hA X. Thus for zij ∈ A ⊗hA X, [zij ]n = lim [φt (m(zij ))]n ≤ [m(zij )]n . t

Thus m is a complete isometry. It is easily seen that m maps onto the essential part of X. 2

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The following result is often useful (although we shall not need it later), since it can help reduce certain calculations involving matrix normed modules to the simple case of Sn1 [A] (see 3.1.5 (3)). Proposition 3.4.7 Let X be a nondegenerate left matrix normed module over an approximately unital matrix normed algebra A. Then there exists a set I, natural numbers ni for i ∈ I, and a complete quotient map u : ⊕1i Sn1 i [A] → X, such that u is also a left A-module map. Hence X is completely A-isometric to a quotient A-module of ⊕1i Sn1 i [A]. Proof As noted in 1.4.13 there is a complete quotient map from ⊕1i Sn1 i onto

X. By the projectivity of ⊗ (mentioned in 1.5.11) we obtain a complete quotient map from A ⊗ (⊕1i Sn1 i ) onto A ⊗ X. By (1.52), the deﬁnition of ⊗A , and 3.4.6, we obtain a complete quotient map ⊕1 S 1 [A] → A ⊗A X ∼ 2 = X. i

ni

3.4.8 (Threefold tensor products) Let B, C be algebras, and suppose that X, Y, Z are operator spaces. Suppose that X is also a right B-module, that Y is a B-C-bimodule, and that Z is a left C-module. We deﬁne X ⊗hB Y ⊗hC Z to be the quotient of X ⊗h Y ⊗h Z by the closure of the linear span of terms of the form xb ⊗ y ⊗ z − x ⊗ by ⊗ z and x ⊗ yc ⊗ z − x ⊗ y ⊗ cz. Then exactly as above, but using the universal property of the threefold Haagerup tensor product (see 1.5.5), one sees that X ⊗hB Y ⊗hC Z has the following universal property: If u : X × Y × Z → W is a completely contractive, balanced (in the sense that u(xb, y, z) = u(x, by, z) and u(x, yc, z) = u(x, y, cz)) trilinear map, then there is a unique completely contractive linear map u ˜ : X ⊗hB Y ⊗hC Z → W such that u ˜(x ⊗ y ⊗ z) = u(x, y, z) for all x ∈ X, y ∈ Y, z ∈ Z. This is a universal property in the same sense as the one discussed at the end of 3.4.2. Similarly there is a threefold tensor product X ⊗B Y ⊗C Z linearizing jointly completely bounded balanced trilinear maps. Indeed for n ≥ 4 there are n-fold variants of both module tensor products, which are deﬁned in an analoguous way. 3.4.9 (Tensor products as bimodules) Let A and C be Banach algebras which are also operator spaces and let B be an algebra. Let X, Y be operator spaces which are respectively an A-B-bimodule and a B-C-bimodule. Suppose that the left action makes X a matrix normed A-module. Then X ⊗ Y is canonically a matrix normed A-module, as noted in 3.1.5 (3). Using 3.1.10, we see that X ⊗B Y is also a left matrix normed A-module. If A acts nondegenerately on X, then it is easy to see using 3.1.10 that A also acts nondegenerately on X ⊗B Y . Likewise if Y is a (resp. nondegenerate) right matrix normed C-module, then X ⊗B Y is a (resp. nondegenerate) right matrix normed C-module as well. Moreover if both hold, then the left and right actions on X ⊗B Y commute hence X ⊗B Y is actually a matrix normed A-C-bimodule. An almost identical argument shows that if X is a left h-module over A and Y is a right h-module over C, then X ⊗hB Y is an h-bimodule over A and C.

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Furthermore, this bimodule is nondegenerate if X and Y are nondegenerate. If we appeal to the CES theorem (or its extension to be proved in 4.6.7) we ﬁnd that the module Haagerup tensor product of left and right nondegenerate operator modules is a nondegenerate operator bimodule. We also remark that in Lemma 3.4.5, if further X1 and X2 are left matrix normed modules (resp. h-modules) over A, and if further Y1 and Y2 are right matrix normed modules (resp. h-bimodules) over C, and if u, v are bimodule maps, then u ⊗ v is a A-C-bimodule map on X ⊗B Y (resp. X ⊗hB Y ). Theorem 3.4.10 (Associativity of the module tensor product) Let A, B, C, D be Banach algebras which are also operator spaces. Assume that X is a matrix normed A-B-bimodule, that Y is a matrix normed B-C-bimodule, and that Z is a matrix normed C-D-bimodule. Then

X ⊗B (Y ⊗C Z) ∼ = (X ⊗B Y ) ⊗C Z ∼ = X ⊗ B Y ⊗C Z completely isometrically (via an A-D-bimodule map). Similarly, if X, Y, Z are h-bimodules, then X ⊗hB (Y ⊗hC Z) ∼ = (X ⊗hB Y ) ⊗hC Z ∼ = X ⊗hB Y ⊗hC Z completely isometrically (via an A-D-bimodule map). One way to prove the last result (see [65, Theorem 2.6]) is to show that the iterated module tensor products have the universal property of the threefold module tensor product (see 3.4.8). 3.4.11 Suppose that A is an algebra, and that X and Y are operator spaces which are, respectively, right and left A-modules. Then it is not hard to extend 1.5.14 (7) to show that there is a canonical completely isometric isomorphism Cm (X) ⊗hA Rn (Y ) ∼ = Mm,n (X ⊗hA Y ) taking [x1 x2 · · · xm ]t ⊗ [y1 y2 · · · yn ] to the matrix [xi ⊗ yj ]. Indeed a variant of 3.4.10 and 1.5.14 (6) show that Cm (X) ⊗hA Rn (Y ) ∼ = (Cm ⊗h X) ⊗hA (Y ⊗h Rn ) ∼ Cm ⊗h (X ⊗hA Y ) ⊗h Rn ∼ = Mm,n (X ⊗hA Y ). = A ‘minimal’ tensor product of operator modules is discussed in the Notes. 3.5 MODULE MAPS In this section we again use the convention from 3.1 that A, B, C are at least Banach algebras with an operator space structure.

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3.5.1 (The category OM OD) If X and Y are left matrix normed A-modules, then we write A CB(X, Y ) for the set of completely bounded A-module maps from X to Y . Evidently A CB(X, Y ) is an operator space, namely a subspace of the operator space CB(X, Y ) (see 1.2.19). Its matrix norms are speciﬁed by the canonical identiﬁcation Mn (A CB(X, Y )) ∼ = A CB(X, Mn (Y )). Similar notations will apply for right B-modules and A-B-bimodules, except that we write CBB (X, Y ) (resp. A CBB (X, Y )) for the completely bounded right B-module maps (resp. the A-B-bimodule maps). We write A OM OD for the category of nondegenerate left operator A-modules, with morphisms the completely bounded left A-module maps. We write OM ODB for the category of nondegenerate right operator B-modules and A OM ODB for the category of nondegenerate operator A-B-bimodules. It is usually convenient for us to view the category A HM OD of Hilbert A-modules (see 3.2.5) as a subcategory of A OM OD. To do this, note that by 3.1.7 every Hilbert A-module is a left operator A-module (with its ‘Hilbert column space’ structure), and by 3.2.5 the morphisms are the same. 3.5.2 (Mapping spaces as modules) Suppose that X and Y are matrix normed bimodules over A and B, and A and C, repectively. Then A CB(X, Y ) becomes a B-C-bimodule with the ‘interior’ left action (bu)(x) = u(xb), and the ‘exterior’ right action deﬁned by (uc)(x) = u(x)c. Here b ∈ B, x ∈ X, c ∈ C and u ∈ A CB(X, Y ). We say that such a map bu is left B-essential. We write ess (X, Y ) for the subset of A CB(X, Y ) consisting of such maps bu, for b ∈ B A CB and u ∈ A CB(X, Y ). It is easy to see that A CB(X, Y ) and A CB ess (X, Y ) above are matrix normed B-C-bimodules. Similarly, CBB (X, W ) is a matrix normed D-A-bimodule for any matrix normed D-B-bimodule W , with the exterior left D-module action, and the inteess (X, W ) for the set rior right A-module action (ua)(x) = u(ax). We write CBB of such right A-essential maps ua. 3.5.3 (Transplanting relations from algebra) In 3.2.6 we have brieﬂy discussed the process of transferring basic concepts and relations from algebra (of the type found in basic algebra texts such as [8, 368]), into the setting of modules over operator algebras. For some results or notions the transferal is routine; others require a great deal of analysis, or have no satisfactory variant. Also, quite often an algebraic notion has several interesting analogues in the operator framework. Such results, while sometimes initially looking a little ‘formal’, are nonetheless the ‘right way’ to approach certain topics, as is testiﬁed to, for example, in the literature on Hilbert modules cited in 3.2.6. In the present text we will not attempt to systematically collect such ‘transferals from pure algebra’, partly because we will not be able to reach the more sophisticated topics that require many such ‘transplanted relations’. We will however give a few simple and useful results of this sort in this section. Indeed we will treat some properties of the ‘Hom(−, −) functor’ in our category; namely the module mapping spaces of the type A CB or A CB ess deﬁned above.

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Lemma 3.5.4 Let X be a nondegenerate left matrix normed module over an approximately unital matrix normed algebra A. Then (1) X ∼ = A CB ess (A, X) ⊂ A CB(A, X) completely isometrically. (2) If A is unital, or if X is a dual space and the map x → ax on X is w ∗ continuous for all a ∈ A, then X ∼ = A CB(A, X) in (1). Proof Let (et )t be a cai for A. For each x ∈ X deﬁne rx (a) = ax for a ∈ A. It is easy to see that the map x → rx is completely contractive. In fact it is completely isometric, as may be seen from the relation [xij ]n = lim [et xij ]n ,

[xij ] ∈ Mn (X).

t

By Cohen’s factorization theorem A.6.2, we may write any x ∈ X as x = a1 x1 with a1 ∈ A, x1 ∈ X. Therefore rx (a) = rx1 (aa1 ) for a ∈ A. Thus rx = a1 rx1 , a map in A CB ess (A, X). Conversely, if u ∈ A CB(A, X) and if a1 ∈ A, then (a1 u)(a) = u(aa1 ) = au(a1 ) for a ∈ A. Hence a1 u = ru(a1 ) . This yields (1). Assertion (2) follows from (1). This is obvious if A is unital. In the other case, consider u ∈ A CB(A, X), and let x be a w∗ -cluster point in X of the bounded 2 net (u(et ))t . Then u(a) = limt au(et ) = ax for any a ∈ A. If X is a Hilbert A-module, or more generally if X is reﬂexive, then the hypothesis in (2) of the above lemma clearly holds, so that X ∼ = A CB(A, X). 3.5.5 (Module maps and multipliers) Suppose that X and Y are left matrix normed modules over an approximately unital matrix normed algebra A, and that the action on X is nondegenerate. If u ∈ A CB(X, Y ) and x ∈ X then by Cohen’s theorem A.6.2 we may write x = ax for an a ∈ A, x ∈ X. Thus u(x) = au(x ). Therefore u maps into the essential part (see A.6.4) Y of Y , and so A CB(X, Y ) = A CB(X, Y ). If, in addition, Y is nondegenerate then by 3.1.11 we may view both X and Y as left modules over LM (A), If η ∈ LM (A) and if u is as above, then by 3.1.11 we have u(ηx) = lim u(η(et )x) = lim η(et )u(x) = ηu(x). t

t

That is, u is a LM (A)-module map. If A is in addition an operator algebra, then it follows that u is also an A1 -module map. Likewise if H is a Hilbert A-module and K is any Hilbert space, then B(K, H) is canonically a left operator LM (A)-module (see 3.1.11) and any A-module map u : X → B(K, H) is an LM (A)-module map. 3.5.6 (Module maps and matrix spaces) We next consider some frequently encountered module map variants of the matrix space relations in 1.2.28 and 1.2.29. For example, if X and Y are matrix normed A-B-bimodules then we have A CBB (X, MI,J (Y )) ∼ = MI,J (A CBB (X, Y )) completely isometrically. This is easily seen from 1.2.29. One needs to check that the isomorphisms in that proof take matrices of bimodule maps to bimodule maps, but this is fairly evident.

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By the idea of the proof as 1.2.28 and the proof of (1.14) that we sketched in the Notes to Section 1.2, one may prove the following commonly encountered ess refers identities. We omit the tedious details. In this result, the expression CBB to the space of right B-essential maps (see 3.5.2); which coincides, for example ess (Cn (B), X) case, with the set of right Mn (B)-essential maps. Of in the CBB course if B is unital, then one may delete the symbol ‘ess’ below. Proposition 3.5.7 Let X be a right operator B-module over an approximately unital matrix normed algebra B. Let I, J be cardinals, and m, n ∈ N. Then we have the following canonical completely isometric isomorphisms: (1) CBB (CJ (B), CI (X)) ∼ = MI,J (CBB (B, X)), and ess (2) CB (Cn (B), Cm (X)) ∼ = Mm,n (X). B

Also, there are canonical complete isometries ess CBB (CI (B), X) → RIw (X) → CBB (CI (B), X).

If B is unital then (1) becomes CBB (CJ (B), CI (X)) ∼ = MI,J (X). 3.5.8 (The role of complete boundedness) The canonical isomorphisms in the result above are perhaps most quickly grasped by looking at the special case of (2) when n = m and X = B. We see that if B is a unital operator algebra, then Mn (B) ∼ = CBB (Cn (B))

completely isometrically.

This isomorphism is the same as that in the fundamental fact from ring theory that Mn (A) ∼ = HomA (A(n) ) (which in turn is a trivial generalization of the elementary linear algebra fact that Mn (C) is isomorphic to the set of linear maps on Cn ). Namely, a matrix b ∈ Mn (B) is taken by this isomorphism to the map on Cn (B) given by left matrix multiplication by b. It should be carefully noted that for a general operator algebra B, the operator algebra M n (B) is not isometrically isomorphic to BB (Cn (B)) via the canonical map (e.g. see [182]). This is an illustration of the importance of the operator space approach to operator algebras. The failure of this elementary relation at the isometric level is the ﬁrst indication of the complete breakdown further down the road if one attempts to generalize certain facts from purely algebraic module theory without using the matrix norms of operator space theory (see, for example, [46]). Proposition 3.5.9 Let A, B and C be Banach algebras which are also operator spaces, suppose that X is a matrix normed A-B-bimodule, that Y is a matrix normed B-C-bimodule, and that Z is a matrix normed A-C-bimodule. Then there are canonical completely isometric isomorphisms A CBC (X

⊗B

Y, Z) ∼ =

B CBC (Y,A

CB(X, Z)) ∼ =

A CBB (X, CBC (Y, Z)).

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Proof We prove just the ﬁrst isomorphism, the other is similar. By deﬁnition, X ⊗B Y is a complete quotient of X ⊗ Y , hence we have complete isometries

CB(X ⊗B Y, Z) −→ CB(X ⊗ Y, Z) ∼ = CB(Y, CB(X, Z)). The last relation is simply (1.50), and the ﬁrst ‘arrow’ is ‘composition with q’, where q : X ⊗ Y → X ⊗B Y is the canonical quotient map. Let r be the composition of the two maps in the sequence. If u ∈ CB(X ⊗B Y, Z) then (r(u)(y))(x) = u(x ⊗ y),

x ∈ X, y ∈ Y.

(3.10)

From this it is clear that if u is an A-C-bimodule map, then r(u)(y) ∈ A CB(X, Z) for any y ∈ Y , and r(u) ∈ B CBC (Y,A CB(X, Z)). Thus r is a complete isometry from A CBC (X ⊗B Y, Z) to B CBC (Y,A CB(X, Z)). If Ψ ∈ B CBC (Y,A CB(X, Z)), then in particular, Ψ ∈ CB(Y, CB(X, Z)). Thus there is by (1.50), a map v in CB(X ⊗ Y, Z) with v(x ⊗ y) = Ψ(y)(x) for x ∈ X and y ∈ Y . Then v(xb ⊗ y) = Ψ(y)(xb) = (bΨ(y))(x) = Ψ(by)(x) = v(x ⊗ by).

Using 3.4.2, we obtain a map u ∈ CB(X ⊗B Y, Z) with v = uq. Then Ψ = r(u), and it is simple algebra using (3.10) to check that u is an A-C-bimodule map. 2 Thus r maps A CBC (X ⊗B Y, Z) onto B CBC (Y,A CB(X, Z)). The following is a special case corresponding to Z = A = C = C. Corollary 3.5.10 Let B be a Banach algebra which is also an operator space, and suppose that X and Y are respectively right and left matrix normed modules over B. Then there are canonical completely isometric isomorphisms

(X ⊗B Y )∗ ∼ = In particular,

B CB(Y, X

∗

B CB(Y, X

∗

) ∼ = CBB (X, Y ∗ ).

) is a w∗ -closed subspace of CB(Y, X ∗ ).

Corollary 3.5.11 If H and K are Hilbert modules over A and B respectively, and if X is a matrix normed A-B-bimodule, then A CBB (X, B(K, H)) is a dual operator space with predual

¯ r ⊗A X ⊗B K c = H ¯ r ⊗hA X ⊗hB K c . H

(3.11)

¯ r ⊗h X = H ¯ r ⊗X, so that H ¯ r ⊗hA X = H ¯ r ⊗A X. Proof By 1.5.14 (1) we have H ¯ r ⊗hA X) ⊗hB K c = (H ¯ r ⊗A X) ⊗B K c . Then (3.11) follows Similarly, we have (H from the associativity of the two tensor norms. From 3.5.10, (1.15), and 3.5.9, we have

¯ r ⊗A (X ⊗B K c ))∗ ∼ (H =

A CB(X

⊗B K c , H c ) ∼ =

A CBB (X, CB(K

and the latter space is just A CBB (X, B(K, H)), by (1.14).

c

, H c )), 2

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3.6 MODULE MAP EXTENSION THEOREMS In this section all algebras are C ∗ -algebras, and all operator modules are assumed nondegenerate for simplicity. 3.6.1 (The bimodule Paulsen system) Let X be an operator A-B-bimodule over unital C ∗ -algebras A and B. By choosing a nondegenerate faithful CESrepresentation, we may assume that A and B are unital-subalgebras of B(H) and B(K) respectively, and that X is an A-B-submodule of B(K, H). According to 3.1.16, we regard X ⊂ B(K, H) as a B-A-bimodule. We deﬁne the bimodule Paulsen system to be the set S of formal matrices a x y∗ b for a ∈ A, b ∈ B, x, y ∈ X. Clearly this may be viewed as a unital selfadjoint (A ⊕ B)-(A ⊕ B)-submodule of B(H ⊕ K). Next suppose that we have unital ∗-representations θ : A → B(H1 ) and π : B → B(K1 ), and suppose that u : X → B(K1 , H1 ) is a completely contractive A-B-bimodule map. That is, u(axb) = θ(a)u(x)π(b) for a ∈ A, b ∈ B, x ∈ X. A fairly literal modiﬁcation of the proof of Lemma 1.3.15 shows that the map a x θ(a) u(x) Θ: → (3.12) y∗ b u(y)∗ π(b) from S to B(H1 ⊕ K1 ) is completely positive. By 1.3.3, we have that Θ is completely contractive. By symmetry, if in addition u is a complete isometry and θ and π are one-to-one, then we see that Θ is a unital complete order isomorphism, and a complete isometry. This shows that the bimodule Paulsen system may be deﬁned as an operator system independently of the particular nondegenerate faithful CES-representation of the bimodule chosen in the beginning of this section. Theorem 3.6.2 (Wittstock) Let Y be an operator A-B-bimodule over unital C ∗ -algebras A and B, and suppose that X is an A-B-submodule of Y . We suppose that θ : A → C and π : B → C are unital ∗-homomorphisms, where C is an injective unital C ∗ -algebra. Suppose that u : X → C is a completely contractive A-B-bimodule map (i.e. u(axb) = θ(a)u(x)π(b) for a ∈ A, b ∈ B, x ∈ X). Then there exists a completely contractive A-B-bimodule map uˆ : Y → C extending u. Proof Suppose that C is represented as a unital-subalgebra of B(L) for a Hilbert space L. Since C is injective, there is a unital completely contractive projection from B(L) onto C, which is a θ(A)-π(B)-bimodule map by 1.3.12. Thus it is clear that we may assume that C = B(L). On the other hand we may assume that A and B act faithfully and nondegenerately on some Hilbert spaces H and K respectively, that X ⊂ B(K, H) is an A-B-submodule, and that Y = B(K, H). Let S ⊂ B(H ⊕ K) be the bimodule Paulsen system considered above. By 3.6.1,

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we obtain a completely positive Θ : S → M2 (B(L)) = B(L ⊕ L) deﬁned by (3.12). By Arveson’s extension theorem 1.3.5, Θ extends to a completely posiˆ : B(H ⊕ K) → B(L ⊕ L). Clearly this map is a ∗-homomorphism on tive map Θ ˆ is cornerthe diagonal C ∗ -algebra CIH ⊕ CIK . Thus by 1.3.12 we have that Θ ˆ preserving in the sense of 2.6.15 and 2.6.16. Indeed since Θ is a ∗-homomorphism ˆ is an (A ⊕ B)-bimodule map. If on the diagonal A ⊕ B, we have by 1.3.12 that Θ ˆ u ˆ : B(K, H) → B(L) is the 1-2-corner of Θ (see 2.6.16), then u ˆ clearly extends ˆ is a (A ⊕ B)-bimodule map it follows u and is completely contractive. Since Θ that θ(a)ˆ u(y)π(b) = uˆ(ayb) for all a ∈ A, b ∈ B, y ∈ B(K, H). 2 Corollary 3.6.3 Let A and B be C ∗ -algebras, and let H and K be Hilbert modules over A and B respectively. Suppose also that Y is an operator A-B-bimodule, and that X is a closed A-B-submodule of Y . Given any completely contractive A-B-bimodule map u : X → B(K, H), then there exists a completely contractive A-B-bimodule map u ˆ : Y → B(K, H) extending u. Proof We let θ : A → B(H) and π : B → B(K) be the nondegenerate contractive homomorphisms (and hence ∗-homomorphisms by A.5.8) associated to the Hilbert modules. Case 1: H = K, A and B are unital. Here π and θ are unital, and the result follows by applying Theorem 3.6.2 with C = B(H). Case 2: H = K, and π and θ are possibly nonunital. In this case, let A1 and 1 B be the unitizations of A and B. By 3.1.11 we have that Y is an operator A1 -B 1 -bimodule, and X is an A1 -B 1 -submodule. We may extend θ and π to unital ∗-representations of A1 and B 1 respectively on H and K. By 3.5.5, u is an A1 -B 1 -bimodule map. Thus by Case 1, there exists a completely contractive A1 -B 1 -bimodule map from Y into B(H) extending u. Case 3: H = K. Choose a cardinal γ and a unitary V as in the proof of 3.3.1. Let θ = V θ(·)γ V ∗ and u = V u(·)γ . Note that u is a completely contractive A-B-bimodule map from X into B(K (γ) ). By Case 2, there is a completely contractive A-B-bimodule map from Y into B(K (γ) ) extending u . Multiply this map by V ∗ . Pre- (resp. post-) multiplying the resulting map by the canonical inclusion (resp. projection) map from H → H (γ) (resp. K (γ) → K) gives a completely contractive bimodule map from Y into B(K, H) extending u. 2 3.6.4 (Injective bimodules) Assume for simplicity that A and B are unital C ∗ -algebras. One may rephrase the last result as the statement that for Hilbert modules H and K over A and B respectively, the space B(K, H) is injective in the category A OM ODB . As a special case (taking B = K = C) we see that any Hilbert module H over a C ∗ -algebra A is injective in the category A OM OD. Theorem 3.6.5 Suppose that A is a C ∗ -algebra, and that X (resp. Y ) is an A-submodule of a right (resp. left) operator A-module W (resp. Z). (1) If L is a Hilbert space and if u : X × Y → B(L) is a completely contractive A-balanced bilinear map, then u has a completely contractive A-balanced bilinear extension u ˆ : W × Z → B(L). (2) The canonical map X ⊗hA Y → W ⊗hA Z is a complete isometry.

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Proof (1) By the last remark in 3.1.11 we may assume that A is unital (by replacing A by A1 if necessary). By Lemma 3.3.5 we may assume that there is a single C ∗ -algebra B = B(H) containing A as a unital C ∗ -subalgebra and W and Z as A-A-submodules. It clearly suﬃces to extend u to a completely contractive A-balanced bilinear map B × B → B(L). We will use the well-known techniques seen in the proof of 3.3.1. By 1.5.7 (2) we may write u(x, y) = Rθ(x)Sπ(y)T for Hilbert spaces H1 , K1 , ∗-representations π and θ of B on K1 and H1 respectively, and contractions R, S, T between these Hilbert spaces. Let K2 = [π(Y )T L], and let P be the projection of K1 onto K2 . Since AY ⊂ Y , the space K2 is obviously π(A)-invariant. Let H2 = [θ(X)∗ R∗ L] and let Q be the projection of H1 onto H2 . Since XA ⊂ X, we have [θ(A)∗ H2 ] = [θ(A)∗ θ(X)∗ R∗ L] = [θ(XA)∗ R∗ L] ⊂ H2 . Since θ is a ∗-representation, H2 is θ(A)-invariant. Let Θ(x) = Rθ(x)|H2 and let Ψ(y) = P π(y)T = π(y)T . Let S = QS|K2 viewed as a map from K2 to H2 . Then by standard arguments of the type found in 3.3.1 we have u(x, y) = Rθ(x)Sπ(y)T = Rθ(x)QSΨ(y) = Θ(x)S Ψ(y). Let θ : A → B(H2 ) (resp. π : A → B(K2 )) be deﬁned by restricting θ(A) (resp. π(a)) to its invariant subspace H2 (resp. K2 ) for any a ∈ A. Then θ (a)S π(y)T ζ, θ(x)∗ R∗ η = Rθ(xa)Sπ(y)T ζ, η = u(xa, y)ζ, η, for ζ, η ∈ L and a ∈ A. A similar calculation shows that S π (a)π(y)T ζ, θ(x)∗ R∗ η = u(x, ay)ζ, η. Since u is balanced these are equal. By linearity and density considerations, θ (a)S = S π (a),

a ∈ A.

(3.13)

For a ∈ A, x ∈ X, and ζ ∈ H2 , η ∈ L we have Θ(xa)ζ, η = Rθ(x)θ(a)ζ, η = Θ(x)θ (a)ζ, η, so that Θ : X → B(H2 , L) is an A-module map. A similar but easier calculation shows that Ψ : Y → B(L, K2 ) is an A-module map. Thus by 3.6.3, Ψ and Θ have completely contractive A-module map extensions Ψ : B → B(L, K2 ) and ˆ(b1 , b2 ) = Θ (b1 )S Ψ (b2 ) is a completely contractive Θ : B → B(H2 , L). Then u bilinear map extending u, and we have using (3.13) that for a ∈ A, b1 , b2 ∈ B, ˆ(b1 , ab2 ). u ˆ(b1 a, b2 ) = Θ (b1 )θ (a)S Ψ (b2 ) = Θ (b1 )S π (a)Ψ (b2 ) = u (2) Let V be the closure in W ⊗hA Z of the span of the ﬁnite rank tensors x ⊗ y for x ∈ X, y ∈ Y . It is straightforward to show using (1) that V has the universal property discussed at the end of 3.4.2. Thus V is completely isometric 2 to X ⊗hA Y .

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3.6.6 (Extensions of multilinear module maps) Theorem 3.6.5 says that the module Haagerup tensor product over C ∗ -algebras is injective. From this we may make the following deduction (generalizing 3.6.5 (1)). Suppose in addition to the hypotheses of 3.6.5 (1), that we have two other C ∗ -algebras B and C which W and Z are respectively left and right operator modules over, and L is both a Hilbert B-module and a Hilbert C-module. Suppose that X and Y are subbimodules of W and Z respectively, and suppose that u : X × Y → B(L) is as in 3.6.5 (1) but with the further property that u(bx, yc) = bu(x, y)c for all x, y, b, c. Then we may choose the extension uˆ to also have this further property. This may be proved by using (2) of 3.6.5 to view X ⊗hA Y as a subbimodule of W ⊗hA Z, by noting that the latter is an operator bimodule by 3.4.9, and then using 3.6.3 to extend the associated bimodule map X ⊗hA Y → B(L) to a bimodule map W ⊗hA Z → B(L). We leave it to the reader to make this precise. With the last paragraph in hand, together with the associativity of the module Haagerup tensor product, it follows by induction that there are similar extension theorems for multilinear completely contractive ‘N -balanced’ maps Φ from a product X1 × X2 × · · · × XN of N bimodules, into B(L). The last fact, namely that multilinear completely bounded module maps into B(H) may be extended, is a key tool in completely bounded cohomology theory (see references in the Notes to Section 3.6). 3.7 FUNCTION MODULES Function modules resemble operator modules, except that we consider subspaces of a C(K)-space rather than subspaces of B(K, H). They form a good example of operator modules, but are less interesting as a class. Some of the most important modules over uniform algebras, such as those coming from representations on Hilbert space, are not function modules but operator modules. On the other hand, in this section we shall demonstrate a link between this class of operator modules and the classical theory of ‘M -structure’, that will simultaneously facilitate a deeper understanding of function modules, and point us toward the methods in Chapter 4 which give similar insights into the structure of general operator modules. Throughout this section E is a Banach space, with E = {0}. We will also consistently write KE for the topological space of extreme points of Ball(E ∗ ) equipped with the w∗ -topology. We write Cb (KE ) for the commutative C ∗ algebra of bounded continuous functions on KE , and j : E → Cb (KE ) for the canonical map given by j(x)(ψ) = ψ, x for x ∈ E and ψ ∈ KE . This is an isometry by the Krein–Milman theorem. We will use several times the obvious fact that ψ = 1 for any ψ ∈ KE . 3.7.1 (The function multiplier algebra) This is deﬁned to be, for a Banach space E, the closed unital algebra M(E) = {f ∈ Cb (KE ) : f j(E) ⊂ j(E)}. The centralizer algebra of E is Z(E) = {f ∈ Cb (KE ) : f, f¯ ∈ M(E)}.

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Note that Z(E) is a commutative unital C ∗ -algebra, and M(E) is a uniform algebra. Every Banach space E is a Banach M(E)-module. Indeed there is a canonical homomorphism π : M(E) → B(E), deﬁned by π(f )(x) = j −1 (f j(x)),

x ∈ E, f ∈ M(E).

This unital homomorphism is easily seen to be contractive and in fact it is isometric. To see this note that π(f ) = sup{f j(x) : x ∈ Ball(E)} = sup{|f (ψ)||ψ(x)| : x ∈ Ball(E), ψ ∈ KE } = sup{|f (ψ)|ψ : ψ ∈ KE } = f . The isometry π above maps Z(E) onto a closed unital-subalgebra of B(E). We will usually identify M(E) and Z(E) with their images in B(E) under this homomorphism. One of the main results of this section, is that just as Banach A-module actions on a Banach space E correspond to contractive homomorphisms from A into B(E) (see A.6.1), function module actions on E are in a canonical oneto-one correspondence with contractive homomorphisms into M(E). If A is a C ∗ -algebra then the correspondence is with ∗-homomorphisms into Z(E). Theorem 3.7.2 Let E be a Banach space and recall that K E = ext(Ball(E ∗ )). If T ∈ B(E), then the following are equivalent: (i) T ∈ M(E). (ii) There is a nonnegative constant C such that |ψ(T (x))| ≤ C|ψ(x)|,

x ∈ E, ψ ∈ KE .

(iii) ψ is an eigenvector for T ∗ for all ψ ∈ KE . (iv) There is a compact space Ω, a linear isometry σ : E → C(Ω), and an f ∈ C(Ω), such that σ(T x) = f σ(x), for all x ∈ E. The least C in (ii), and least f ∞ in (iv), coincides with the usual norm of T . Proof Given (iii), let aT (ψ) be the eigenvalue associated with ψ. Then |ψ(T (x))| = |aT (ψ)||ψ(x)|,

x ∈ E.

Taking the supremum over x ∈ Ball(E) gives |aT (ψ)| = ψ ◦ T ≤ T . From the last two equations, (ii) is obvious. On the other hand, if (ii) holds then the kernel of ψ ◦ T contains the kernel of ψ. Hence by linear algebra, ψ ◦ T is a scalar multiple of ψ. Thus (ii) implies (iii). Suppose again that (iii) holds, and deﬁne aT as above. We noticed that aT : KE → C is bounded. Let (ψi )i be a net in KE converging in the w∗ topology to some ψ ∈ KE . Choose x ∈ E with ψ(x) = 0. Then ψi (x) = 0 for large enough i, and

Basic theory of operator modules aT (ψi ) =

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ψ(T (x)) ψi (T (x)) −→ = aT (ψ). ψi (x) ψ(x)

Thus aT belongs to Cb (KE ). Since aT j(x) = j(T (x)), it is clear that (i) holds. Given (i), we appeal to Gelfand theory (see A.5.4) to see that Cb (KE ) ∼ = C(Ω) for some compact Ω. This yields (iv). That (iv) implies (iii) follows by the well-known consequence of the Krein– Milman theorem, that extreme points of Ball(σ(E)∗ ) extend to extreme points of Ball(C(Ω)∗ ) (e.g. see the proof of Lemma 4.1.2 below), and the fact that the latter extreme points are the obvious ones, namely point evaluations multiplied by a unimodular constant. If ψ ∈ KE = ext(Ball(E ∗ )), then ψ ◦ σ −1 is an extreme point of Ball(σ(E)∗ ), so that ψ(·) = ασ(·)(w) for some w ∈ Ω, and some α ∈ C, |α| = 1. This gives (iii), since for any x ∈ E, ψ(T (x)) = ασ(T (x))(w) = αf (w)σ(x)(w) = f (w)ψ(x). Thus (i)–(iv) are equivalent. If T ∈ M(E), with associated function g on K, and if f is obtained as in the proof of ‘(i) ⇒ (iv)’, then f ∞ = g∞ = T . With this same f , and tracing through the argument that (iv) implied (ii), we see that C in (ii) may be chosen to be T . If C satisﬁes (ii), then taking the supremum over ψ ∈ ext(Ball(E ∗ )) and x ∈ Ball(E), we see that T ≤ C. Thus T is the least constant possible in (ii). Similarly, if f is as in (iv), then it is 2 clear that f ∞ ≥ T since σ is an isometry. Since Z(E) ⊂ M(E), by the theorem above we may associate to any centralizer T ∈ Z(E) and ψ ∈ ext(Ball(E ∗ )), an eigenvalue aT (ψ) for T ∗ . Theorem 3.7.3 For T ∈ B(E) the following are equivalent: (i) T ∈ Ball(Z(E)) and aT ≥ 0. (ii) There is a compact space Ω and an f ∈ C(Ω) with 0 ≤ f ≤ 1, and a linear isometry σ : E → C(Ω) such that σ(T x) = f σ(x) for all x ∈ E. (iii) T (x) + y − T (y) ≤ max{x, y} for all x, y ∈ E. Proof That (i) implies (ii) is exactly as in the proof that (i) implied (iv) in 3.7.2. Assuming (ii), we have T (x) + y − T (y) = σ(T x) + σ(y) − σ(T (y)) = f σ(x) + (1 − f )σ(y). Pointwise, f σ(x) + (1 − f )σ(y) is a convex combination of σ(x) and σ(y). Thus T (x) + y − T (y) ≤ max{σ(x), σ(y)} = max{x, y}, which is (iii). Assume (iii). That is, the map ν : (x, y) → T (x) + y − T (y) is a contraction from E ⊕∞ E → E. Hence ν ∗ is a contraction. Thus if ψ ∈ ext(Ball(E ∗ )) then T ∗ (ψ) + ψ − T ∗ (ψ) ≤ 1.

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This implies that 1 = ψ = T ∗ (ψ) + (ψ − T ∗ (ψ)) ≤ T ∗ (ψ) + ψ − T ∗ (ψ) ≤ 1.

(3.14)

If T ∗ (ψ) is neither 0 nor ψ, (3.14) yields a convex combination ψ = T ∗ (ψ)

T ∗ (ψ) ψ − T ∗ (ψ) + ψ − T ∗ (ψ) . ∗ T (ψ) ψ − T ∗ (ψ)

Since ψ is an extreme point, we obtain T ∗ (ψ) = T ∗ (ψ)ψ and of course this equality holds as well if T ∗ (ψ) is 0 or ψ. Thus T satisﬁes (iii) of 3.7.2, with 2 aT (ψ) = T ∗ (ψ). This yields (i). 3.7.4 (Centralizers and M -projections) An M -projection on a Banach space E is an idempotent linear map P : E → E such that P (x) + y − P (y) = max{P (x), (I − P )(y)},

x, y ∈ E.

(3.15)

This is simply saying that E = P (E) ⊕∞ (I − P )(E). These M -projections are related to the important notion of M -ideals that we shall meet again in Section 4.8. If P is an M -projection then setting x = y shows that P and I − P are contractive, from which it is clear that 3.7.3 (iii) holds. Hence P is an idempotent in the centralizer algebra Z(E). Conversely, any idempotent P in Z(E) is an M projection, as may be seen by the following argument. An idempotent P in Z(E) is contractive, as is I − P , since Z(E) is a function algebra. Applying P to the quantity P (x) + y − P (y) shows that P (x) ≤ P (x) + y − P (y). Similarly, y − P (y) ≤ P (x) + y − P (y), and now we have one direction of (3.15). To obtain the other, replace x by P x and y by y − P y in 3.7.3 (iii). From this we see that the set of M -projections on E is exactly the lattice of projections in the centralizer algebra Z(E). The following is the main result of this section. We will not prove all the implications, since the purpose of this section is mostly to illustrate the connections between the main ideas. However we remark that the ‘missing proof’ also follows easily from a formulation of (v) given in 3.7.7, together with results in Chapter 4 (for example, from 4.6.2 in conjunction with 4.5.10). Theorem 3.7.5 Let A be a unital Banach algebra and let E be a nondegenerate left Banach A-module. The following are equivalent: (i) There exists a compact space Ω, a contractive and unital homomorphism θ : A → C(Ω), and an isometric linear map Φ : E → C(Ω), such that Φ(ax) = θ(a)Φ(x) for all a ∈ A, x ∈ E. (ii) There is a contractive unital homomorphism θ : A → Cb (KE ) such that j(ax) = θ(a)j(x) for all a ∈ A, x ∈ E (see 3.7.1). (iii) The canonical homomorphism A → B(E) given by the module action, maps into M(E) (indeed into Z(E) if A is a C ∗ -algebra).

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ˇ → E, where (iv) The module action A × E → E extends to a contraction A⊗E ˇ is the injective Banach space tensor product (see A.3.1 and 1.5.3). ⊗ (v) The module action A × E → E extends to a contraction A ⊗ g2 E → E, where g2 is the tensor norm in (A.6). Proof That (iii) implies (ii) is clear from the discussion of multipliers in 3.7.1. Since Cb (KE ) is a C(Ω) for some compact space Ω, we see that (ii) implies (i). If (i) holds, and if a1 , . . . , an ∈ A, x1 , . . . , xn ∈ E, then (3.16) ak xk = Φ ak xk = θ(ak )Φ(xk ). k

k

k

If ω ∈ Ω then θ(·)(ω) and Φ(·)(ω) are contractive functionals on A and E respectively. By the deﬁnition in A.3.1 it follows that θ(ak )(ω) Φ(xk )(ω) ≤ ak ⊗ xk , k

k

ˇ the last norm taken in A⊗E. Together with (3.16), this yields (iv). Since g2 dominates the injective tensor norm, (iv) implies (v). We omit the proof that (v) implies (iii), instead referring the reader to [63] for the argument, which is a reprise of an earlier argument of Tonge [411, 120], and uses the Pietsch factorization theorem and a little measure theory. For the last assertion in (iii): if A is a C ∗ -algebra then since the canonical homomorphism from A to B(E) is a contractive homomorphism into M(E), it is a ∗-homomorphism by the last paragraph in 2.1.2, and maps into Z(E). 2 3.7.6 (Characterization of function modules) A module E satisfying one of the equivalent conditions of Theorem 3.7.5, will be called a function module over A, or a function A-module. The most interesting point, perhaps, is the following. For any Banach space E, ﬁrst, E is a function M(E)-module, and second, by 3.7.5 (iii), any function A-module action on E is a prolongation (see 3.1.12) of the M(E) action. We have also established the correspondences mentioned in the last paragraph of 3.7.1. 3.7.7 Let A be a unital Banach algebra equipped with an operator space structure, and let E be a nondegenerate left Banach A-module. Claim: E is a function A-module if and only if Min(E) is an operator A-module. The ‘only if’ assertion is clear from condition (i) in 3.7.5. Conversely, if Min(E) is an operator A-module, then it is an h-module (see 3.1.3). Hence the module action induces a complete contraction from Max(A) ⊗h Min(E) to Min(E). But this is equivalent to condition (v) in 3.7.5 by (1.46) and (1.10). Hence E is a function A-module. 3.7.8 (Examples) (1) We show, as asserted at the start of this section, that there are no interesting function module representations on a Hilbert space H. Indeed ext(Ball(H ∗ )) is precisely the unit sphere of H ∗ . Thus if T is a multiplier of H then every nonzero vector is an eigenvector for T ∗ . If {ζ, η} is an orthonormal set

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in H, and if λ, ν are the corresponding eigenvalues, then for some µ we have T ∗ (ζ + η) = µ(ζ + η) = λζ + νη, showing that λ = µ = ν. Therefore T ∗ and T are scalar multiples of the identity operator. Thus H has no nontrivial multipliers. Hence there are no nontrivial function module actions on H. (2) In contrast, one can show that if A is a uniform algebra then M(A) ∼ =A isometrically isomorphically (as Banach algebras). We omit the details since these facts are generalized in 4.5.11. (3) If E is a closed subspace of C(Ω) and A is a uniform algebra on Ω, then AE is a function A-module. Thus there are plenty of examples of nontrivial function modules. Corollary 3.7.9 (Tonge–Kaijser) If A is a unital Banach algebra, with multiplication m viewed as a map A ⊗ A → A, then the following are equivalent: (i) A is isometrically isomorphic to a uniform algebra, ˇ −→ A is contractive, (ii) the mapping m : A⊗A (iii) the mapping m : A ⊗g2 A −→ A is contractive. Proof These follow by applying 3.7.5 with E = A regarded as an A-module. Indeed if this module satisﬁes condition (i) in 3.7.5 then we have a = Φ(a) = θ(a)Φ(1A ) ≤ θ(a)Φ(1A ) ≤ θ(a) for any a ∈ A, hence the contractive homomorphism θ : A → C(Ω) is an isometry. Thus A is (isometrically isomorphic to) a uniform algebra. 2 Corollary 3.7.10 If A is a unital operator algebra, then A is completely isometrically isomorphic to a uniform algebra if and only if A is a ‘minimal operator space’ (see 1.2.21). Proof The one direction is obvious. Conversely, assume that A is a ‘minimal operator space’ and an operator algebra. Then the product is contractive as a map from Min(A) ⊗h Min(A) to A, by 2.3.2. Hence it is contractive as a map from Max(A) ⊗h Min(A) to A. Now use (1.46) and 3.7.9 (iii). 2 An alternate proof of this result will be given in the Notes to Section 4.6. 3.8 DUAL OPERATOR MODULES In this section, M and N are at least unital Banach algebras which are also dual operator spaces. We discuss M -N -bimodules below, this contains the onesided module case by taking either M or N equal to C. Many of the results in this section are due to Eﬀros and Ruan in the case that M and N are von Neumann algebras. 3.8.1 (Dual operator bimodules) A concrete dual operator M -N -bimodule is a w∗ -closed subspace X of B(K, H) such that θ(M )Xπ(N ) ⊂ X, where θ and π are unital w∗ -continuous completely contractive representations of M and N on

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H and K respectively. An (abstract) dual operator M -N -bimodule is deﬁned to be an operator M -N -bimodule X, which is also a dual operator space, and which possesses a W ∗ -representation. By the latter term we mean a CES-representation (Φ, θ, π) in the sense of 3.3.3, with Φ, θ and π w ∗ -continuous, and with θ, π unital. By A.2.5, if there exists such a triple, then Φ is a w ∗ -homeomorphism onto its w∗ -closed range, and indeed Φ(X) is a concrete dual operator M -N -bimodule. If M and N are dual operator algebras, a W ∗ -representation is called faithful if θ and π are completely isometric. We remark that one may always choose a W ∗ -representation of a dual operator bimodule acting on a single Hilbert space. This may be accomplished, for example, by taking a large enough direct sum of the maps involved, as in the third last paragraph of the proof of 3.3.1. 3.8.2 (Normal dual modules) An M -N -bimodule X which is also a dual operator space is called a normal dual bimodule if the trilinear mapping (a, x, b) → axb from M × X × N to X is separately w ∗ -continuous. Since the product on B(H) is separately w∗ -continuous, it is clear that every dual operator bimodule is a normal dual bimodule. The converse of this is true too (see 4.7.7 and 4.7.6). For now we just state a preliminary form of this converse in the important operator algebra case. Theorem 3.8.3 Let M and N be unital dual operator algebras, and let X be a nondegenerate operator bimodule (or, equivalently, an h-bimodule) over M and N . Assume that X is a normal dual M -N -bimodule. (1) X has a faithful W ∗ -representation (Φ, θ, π) on a single Hilbert space H. Thus X is a dual operator M -N -bimodule. (2) If M = N , then the representation in (1) may be accomplished with π = θ. Proof We follow the proof of 3.3.1, except that we suppose that X ⊂ B(L) is a w∗ -closed subspace. As in that proof we may write axb as a product of maps. Indeed there are three completely contractive maps α : M → B(H0 , L), and v : X → B(K0 , H0 ), and β : N → B(L, K0 ), such that axb = α(a)v(x)β(b) for a ∈ M, b ∈ N and x ∈ X. Since (a, x, b) → axb is assumed to be separately w ∗ continuous, it follows as in 1.6.10 that the three maps α, v, β can be chosen to be w∗ -continuous. By a slight modiﬁcation of 2.7.10, we can then factor α and β as follows. There exist a Hilbert space H1 , a w∗ -continuous unital homomorphism θ1 : M → B(H1 ) and two contractions T1 : H0 → H1 and R : H1 → L such that α(a) = Rθ1 (a)T1 . Likewise, there exist a Hilbert space K1 , a w∗ -continuous unital homomorphism π1 : N → B(K1 ), and two contractions S : L → K1 and T2 : K1 → K0 such that β(b) = T2 π1 (b)S. We deﬁne Ψ : X → B(K1 , H1 ) by Ψ(x) = T1 v(x)T2 . Then Ψ is w∗ -continuous. Moreover, axb = Rθ1 (a)Ψ(x)π1 (b)S,

a ∈ M, x ∈ X, b ∈ N.

The rest of the proof is identical to that of 3.3.1, except that we need to check that all maps are w∗ -continuous. This is clear by the principle mentioned in the second paragraph of 1.6.10. 2

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3.8.4 (Examples) All of the examples in 3.1.2 have simple dual bimodule variants. For instance, a unital dual operator algebra M is canonically a dual operator M -M -bimodule. Since a w∗ -closed M -N -submodule of a dual operator M -N -bimodule is clearly a dual operator M -N -bimodule, it is easy to see that if M is a unital dual operator algebra, and if p and q are projections in M , then qM p is a dual operator (qM q)-(pM p)-bimodule. As in 3.3.4, one may view dual operator bimodules over unital dual operator algebras, as the corners pR(1 − p) in a unital dual operator algebra R. We leave the details as an exercise. Another important class of dual bimodules are the ‘normal rigged bimodules’ which we shall meet in Section 8.5. 3.8.5 (Normal Hilbert modules) If π : M → B(H) is a w ∗ -continuous unital completely contractive representation, then H (with its Hilbert column space structure) is a left operator M -module, by 3.1.6. Clearly H is a dual operator M -module. We call such H a normal Hilbert module. The category M N HM OD of normal Hilbert M -modules is a subcategory of M HM OD. An obvious (but important) fact is that M N HM OD is closed under Hilbert space direct sums: this is just the fact that a direct sum of w ∗ -continuous representations of M is a w∗ -continuous representation. If M is a W ∗ -algebra, then there is a ‘most important’ module in M N HM OD, namely the standard form L2 (M ). We shall not really need this module in this book, although it is mentioned brieﬂy in 8.5.39. It is worth pointing out that if M is a von Neumann algebra acting on a separable Hilbert space H, then H is (up to spatial equivalence) the standard form if and only if M has a ‘separating cyclic vector’ ξ ∈ H (that is, H = [M ξ], and the map x → xξ on M is one-to-one). See [175] and [408, Chapter IX]. It follows from 2.5.9 that there is a bijective correspondence between nondegenerate completely contractive representations of an approximately unital operator algebra A, and w∗ -continuous completely contractive unital representations of A∗∗ . Thus as objects, A HM OD = A∗∗ N HM OD. If T : H1 → H2 is a morphism in A HM OD, then by the separate w∗ -continuity of operator multiplication, T is also a morphism in A∗∗ N HM OD. The converse is even easier, so that we see that A HM OD and A∗∗ N HM OD are equal as categories. Thus, for example, H is a generator (see 3.2.7) for A HM OD if and only if H (considered as an A∗∗ -module in this canonical way) is a generator for A∗∗ N HM OD. Proposition 3.8.6 Let ρ : M → B(H) be a normal representation of a W ∗ algebra M , and view H ∈ M N HM OD. Then ρ is faithful if and only if H is a generator for M N HM OD (and if and only if H is a cogenerator). Proof Suppose that H is a generator, and that x ∈ M with ρ(x) = 0. Suppose that M is a von Neumann algebra in B(K), say. For any T ∈ M B(H, K), we have xT = T ρ(x) = 0. Hence x = 0 on the closure L of the joint span of the ranges of all such maps T . Clearly L is an M -submodule of K, and hence so is L⊥ . Hence L⊥ = (0), by deﬁnition of L and of ‘generator’. Thus x = 0, and so ρ is faithful.

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For the converse, we may suppose that M is a von Neumann algebra in B(H). To show that H is a generator, it suﬃces to show that M B(H, K) = (0), for any nontrivial K in M N HM OD (since Ker(S)⊥ is an M -module for any morphism S of M N HM OD). Indeed, it suﬃces to show that there exists a nonzero map T in M B(H (∞) , K), where H (∞) is the countably inﬁnite multiple of H. For if this were the case, and if i is the inclusion of H into H (∞) as the ith summand, then T ◦ i = 0 for some i. Fix ξ ∈ K, ξ = 1, and let π be the representation of M on K. Then π(·)ξ, ξ is a normal state on M . By the proof of 3.6.6 in [320], there exists η ∈ H (∞) such that π(x)ξ, ξ = xη, η, for all x ∈ M . Deﬁne T : M η → K by T (xη) = π(x)ξ. By the above, T is isometric, and hence extends to an isometric M -module map from [M η] onto [π(M )ξ]. Let P be the projection onto [M η], this is also an M -module map by 3.2.3. Then 0 = T P ∈ M B(H (∞) , K). The cogenerator assertion follows as in 3.2.11 (7). 2 We remark in passing that a simple Zorn’s lemma argument applied to the spaces [π(M )ξ] in the last proof, will show that any K in M N HM OD is spatially equivalent to a direct sum of submodules of H (∞) . It follows that K is spatially equivalent to a submodule of a multiple of H. This shows that every generator H for M N HM OD is ‘universal for M N HM OD’. Conversely, the proof of 3.2.11 (6) shows that any ‘universal for M N HM OD’ is a generator. From this, and the remarks above 3.8.6, it is an easy exercise to prove the matching facts in 3.2.8. 3.8.7 (Direct sums of dual modules) For a family {Xi : i ∈ I} of dual operator Mi -Ni -bimodules, the direct sum ⊕∞ i Xi is a dual operator bimodule over ∞ the algebras ⊕∞ M and ⊕ N . To see this, suppose that (Φi , θi , πi ) are W ∗ i i i i representations of Xi , for each i, and suppose (as we may) that Φi , θi , πi map into Bi = B(Hi ). Then by the last paragraph in 1.4.13, we obtain w ∗ -continuous ∞ complete contractions Φ = (Φi ), θ = (θi ), and π = (πi ), from ⊕∞ i Xi , ⊕ i M i , ∞ ∞ and ⊕i Ni , respectively, into ⊕i Bi ⊂ B(⊕i Hi ). Moreover Φ is completely isometric. It is easy to check that θ and π are homomorphisms, and that (Φ, θ, π) is a W ∗ -representation for the bimodule ⊕∞ i Xi . This yields the desired result. 3.8.8 (Quotients of dual modules) Let M and N be unital dual operator algebras for simplicity. Suppose that X is a dual operator M -N -bimodule and that Y is a w∗ -closed M -N -submodule. In 3.1.10 and 3.3.2 we said that X/Y is a nondegenerate operator M -N -bimodule, and we said that the canonical quotient map q : X → X/Y is an M -N -bimodule map. By the duality of subspaces and quotients (see 1.4.4), X/Y is the dual operator space of Y⊥ , and q is w∗ continuous. From this and the fact that X is a normal bimodule, it is easy to see that the bimodule actions on X/Y are separately w ∗ -continuous. That is, X/Y is a dual operator M -N -bimodule by Theorem 3.8.3. 3.8.9 (The second dual of a bimodule) Suppose that A and B are approximately unital operator algebras. Then A∗∗ and B ∗∗ are unital dual operator algebras by 2.5.8 and 2.7.3. If X is a nondegenerate operator A-B-bimodule,

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then X ∗∗ is a dual operator A∗∗ -B ∗∗ -bimodule in a canonical way. To see this, ﬁrst notice that by 1.6.7, there is a unique separately w ∗ -continuous extension from A∗∗ × X ∗∗ to X ∗∗ , of the A-module action on X. Similarly we obtain a separately w∗ -continuous map from X ∗∗ × B ∗∗ to X ∗∗ . Using the separate w∗ continuities and routine w∗ -approximation arguments, it is easy to check that these are nondegenerate module actions, and that they commute. Thus X ∗∗ is an A∗∗ -B ∗∗ -bimodule for these canonical second dual actions. Since these actions A∗∗ × X ∗∗ → X ∗∗ and X ∗∗ × B ∗∗ → X ∗∗ are completely contractive (see 1.6.7), X ∗∗ is an h-bimodule. It therefore follows from Theorem 3.8.3 that X ∗∗ is a dual operator A∗∗ -B ∗∗ -bimodule. 3.8.10 (W ∗ -converging inﬁnite sums) Let H be a Hilbert space, let I be a cardinal, and suppose that a = (ai )i ∈ RIω (B(H)) and b = (bi )i ∈ CIω (B(H)). According to (1.19), we may identify RIω (B(H)) and CIω (B(H)) with B(H (I) , H) and B(H, H (I) ) respectively. Let S : H (I) → H and T : H → H (I) correspond to a andb in these identiﬁcations. For each ﬁnite subset I0 of I consider the ﬁnite sum i∈I0 ai bi ∈ B(H). We obtain a bounded net indexed by such I0 and it is easy to check that this net converges in the w ∗ -topology of B(H) to ST . In the ∗ sequel, we will denote this w -limit by i∈I ai bi ∈ B(H). Proposition 3.8.11 (Dual matrix modules) If X is a dual operator bimodule over M and N , and if I, J are sets, then MI,J (X) is also a dual operator bimodule over M and N . If M and N are unital dual operator algebras then MI,J (X) is a dual operator MI (M )-MJ (N )-bimodule. Proof Fix a W ∗ -representation (Φ, θ, π) of X on a Hilbert space H, and ﬁx cardinals I and J. Then by 1.6.3 (2), the ampliﬁcation Ψ = ΦI,J : MI,J (X) −→ MI,J (B(H)) ∼ = B(H (J) , H (I) ) is a w∗ -continuous complete isometry. Also, it is easy to see that the multiples (I) (J) ) and π (J) : N → ∞ ) are θ(I) : M → ∞ I (B(H)) ⊂ B(H J (B(H)) ⊂ B(H ∗ (I) w -continuous completely contractive homomorphisms. Since (Ψ, θ , π (J) ) is evidently a W ∗ -representation of the M -N -bimodule MI,J (X), we obtain the ﬁrst assertion. For the second, we may assume by Theorem 3.8.3 that M, N are unital w ∗ closed subalgebras of B(H), and that X ⊂ B(H) is a w ∗ -closed submodule of B(H). Using (1.19), we see that MI,J (B(H)) ∼ = B(H (J) , H (I) ) is a dual operator MI (B(H))-MJ (B(H))-bimodule, and hence it is clearly a dual operator MI (M )-MJ (N )-bimodule. Thus we only need to check that the w ∗ -closed subspace MI,J (X) ⊂ MI,J (B(H)) is a submodule. This follows from the ﬁrst paragraph of 3.8.10. Indeed if a = [aik ] ∈ MI (M ) and x =[xkj ] ∈ MI,J (X), then the i-j entry of ax ∈ MI,J (B(H)) latter is the (inﬁnite) sum k∈I aik xkj . The ∗ ∗ -limit of ﬁnite sums a x of terms in X. Since X is w -closed, is the w k∈I0 ik kj 2 k∈I aik xkj ∈ X. The argument for the right action is the same. Henceforth in this section M and N are unital dual operator algebras.

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3.8.12 Let X and Y be dual operator M -N -bimodules, and suppose that u : X → Y is a w∗ -continuous completely bounded M -N -bimodule map. Then it is not hard to deduce from the proof of 3.8.11 that the w ∗ -continuous completely bounded map uI,J : MI,J (X) → MI,J (Y ) is a MI (M )-MJ (N )-bimodule map. In fact a surprising stronger statement holds in the case that M and N are selfadjoint. We assume that I = J (the contrary case is derivable from this one): Theorem 3.8.13 (Eﬀros and Ruan) Let X and Y be dual M -N -bimodules over W ∗ -algebras M and N , and suppose that u : X → Y is a given completely bounded M -N -bimodule map (we are not assuming that u is w ∗ -continuous). Let I be a cardinal. Then the map uI : MI (X) → MI (Y ) is a completely bounded MI (M )-MI (N )-bimodule map. Proof By 3.8.3, we may assume that X ⊂ B(K, H) and Y ⊂ B(L, G) are concrete dual operator M -N -bimodules, with M faithfully represented as a unital von Neumann algebra on both H and on G, and N faithfully represented as a unital von Neumann algebra on both K and on L. Form the bimodule Paulsen systems S 1 ⊂ B(H ⊕ K) and S 2 ⊂ B(G ⊕ L) (see 3.6.1) of X and Y respectively. Then the two systems MI (S 1 ) ⊂ MI (B(H ⊕ K)) = B(H (I) ⊕ K (I) ) and MI (S 2 ) ⊂ B(G(I) ⊕ L(I) ), may respectively be thought of as the bimodule Paulsen systems of the MI (M )-MI (N )-bimodules MI (X) ⊂ B(K (I) , H (I) ) and MI (Y ) ⊂ B(L(I) , G(I) ) (see 3.8.11 and its proof). By 3.6.1, we obtain a completely positive unital map Θ : S 1 → S 2 taking a u(x) a x −→ , a ∈ M, b ∈ N, x, y ∈ X. u(y)∗ b y∗ b Then Ψ = ΘI : MI (S 2 ) → MI (S 2 ) is completely contractive, unital, and completely positive by 1.3.3. Since Ψ is a ∗-homomorphism on MI (M ) ⊕ MI (N ), it is a bimodule map by 1.3.12. Thus the ‘1-2-corner’ of Ψ, which is uI , is an 2 MI (M )-MI (N )-bimodule map. 3.8.14 (A module version of the weak* Haagerup tensor product) We will be very brief, and omit proofs, since we are leaving the realm of ‘basic theory’ here. If X and Y are respectively right and left dual operator M -modules, then one may deﬁne the weak* module Haagerup tensor product X ⊗w∗ hM Y , to be the quotient of the weak* Haagerup tensor product X ⊗w∗ h Y (see 1.6.9) by the smallest norm closed subspace containing terms of form xr y − x ry. Here w w x∈R I (X), y ∈ CI (Y ), and r ∈ MI (M ). Also, the expression x y means the such as xr sum i∈I xi ⊗ yi in X ⊗w∗ h Y , in the sense of 1.6.9. The expressions above make sense by virtue of 3.8.11. We continue to write i∈I xi ⊗ yi and x y ∗ for the equivalence class in the quotient X ⊗w∗hM Y . Every w ∈ X ⊗w hM Y may be written in such a way, and we call the expression i∈I xi ⊗ yi a weak representation for w. It is not too diﬃcult to check that some of the properties of the module tensor product that we saw in Section 3.4, have appropriate versions for this

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new tensor product. For example, this tensor product is functorial. Indeed, one can show easily that if u : X1 → X2 and v : Y1 → Y2 are completely bounded M -module maps between dual operator M -modules, and if either (a) M is a W ∗ -algebra, or (b) u and v are w∗ -continuous, then there is a well deﬁned linear ∗ ∗ completely bounded map u⊗vfrom X1 ⊗w hM Y1 to X2 ⊗w hM Y2 taking any weak representation i xi ⊗ yi to i u(xi )⊗ v(yi ). Also, u ⊗ vcb ≤ ucbvcb . It is also easy to see that if X is a left dual operator M -module then M ⊗w∗ hM X ∼ =X completely isometrically. Other, more sophisticated, properties of the module weak* Haagerup tensor product, such as associativity (in the sense of 3.4.10), are much harder to prove. Indeed, for these it is preferable to use equivalent formulations of this tensor product, which have been introduced and studied by Magajna in the case that M is a W ∗ -algebra (see the Notes section). One attractive feature of this tensor product, is that it can be quite useful in the manipulation of certain dual space identities. To illustrate this, suppose that M is a W ∗ -algebra, and that X is a dual operator M -module. One can show using (1.59), and the just mentioned associativity, that CIw (M ) ⊗w∗ hM X ∼ = CI ⊗w∗ h M ⊗w∗ hM X ∼ = CI ⊗w∗ h X ∼ = CIw (X). Setting X = RJw (M ) and using (1.20) yields CIw (M ) ⊗w∗ hM RJw (M ) ∼ = CIw (RJw (M )) ∼ = MI,J (M ). 3.9 NOTES AND HISTORICAL REMARKS Linear spaces of Hilbert space operators which are bimodules over operator algebras have been studied for decades (e.g. see [108]). As usual we have restricted our attention to topics particularly connected with operator spaces. Operator modules over C ∗ -algebras were ﬁrst deﬁned by Wittstock around 1980 in [432], where amongst other things he gave module map extension theorems of the type in 3.6.2. See also [431, 273]. Other early theory of operator modules and their tensor products appears in papers of Eﬀros, alone or with coauthors Kishimoto and Exel (e.g. see [135] and [138]). These were partly inspired by some work of Haagerup [176, 177]. Christensen, Eﬀros, and Sinclair characterized operator modules over unital C ∗ -algebras in [92], and used this to solve some questions about the cohomology of operator algebras. See also [385], and references therein, for more on bounded cohomology of operator algebras; there is also the school of Helemskii (e.g. see [199, 200]). Operator modules were also studied in the 1980s (under that name) by Eﬀros and Ruan (see [143]). However later they changed notation and refer to what we call ‘matrix normed modules’ by the name ‘operator module’. The objects which we call ‘matrix normed modules’ appear ﬁrst in Ruan’s work on operator amenability (see [149,377], and references therein), and later played a role in [236, 374, 375], for example. The development of operator modules was of course inﬂuenced by important earlier work on Hilbert modules (see references in 3.2.6). Another early source for operator modules and their tensor products are the papers of Smith with various

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coauthors (e.g. see [391, 85]), and of Ara and Mathieu [12]. These date to the ﬁrst few years of the 1990s decade, and contain many very useful and powerful techniques that were beyond the scope of our book. Much of this work was on the ‘central Haagerup tensor product’, that is the module Haagerup tensor product R ⊗hZ R of a von Neumann algebra R over its center Z. This is related to the concept of ‘elementary operators’; for example the operators on a von Neumann algebra R of the form x → k ak xbk . Here ak , bk ∈ R. See also [13]. The associated map R ⊗Z R → CB(R) whose range is the set of such operators extends to a map R ⊗hZ R → CB(R). It is shown in [85] that this map is an isometry, generalizing an older result of Haagerup [176]. The memoir [65] of Blecher, Muhly, and Paulsen (circulating since the ﬁrst half of the 1990s) is a standard reference for operator modules and the module Haagerup tensor product. The ideas in Section 3.4 (and many of the ideas in other sections) may be found here. Some of these facts were found independently in [261]. Some of the algebraic aspects of the theory have been inspired by Muhly’s consistent advocation (for example, in his CBMS lectures of May 1990, or [279]) of the transferal of the classical algebraic theory of rings and modules to the framework of modules over general operator algebras. Of course there are several natural obstacles, for example the dual of an operator module is usually not an operator module (although it is a matrix normed module), and direct sums can be problematic. Nonetheless this is a very important and useful perspective. 3.1: Many of the basic constructions with operator modules discussed in Section 3.1 are well-known (e.g. see [65]). Although we will not take the time to do so here, one may deﬁne, for example, limsup’s, direct limits, and ultraproducts, of operator bimodules. See 2.3.3, 2.3.8, 1.2.31, and 2.2.13 for the operator algebra versions of these constructions which are easily adapted. We are not sure whom to attribute 3.1.7 to—perhaps [65]. The Banach algebra version of 3.1.11 is due to Johnson (see [106, 2.9.50]). 3.2: For historical sources of Hilbert modules, we refer to [21, 22] in addition to those cited in 3.2.6. Item 3.2.2 is from [381]. The second half of the section comes from [72], and is also valid for contractive as opposed to completely contractive representations. We also study there a suﬃcient condition for the double commutant property to hold which is more general than the notion of ‘generator’. See that paper for more information. The double commutant theorem here is an important tool, for example in the study of Morita equivalence of operator algebras. The notions of A-universal, generator, cogenerator, in 3.2.7 are functorial. The facts in 3.2.8 are from [361]. The most common tensor product for Hilbert modules is given by the procedure involving π ⊗ θ in 2.2.2. Another very important tensor product is the so-called ‘fusion’ or ‘relative tensor product’ of Connes’ bimodules (the ‘correspondences’ brieﬂy discussed in 8.5.39). E.g. see [101, Section V.B] or [408, IX.3]. 3.3: Theorem 3.3.1 is a remarkable result due to Christensen–Eﬀros–Sinclair [92]. Its proof still works if A and B are not necessarily approximately unital, and without any nondegeneracy condition, if X is merely ‘normed by A and B’.

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By this we mean [xij ]n = sup{[axij b]n : a ∈ Ball(A), b ∈ Ball(B)},

[xij ] ∈ Mn (X), n ∈ N.

Magajna, Johnson, and Pop have given isometric characterizations of Banach bimodules which admit an operator module structure (e.g. see [267, 345]). 3.4. See references at the beginning of this Notes section. Some other facts mentioned in this section are from [19]. It is easy to see by an argument almost identical to the proof of the projectivity of the Haagerup tensor product in 1.5.5, that the module Haagerup, and module projective, tensor products, are also projective in an appropriate sense (for complete quotient module maps). If X and Y are left operator modules over operator algebras A and B respectively, then the minimal tensor product X ⊗min Y is an operator module over A ⊗min B, with the ‘canonical action’ (a ⊗ b)(x ⊗ y) = (ax) ⊗ (by). An easy way to see that this is an operator module is to use 3.3.1 to choose two Hilbert spaces H and K with X → B(H) and Y → B(K), and completely contractive representations θ and π of A and B on H and K respectively, such that θ(A)X ⊂ X and π(B)Y ⊂ Y . Then X ⊗min Y may be identiﬁed with a subspace of B(H ⊗ K), and by 2.2.2 there is a completely contractive homomorphism θ ⊗ π : A ⊗min B → B(H ⊗ K). Clearly X ⊗min Y is a concrete left operator (A ⊗min B)-module for the action induced by θ ⊗ π. It is easy to see that this tensor product is ‘functorial’ (in a sense similar to 3.4.5) for completely ¯ of the bounded module maps. Similarly one may deﬁne a weak* version X ⊗Y minimal tensor product of dual operator bimodules using 1.6.5 and 3.8.3 (see 8.2.17 or [48] for a special case of this). Another kind of module tensor product A⊗Z B of C ∗ -algebras over a commutative C ∗ -algebra Z may be found discussed in [37, 262]. 3.5: Again [65] is a basic reference here. The space A CB(A, X) considered in 3.5.4 sometimes appears in the literature (e.g. see [19,51,132]) as a useful module version of the multiplier algebra of X. See also the last paragraph of the Notes to Section 8.1. One can show quite easily that if X is a left operator A-module, then so is A CB(A, X). Variants of 3.5.11 appear in [135, 288]. Some other ideas in 3.5 (in some form) appear in e.g. [46–48, 51, 52]. Some of the module mapping space results in this section have been observed by others too, for example by Aristov [19], Peters [323], and Neufang (private communication). Magajna has some results, in [265, 268] for example, for the module Haagerup tensor product corresponding to 3.5.9 and some of the relations in 1.5.4, 1.5.5, 1.5.7, 1.6.9. A Hilbert module H over A is called locally cyclic if any ﬁnite subset of H is contained in [Aξ], for some vector ξ ∈ H. If H and K are locally cyclic Hilbert modules over C ∗ -algebras A and B respectively, and if X is a given operator A-B-bimodule, then Smith has shown that T = T cb for any A-B-bimodule map T : X → B(K, H). There are also versions of this result valid for bimodule maps. See [391, Section 2], [385, Section 1.6], [7]. There are some relations between modules over an approximately unital operator algebra A, and modules over a C ∗ -cover B of A, which are often useful.

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It is shown in [50], using A.5.9, that a Banach A-module E has at most one Banach B-module action extending the A-module action. Thus if T : E1 → E2 is a surjective isometric A-isomorphism between two Banach D-modules, then T is a B-isomorphism too. It follows from this that the category of Banach modules over B is a subcategory (via the ‘forgetful functor’) of the category of Banach modules over A. Similarly, B OM OD is a subcategory of A OM OD, and B HM OD is a subcategory of A HM OD. This is not as obvious as it looks at ﬁrst sight. There is a natural variant for modules of the notion of C ∗ -cover from Chapter 2. This is the dilation or adjunct, and was studied in [288, 280, 50]. It may be deﬁned to be B ⊗hA X, for an operator A-module X and a C ∗ -cover B of A. In module theory this is called a change of rings. This construction has many useful properties which are noted in the cited references. In particular, its universal property provides another way to transfer certain problems into the C ∗ -algebra world where they may be solved (e.g. see [52]). In the language of algebra, this dilation, and the forgetful functor discussed in the last paragraph, form a pair of adjoint functors between B OM OD and A OM OD. 3.6: After Wittstock’s work on module extensions cited above, Suen gave in [402] the pretty proof of 3.6.2. Muhly and Solel consider various kinds of injectivity of Hilbert modules in [281] (e.g. see Theorem 3.1 there). Another important reference is Paulsen’s work on injectivity in various module categories, and relations to cohomology (e.g. see [311, 163, 217, 314]). See also, for example, [68, 50], and papers of Christensen, Sinclair, and Smith cited here. For 3.6.5 (i) see [386, Theorem 3.1]; the main idea of the proof is essentially a remark in [318]. The injectivity of the module Haagerup tensor product over C ∗ -algebras appears in [7]. None of the results in Section 3.6 are valid in general with A replaced by a nonselfadjoint operator algebra. See, for example, Proposition 7.2.11, and Example 3.5 in [391]. If X is a nondegenerate operator module over a C ∗ -algebra, and Φ is a linear contractive projection from X onto a closed submodule, then Φ is a module map. This is proved in the paper in preparation referred to in the Notes to 4.7. 3.7: Multipliers of Banach spaces were ﬁrst considered by Cunningham [104], but Alfsen and Eﬀros’ paper [4] is the deﬁnitive source for most of the basic ideas of this theory. Behrend’s contributions are also quite signiﬁcant [31]. See [195] for more detailed references, and an excellent survey of Banach space multipliers, including several other important characterizations of multipliers and centralizers. For example, Z(A) for a C ∗ -algebra A is the center of M (A). This old fact essentially dates back to [4]; it generalizes to approximately unital operator algebras but now one needs the diagonal of the center of M (A) (see [73, Section 7]). Jarosz proves in [204] that a Banach space E not containing a isometric copy of c0 (and in particularly any reﬂexive Banach space) has Z(E) = M(E) ∼ = ∞ n for some n ∈ N. Hence the function A-module actions on such E may be completely characterized: the n minimal projections in Z(E) give an isomorphism E∼ = ⊕nk=1 Ek , and each Ek has only a ‘scalar’ A-module action.

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Many of the results in this section are taken from [195], the main exception being the main theorem, which is a slightly weaker version of a result in [63], which in turn relies mainly on Tonge’s result from [411]. Corollary 3.7.9 is from [216,411], whereas 3.7.10 is from [40]. Some applications of 3.7.5 are given in [63], and in Part A of [53]. 3.8: The material in this section developed out of Eﬀros and Ruan’s excellent paper [143], treating dual operator modules over von Neumann algebras. They proved many of the results here (for example, 3.8.3 and 3.8.11) in that case. The case of 3.8.3 for A and B dual operator algebras is from [63]. More general results will be met later at the end of 4.7. For W ∗ -algebras, the category N HM OD was studied under the name N ormod by Rieﬀel [361]. Item 3.8.6, and the remark after it, are from that paper, as are observations such as the fact that A HM OD = A∗∗ N HM OD. For a discussion of the second dual of Banach bimodules, e.g. see 2.6.15 and A.3.51–54 in [106]; these results extend easily to matrix normed modules. Many of the earlier results in this section have simple ‘normal matrix normed module’ versions. The result 3.8.9 (or a small variant of it) was proved in [311]. Second duals have natural w ∗ -continuous extension properties. For example, let X be an operator bimodule over approximately unital operator algebras A and B, and let Y ∗ be a dual matrix normed bimodule (in the sense of 3.1.5(2)) over A and B. Then if u : X → Y ∗ is a completely ˜ : X ∗∗ → Y ∗ is easily bounded A-B-bimodule map, its w∗ -continuous extension u seen to be an A-B-bimodule map as well. Thus in analogy with (1.28), we have an isometric identity A CBB (X, Y ∗ ) = w∗ -A CBB (X ∗∗ , Y ∗ ). A special case of the result 3.8.13 was ﬁrst proved by May [272], the general case is from [143] (see also [273]). The module weak* Haagerup tensor product was mentioned in [139], and comprehensively explored in [263]. It is also known as the extended module Haagerup tensor product. It is not obvious that the deﬁnitions from those two papers coincide; Magajna has pointed out to us that one needs to use Lemma 3.2 (and Remark 3.7) from [263]. Magajna treats the more general case of ‘strong operator modules’ over W ∗ -algebras; unfortunately we are not able to develop these interesting modules here. Projection techniques in the W ∗ -algebras allow Magajna to show that the module weak* Haagerup tensor product behaves very similarly to the usual weak* Haagerup tensor product. In [48] this theory was applied to selfdual C ∗ -modules (see 8.5.40). For more recent results on this tensor product, see [268] and references therein. A sampling of other interesting papers on operator modules not mentioned above, or elsewhere in our text, includes: [7, 18, 95, 202, 253, 288, 292, 345]. More references may be found within these papers. The reader is also directed to the long series of important papers by Magajna on operator modules, we have only cited a sample in our bibliography. Several recent Ph.D. theses have also focused on aspects of operator modules with applications to cohomology or harmonic analysis, for example, those of Neufang [291], Spronk [396], and Wood [434]. The reader might also consult [377, 378] for the work of Runde, and for other references.

4 Some ‘extremal theory’

4.1 THE CHOQUET BOUNDARY AND BOUNDARY REPRESENTATIONS Many problems in functional analysis are best tackled via extreme points. Extreme points, particularly in the guise of the Choquet or Shilov boundaries, play a substantial role in the theory of uniform algebras. Since uniform algebras are particular examples of operator algebras (see 2.2.5) one might expect extreme points to play a large role for us. Instead, in our book we use a variant of the ‘extremal approach’, focused on the ‘noncommutative Shilov boundary’ and the ‘injective envelope’. This theory, and its many applications, are developed in this chapter, and in Section 8.4. We already saw in Section 3.7 that function algebras and function modules may be completely characterized in terms of certain sets of extreme points of the unit ball in the dual space. Later in this chapter we shall see a noncommutative version of this, which is tightly connected to the ‘noncommutative Shilov boundary’, and which gives deep insights into the structure of operator algebras and their modules. For motivational purposes, we ﬁrst look at the classical Choquet and Shilov boundaries. The material in this ﬁrst section of the chapter will not be relied on later, except for inspiration and comparison. 4.1.1 We recall from 2.2.5 that a uniform algebra is a closed unital-subalgebra A of C(Ω), for a compact Hausdorﬀ space Ω. Some authors suppose that A separates points of Ω too. There are several good texts on uniform algebras, e.g. see [167, 400]. We use the term function space for a closed subspace of C(Ω). A unital function space is a closed subspace F of C(Ω) which contains constant functions. More generally, we also use this term for a Banach space E with a distinguished element e ∈ E, such that E is isometrically isomorphic to an F as above, with the isomorphism taking e to 1. A map with the latter property will be called ‘unital’. For a unital function space (E, e) we write S(E) for the set of functionals ϕ ∈ E ∗ with ϕ(e) = ϕ = 1. Thus S(C(Ω)) is the set of probability measures on Ω. We write δω for the point mass at ω ∈ Ω, and ω for its restriction to E, if E ⊂ C(Ω). A Choquet boundary point for E in Ω is deﬁned to be a point ω ∈ Ω satisfying one of the equivalent conditions in the next result:

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Lemma 4.1.2 Suppose that E is a nonzero closed subspace of C(Ω), for a compact Hausdorﬀ space Ω. Suppose that E contains constant functions and separates points of Ω. For a point ω ∈ Ω the following are equivalent: (i) ω ∈ ext(Ball(E ∗ )), (ii) ω ∈ ext(S(E)), (iii) δω is the only probability measure on Ω extending ω . Moreover, if ψ ∈ ext(S(E)), then ψ = ω for some (necessarily unique) ω ∈ Ω. Proof The equivalence of (i) and (ii) is an elementary exercise. To prove the equivalence of (ii) and (iii), we will use the well-known fact that ext(S(C(Ω))) = {δω : ω ∈ Ω}.

(4.1)

Suppose that ω satisﬁes (iii), and that ω = 12 (ϕ + ψ), with ϕ, ψ ∈ S(E). By the Hahn–Banach theorem, we may extend ϕ, ψ to probability measures on Ω. Then the average of these measures, by hypothesis, equals δω . By (4.1), both probability measures coincide with δω . So ϕ = ψ = ω . This yields (ii). Let ψ ∈ ext(S(E)), and consider the set Pψ of probability measures on Ω extending ψ. This is a nonempty (by the Hahn–Banach theorem) compact convex subset of S(C(Ω)), which has an extreme point by the Krein–Milman theorem. An obvious argument shows that such an extreme point is an extreme point of S(C(Ω)), and is hence by (4.1) a point mass δσ for some necessarily unique σ ∈ Ω. This proves the ﬁnal assertion. If ω satisﬁes (ii), and ψ = ω , then the 2 above argument shows that σ = ω, and Pψ is a singleton. This yields (iii). 4.1.3 (The Choquet boundary) For any unital function space (E, e), we consider the collection of pairs (Ω, j) consisting of a compact Hausdorﬀ space Ω, and a unital linear isometry j : E → C(Ω), such that j(E) separates points of Ω. We shall call such a pair a function-extension of E. The above lemma shows that the set of Choquet boundary points in Ω does not depend essentially on the function-extension of E, and that this set is homeomorphic to ext(S(E)) as topological spaces. Note that ext(S(E)) is not empty by the Krein–Milman theorem. Thus we may refer to either ext(S(E)), or to the set of Choquet boundary points in Ω, as the Choquet boundary of E. We write Ch(E) for either set. 4.1.4 (The Shilov boundary) As noted in 1.2.21, any Banach space E is linearly isometric to a closed subspace of C(Ω), for some compact Ω. Indeed, this may be done so that E separates points. The question arises of ﬁnding the smallest Ω which works, that is, what is the minimal or ‘essential’ topological space on which E can be supported in this way? For unital function spaces E such a minimal Ω does exist, and it is called the Shilov boundary ∂E of E. To be a little more careful, we declare two function-extensions (Ω, j) and (Ω , j ) to be E-equivalent, if there exists a surjective homeomorphism τ : Ω → Ω such that j (x) ◦ τ = j(x), for all x ∈ E. This is an equivalence relation on the collection of function-extensions of E. We deﬁne a Shilov boundary for a unital

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function space (E, e) to be a pair (∂E, i) having the universal property of the next theorem. Theorem 4.1.5 (The boundary theorem for unital function spaces) Let (E, e) be a unital function space. Then there exist a compact Hausdorﬀ space, written ∂E, and an isometric unital map i : E → C(∂E) such that i(E) separates points of ∂E, with the following universal property: Given any function-extension (Ω, j) of E, there exists a (necessarily unique) topological embedding τ : ∂E → Ω, such that j(x) ◦ τ = i(x) for all x ∈ E. 4.1.6 (Remarks on the universal property) Before proving the theorem, we will make ﬁve important observations that ﬂow from the universal property in 4.1.5. First, note that any space (∂E, i) having this universal property, has the property that there is no proper closed subset Ω of ∂E such that i(·)|Ω is still an isometry on E. This may be seen by letting j = i(·)|Ω and appealing to the universal property. One obtains a map τ : ∂E → Ω, which is immediately seen to be the identity map from the fact that i(E) separates points. Hence Ω = ∂E. Second, we remark that it is a simple exercise to see that a function-extension (Ω, j) of E is E-equivalent to (∂E, i) if and only if (Ω, j) also has the universal property of the theorem. Thus the set of function-extensions having the universal property of the theorem, is one equivalence class of the relation we called E-equivalence above. Third, we point out that if (Ω, j) is a function-extension of E, and if τ is the associated topological embedding τ : ∂E → Ω coming from the universal property in the theorem, then letting Ω = τ (∂E) and j (x) = j(x)|Ω , we have that (Ω , j ) is E-equivalent to (∂E, i). Hence by the second remark, (Ω , j ) may be taken to be the Shilov boundary of E. Putting these three remarks together we deduce our fourth remark, namely that the Shilov boundary of E may be taken to be any function-extension (Ω, j) of E with the property that there is no closed subset Ω of Ω such that j(·)|Ω is still an isometry on E. Our ﬁfth remark is that nonetheless there is a canonical choice for the Shilov boundary (∂E, i) (that is, a canonical element of the equivalence class). Namely, we deﬁne ∂E to be the closure of Ch(E) (see 4.1.3), the closure taken in E ∗ with respect to the w∗ -topology. Clearly ∂E is compact. Since E is unital, the canonical isometry from E to C(Ball(E ∗ )) restricts to an isometry from E to C(S(E)), hence to an isometry i : E → C(∂E) by the Krein–Milman theorem. The proof of Theorem 4.1.5 will be complete if we can show that this choice of (∂E, i) has the universal property of the theorem. 4.1.7 (Proof of Theorem 4.1.5) Deﬁne ∂E as in the last (ﬁfth) remark, and consider (Ω, j) as in our hypothesis. By Lemma 4.1.2, any element in Ch(E) ‘is’ the restriction to j(E) of an evaluation functional δω , for some unique ω ∈ Ω. Thus we have constructed a function τ : Ch(E) → Ω. The map σ taking ω to ω is a topological embedding of Ω into S(j(E)) ∼ = S(E). Indeed since Ω is compact, σ is a homeomorphism onto a closed subset. Also, σ ◦ τ is the identity map on Ch(E), so that this closed subset contains Ch(E). Moreover, τ is a restriction

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of the homeomorphism σ −1 . Taking closures, we see that a restriction of σ −1 is the required embedding of ∂E into Ω. 4.1.8 (Representation on the maximal ideal space) For a uniform algebra A, consider the Gelfand representation, which is an isometric unital homomorphism of A into C(MA ), where MA is the maximal ideal space (see A.4.2). Clearly A separates points of MA . By the third (or ﬁfth) observation in 4.1.6, the Choquet and Shilov boundaries of A may be thought of as subspaces of MA . 4.1.9 (Examples) (1) We denote as usual by T the compact space of complex numbers with modulus equal to one, and we write z = (z1 , . . . , zn ) for an element of Tn . Let E = 1n+1 , with its canonical basis (ek )nk=0 . Consider the linear function i : E → C(Tn ) which takes e0 to 1, and otherwise takes ek to the ‘kth coordinate function’ z → zk on Tn . This is an isometry such that i(E) separates points of Tn . However there is no proper closed subset Ω of Tn so that i(·)|Ω is still isometric. Indeed if ω = (ωk ) ∈ Tn \ Ω, we consider x = (1, ω1 , . . . , ωn ) ∈ E. Then x = n + 1. But if |i(x)(z)| = n + 1 for some z ∈ Tn , then we must have z = ω. Hence i(·)|Ω is not an isometry. By the fourth remark in 4.1.6, this shows that ∂E = Tn . (2) Another important example is the disc algebra A(D) of 2.2.6. As we noted there, we may regard A(D) ⊂ C(T) isometrically. However no proper compact subset of T cannot ‘support A(D) isometrically’, since one may always ﬁnd a function in A(D) which peaks oﬀ this subset. Thus by the fourth remark in 4.1.6 again, we have ∂A(D) = T. (3) The Shilov boundary of the Hardy space H ∞ (D) (see 2.7.5 (4)) is the maximal ideal space of L∞ (T). This is much more diﬃcult to show (e.g. see [201]). We will see a noncommutative variant of this in 4.3.10. 4.1.10 (C ∗ -extensions) For a unital-subspace X of a unital C ∗ -algebra A, the noncommutative analogue of ‘point separation’ is the assertion that X generates A as a C ∗ -algebra. Indeed by the Stone–Weierstrass theorem, to say that a unitalsubspace X ⊂ C(Ω) generates C(Ω) as a C ∗ -algebra, is the same as saying that X separates points of Ω. Thus we deﬁne a C ∗ -extension of a unital operator space X (see 1.3.1) to be a pair (A, j) consisting of a unital C ∗ -algebra A, and a unital complete isometry j : X → A, such that j(X) generates A as a C ∗ -algebra. 4.1.11 (Boundary representations) These are Arveson’s noncommutative generalization of Choquet boundary points. If A is a C ∗ -algebra generated by a unital-subspace X, then a boundary representation of A for X is an irreducible ∗-representation π : A → B(H) such that π is the unique completely positive map A → B(H) extending π|X . This is the noncommutative analogue of Lemma 4.1.2 (iii). For convenience below, we shall omit the requirement that boundary representations be irreducible. One point about boundary representations, which generalizes what we saw in 4.1.3, is that they only depend on the unital operator space structure of X, and

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not on the particular C ∗ -algebra A containing X. In order to see this we recall that a map v : X → B(K) is a dilation of u : X → B(H) if there is an isometry V : H → K such that u = V ∗ v(·)V on X. This dilation is said to be reducing if v(X)V H ⊂ V H and v(X)∗ V H ⊂ V H. We say that a completely contractive unital map u : X → B(H) is in class B X if every completely contractive unital dilation of u is reducing. We observe that B X is by deﬁnition ‘invariant under unital completely isometric isomorphism’. That is, if σ : X1 → X2 is a unital completely isometric isomorphism between unital operator spaces, then u ◦ σ is in BX1 if and only if u is in B X2 . Proposition 4.1.12 Let X be a unital-subspace of a unital C ∗ -algebra A, which generates A as a C ∗ -algebra, and let u : X → B(H). Then u is the restriction to X of a boundary representation π : A → B(H), if and only if u ∈ B X . Proof If u ∈ B X , then by the extension theorems 1.2.10, and 1.3.3, we may extend u to a completely positive unital map uˆ : A → B(H). By Stinespring’s theorem 1.3.4, we may write uˆ = V ∗ θ(·)V , for a ∗-representation θ : A → B(K) and an isometry V : H → K. Thus θ|X dilates u, and hence it is reducing. So V H reduces θ(A), forcing u ˆ to be a ∗-representation. Therefore u ˆ is uniquely determined by u. The above argument shows that uˆ is a boundary representation. Conversely, suppose that π : A → B(H) is a boundary representation, and that v : X → B(K) dilates u = π|X . Thus there is an isometry V : H → K such that π = V ∗ v(·)V on X. Since v is a completely contractive unital map, as before we may extend v to a completely positive map Ψ : A → B(K). Then V ∗ Ψ(·)V is completely positive on A, so that by hypothesis we have π = V ∗ Ψ(·)V on A. Thus for any a ∈ X, the Kadison–Schwarz inequality 1.3.9 gives π(aa∗ ) = V ∗ Ψ(aa∗ )V ≥ V ∗ Ψ(a)Ψ(a∗ )V ≥ V ∗ Ψ(a)V V ∗ Ψ(a∗ )V = π(a)π(a)∗ . Since the last expression is again π(aa∗ ), the second inequality is an equality. This implies, as in the proof of 1.3.11, that V V ∗ Ψ(a∗ )V = Ψ(a∗ )V . That is, V H is invariant under Ψ(a∗ ). An identical argument with a∗ a in place of aa∗ shows that V H is invariant under Ψ(a), so that V H is reducing for Ψ(a) = v(a). 2 4.1.13 (Remarks on boundary representations) (1) Looking at the proof above yields the following. If A is a C ∗ -cover in the sense of 2.1.1 of a unital operator algebra B, then a completely contractive unital homomorphism θ : B → B(H) is the restriction to B of a boundary representation π : A → B(H), if and only if every completely contractive unital homomorphism dilating θ is reducing. (2) From Proposition 4.1.12 and the remark before it, we see that a map u : X → B(H) is in B X if and only if for any (or every) C ∗ -extension (A, j) of X, the set of completely positive maps from A to B(H) extending u ◦ j −1 is a singleton, and that unique extension is a ∗-representation. (3) One may construct the ‘noncommutative Shilov boundary’ using boundary representations. See the Notes on Section 4.1 for references. We will instead use the approach developed in the next few sections.

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4.2 THE INJECTIVE ENVELOPE We now quickly describe Hamana and Ruan’s injective envelope of an operator space X. This is an injective operator space Z (see 1.2.9 and 1.2.11) containing (a completely isometric copy of) X, satisfying one of several equivalent additional properties which we will spell out momentarily. The next page or so follows closely the classical development of the (Banach space) injective envelope (e.g. see [237] and references therein). 4.2.1 (Projections and seminorms) Suppose that X is a subspace of an operator space W . An X-projection on W is a completely contractive idempotent map Φ : W → W which restricts to the identity map on X. An X-seminorm on W is a seminorm of the form p(·) = u(·), for a completely contractive linear map u : W → W which restricts to the identity map on X. Deﬁne a partial order ≤ on the set of all X-projections, by setting Φ ≤ Ψ if Φ ◦ Ψ = Ψ ◦ Φ = Φ. We clearly have Φ ≤ Ψ if and only if Ran(Φ) ⊂ Ran(Ψ) and Ker(Ψ) ⊂ Ker(Φ). In the following we use the usual ordering on seminorms. Lemma 4.2.2 Let X be a subspace of an injective operator space W . (1) Any decreasing net of X-seminorms on W has a lower bound. Hence there exists a minimal X-seminorm on W , by Zorn’s lemma. Each X-seminorm majorizes a minimal X-seminorm. (2) If p is a minimal X-seminorm on W , and if p(·) = u(·), for a completely contractive linear map on W which restricts to the identity map on X, then u is a minimal X-projection. Proof We sketch enough of the proof to enable the reader to complete it. Suppose that W ⊂ B(H), and that P is a completely contractive idempotent map of B(H) onto W (see the proof of 1.2.11). Take a decreasing net (u t (·))t as in (1). Since ut is completely contractive, and since CB(W, B(H)) is a dual space (see 1.6.1), a subnet of (ut ) converges in the w∗ -topology to a u ∈ CB(W, B(H)), say. It is easy to check from assertions in 1.6.1, that u(·) is a lower bound for the given net. Hence so is the X-seminorm P (u(·)) on W . The other assertions of (1) are evident. Suppose that p, u are as in (2). Claim: u is an idempotent map. Assuming this claim, the desired results follow easily; since if Φ ≤ u in the ordering above on X-projections, then by minimality we must have u(·) = Φ(·). Hence Ker(u) = Ker(Φ), and so Ran(Φ − I) ⊂ Ker(u). Thus u = u ◦ Φ = Φ. To prove the Claim, we consider u as a map into B(H). Let u0 be a w∗ -cluster point (as in the ﬁrst paragraph) of the sequence (u(n) )n of Cesaro averages of the sequence u, u ◦ u, u ◦ u ◦ u, . . .. We have u(n) (·) ≤ u(·) for any n ≥ 1, hence u0 (·) ≤ p. If P is as above, then P (u0 (·)) ≤ u0 (·) ≤ p. By minimality, u0 (·) = p. Then for y ∈ W , u(y) − u2 (y) = p(y − u(y)) = u0 (y − u(y)) ≤ lim sup u(n) (y − u(y)) = 0. n

Thus u is idempotent.

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4.2.3 (Extensions of operator spaces) An extension of an operator space X is an operator space Y , together with a linear completely isometric map i : X → Y . Often we suppress mention of i, and identify X with a subspace of Y . We say that Y is a rigid extension of X if IY is the only linear completely contractive map Y → Y which restricts to the identity map on i(X). We say Y is an essential extension of X if whenever u : Y → Z is a completely contractive map into another operator space Z such that u ◦ i is a complete isometry, then u is a complete isometry. We say that (Y, i) is an injective envelope of X if Y is injective, and if there is no injective subspace of Y containing i(X). Lemma 4.2.4 Let (Y, i) be an extension of an operator space X such that Y is injective. The following are equivalent: (i) Y is an injective envelope of X, (ii) Y is a rigid extension of X, (iii) Y is an essential extension of X. Proof We assume (ii). Suppose that u : Y → Z is a complete contraction such that u restricted to i(X) is a complete isometry. Since Y is injective, the inverse of this restricted map may be extended to a complete contraction v : Z → Y . Clearly vu = IY when restricted to i(X), so that by rigidity vu = IY . Hence u is a complete isometry, which proves (iii). We assume (iii). Suppose that i(X) ⊂ W ⊂ Y , with W injective. Extend IW to a complete contraction Φ : Y → W . Then Φ is idempotent. By hypothesis Φ is also one-to-one, which forces W = Y , and proves (i). We assume (i). Then there can exist no nontrivial i(X)-projections on Y , for the range of such a projection is clearly injective. Thus, if p = u(·) is any minimal i(X)-seminorm on Y , then by 4.2.2 (2) we have that u = IY , so that p is the usual norm on Y . By 4.2.2, there exist minimal i(X)-seminorms, hence the usual norm is a minimal i(X)-seminorm. If v : Y → Y is a complete contraction extending Ii(X) , then v(·) is an i(X)-seminorm on Y dominated by the usual norm. Thus v(·) is a minimal i(X)-seminorm. Hence v = IY by the ﬁrst lines of this paragraph, which shows (ii). 2 Lemma 4.2.5 If (Y1 , i1 ) and (Y2 , i2 ) are two injective envelopes of X, then there exists a surjective complete isometry u : Y1 → Y2 such that u ◦ i1 = i2 . Proof Extend i2 ◦ i−1 and i1 ◦ i−1 to complete contractions u : Y1 → Y2 and 1 2 v : Y2 → Y1 respectively. Since vui1 = i1 , by the rigidity property we have vu = IY1 . Similarly, uv is the identity map. Thus u is a completely isometric surjection by 1.2.7. 2 Theorem 4.2.6 If an operator space X is contained in an injective operator space W , then there is an injective envelope Y of X with X ⊂ Y ⊂ W . Proof Let Y be the image of a minimal X-projection Φ on W (such a projection exists by combining (1) and (2) of Lemma 4.2.2). We claim that Y is an injective envelope of X. For if X ⊂ Z ⊂ Y with Z injective, then there is a completely

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contractive idempotent map Ψ of Y onto Z. Clearly Ψ ◦ Φ ≤ Φ. But Ψ ◦ Φ is an X-projection, and so by the minimality of Φ we must have Ψ ◦ Φ = Φ. Hence 2 Ψ = IY and Z = Y . 4.2.7 (The injective envelope) Combining 1.2.10 and 4.2.6, we see that every operator space has an injective envelope. We will often write it as (I(X), j), or I(X) for short. It is essentially unique by 4.2.5. The following observations, which follow from the last proof and 1.3.13, will be used frequently. Let B be B(H) (or an injective unital C ∗ -algebra). If X ⊂ B, then there is an injective envelope R of X with X ⊂ R ⊂ B, and a completely contractive idempotent map Φ from B onto R. If, further, 1B ∈ X, then Φ is completely positive by 1.3.3, and R is a C ∗ -algebra with the new product ◦Φ deﬁned by a ◦Φ b = Φ(ab) a, b ∈ R. (4.2) If further X is a subalgebra of B, then X is also a subalgebra of R. Indeed in this case Φ(ab) = ab when a, b ∈ X. If X is a unital operator space, and if X + X is the canonical operator system generated by X (see 1.3.7), then every injective envelope of X + X is an injective envelope of X, and vice versa. Indeed suppose that B is an injective envelope of X + X . By the last paragraph we may suppose that B is a unital C ∗ -algebra. Also by the last paragraph there is an injective envelope R of X with X ⊂ R ⊂ B, and a unital completely positive projection Φ from B onto R. Clearly Φ is the identity on X + X , and so Φ = IB by rigidity. Hence R = B. The ‘vice versa’ is similar, but uses also 1.3.6. However we will not use this fact. Corollary 4.2.8 (1) If X is a unital operator space (resp. unital operator algebra, approximately unital operator algebra), then there is an injective envelope (I(X), j) for X, such that I(X) is a unital C ∗ -algebra and j is a unital map (resp. j is a unital homomorphism, j is a homomorphism). (2) If A is an approximately unital operator algebra, and if (Y, j) is an injective envelope for A1 , then (Y, j|A ) is an injective envelope for A. (3) If A is an approximately unital operator algebra which is injective, then A is a unital C ∗ -algebra. Proof The second paragraph of 4.2.7 proves (1) in the unital case. Assume now that A is an approximately unital nondegenerate subalgebra of B(H), and let R ⊂ B(H) and Φ : B(H) → R be as in 4.2.7. Thus R is an injective envelope for A. If (et )t is a cai for A, then et → IH strongly. If ζ ∈ H with ζ = 1, let ϕ(T ) = Φ(T )ζ, ζ, for T ∈ B(H). Then ϕ ≤ 1 and ϕ(et ) → 1. By (1) of 2.1.18, this implies that ϕ(IH ) = 1. Hence Φ(IH ) = IH , and A1 = A + CIH ⊂ R. From this it is clear that R is also an injective envelope for A1 , and (1) and (3) follow at once. For (2), note that the proof in (1) shows that one injective envelope of A is also an injective envelope of A1 . A routine diagram chase using 4.2.5 shows 2 that any injective envelope of A1 is an injective envelope of A.

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The following result is analoguous to a well-known result of Tomiyama [410]: Corollary 4.2.9 If B is a unital-subalgebra of a unital operator algebra A, and if P : A → B is a completely contractive idempotent map onto B, then P is a ‘conditional expectation’. That is, P is a B-bimodule map: P (b1 ab2 ) = b1 P (a)b2 ,

a ∈ A, b1 , b2 ∈ B.

Proof Let P : A → B be the idempotent map. Let i : B → A be the inclusion. Let (I(A), J) and (I(B), j) be injective envelopes of A and B respectively, as in 4.2.8 (1). Thus these are unital C ∗ -algebras, and J, j are unital completely isometric homomorphisms. By injectivity, we may extend j ◦ P ◦ J −1 to a completely contractive unital map Pˆ : I(A) → I(B) with Pˆ ◦ J = j ◦ P . We may also extend J ◦ i ◦ j −1 to a completely contractive unital map ˆi : I(B) → I(A), with ˆi ◦ j = J ◦ i. Thus Pˆ ◦ ˆi ◦ j = Pˆ ◦ J ◦ i = j ◦ P ◦ i = j. Hence Pˆ ◦ ˆi is the identity map on j(B), and so by the rigidity property of the injective envelope, Pˆ ◦ ˆi is the identity map on I(B). Thus Q = ˆi ◦ Pˆ is a unital completely contractive (and hence completely positive) idempotent map on I(A). We have Q(J(a)) = ˆi(Pˆ (J(a))) = ˆi(j(P (a)) = J(P (a)),

(4.3)

for b ∈ B, a ∈ A, and thus J(P (ba)) = Q(J(ba)) = Q(J(b)J(a)) = Q(J(b)Q(J(a))),

(4.4)

the last step by (2) of 1.3.13. By (4.4), and (4.3) used twice, we deduce that J(P (ba)) = Q(J(b)Q(J(a))) = Q(J(bP (a))) = J(P (bP (a))) = J(bP (a)). Hence P (ba) = bP (a), and similarly P (ab) = P (a)b.

2

4.2.10 (Envelopes of matrix spaces) If Z is an injective operator space, then so is Mm,n (Z). This follows from 1.2.11, since if Z ⊂ B(H), and if P is a completely contractive idempotent map of B(H) onto Z, then Pm,n : Mm,n (B(H)) −→ Mm,n (Z) is a completely contractive idempotent map of the injective operator space Mm,n (B(H)) (which may be identiﬁed with B(H n , H m ), by (1.2)) onto Mm,n (Z). More generally, if X is an operator space, then Mm,n (I(X)) is an injective envelope of Mm,n (X). A more general result may be found in 4.6.12, but the assertion may also be argued from what we have done already. (Hint: Use, recursively, the much easier special cases n = 2, m = 1; and n = 1, m = 2). 4.2.11 (The Banach space case) As we mentioned earlier, there is a parallel earlier theory of Banach space injective envelopes. This may be derived formally from what we have done above, by replacing B(H) above by ∞ I , and complete

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contractions and complete isometries by contractions and isometries. It is fairly obvious, by (1.10), that for any Banach space E, Min(IB (E)) = I(Min(E))

completely isometrically,

(4.5)

where IB (E) is the Banach space injective envelope of E. It is well known that the injective Banach spaces are the C(K)-spaces for compact Stonean K (e.g. see [237] for this, and for other related theory and references). 4.3 THE C ∗ -ENVELOPE In 4.1.10 we deﬁned C ∗ -extensions of a unital operator space X. We say that two C ∗ -extensions (B, i) and (B , i ) are X-equivalent if there exists a ∗-isomorphism π : B → B such that π ◦ i = i . We deﬁne a C ∗ -envelope of X to be any C ∗ extension (B, i) with the universal property of the next theorem. Theorem 4.3.1 (Arveson–Hamana) If X is a unital operator space, then there exists a C ∗ -extension (B, i) of X with the following universal property: Given any C ∗ -extension (A, j) of X, there exists a (necessarily unique and surjective) ∗-homomorphism π : A → B, such that π ◦ j = i. 4.3.2 (Remarks on the universal property) Before we begin the proof of the Arveson–Hamana theorem, we will make a series of important but simple remarks concerning the universal property of 4.3.1. First, if (B, i) is a C ∗ -extension of X with this universal property, then there exists no nontrivial closed two sided ideal I of B such that q ◦ i is a complete isometry on X, where q : B → B/I is the natural quotient map. This assertion follows by applying the universal property with j = q ◦ i. One obtains a ∗-homomorphism π : B/I → B with π ◦ q ◦ i = i. Since i(X) is generating we see that π ◦ q = IB , which implies that q is one-to-one. Thus I = (0). The second remark is that the set of C ∗ -extensions (B, i) satisfying the universal property of the theorem, is one entire equivalence class of the relation of X-equivalence deﬁned above 4.3.1. This is easily seen by a routine diagram chase. The third remark is that if (A, j) is any C ∗ -extension of X, if π : A → B is the ∗-epimorphism provided by the universal property, and if I = Ker(π), then (A/I, q ◦ j) is clearly X-equivalent to (B, i) (from which it follows that q ◦ j is completely isometric). Here q is the quotient map from A to A/I. Thus by the second remark, (A/I, q◦j) may be taken to be a C ∗ -envelope of X. Putting these remarks together, we get our fourth and ﬁnal remark, namely that the C ∗ -envelope of X may be taken to be any C ∗ -extension (A, j) of X for which there exists no nontrivial closed two-sided ideal I of A such that q ◦ j is completely isometric on X, where q is the quotient map from A to A/I. 4.3.3 (Proof of Theorem 4.3.1) By 4.2.8, we may choose an injective envelope (I(X), i) of X, with I(X) a unital C ∗ -algebra and i a unital map. We deﬁne Ce∗ (X) to be the C ∗ -subalgebra of I(X) generated by i(X); then (Ce∗ (X), i) is a C ∗ -extension of X.

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Suppose now that (A, j) is any C ∗ -extension of X, and suppose that A is a unital ∗-subalgebra of B(H). Then j(X) ⊂ B(H), and by 4.2.7, there is a completely positive idempotent map on B(H) whose range is an injective envelope R of j(X), and R is a C ∗ -algebra with respect to a new product. With respect to the usual product on B(H), the C ∗ -subalgebra of B(H) generated by R contains A, the C ∗ -subalgebra of B(H) generated by j(X). We let B be the C ∗ -subalgebra (in the new product) of R generated by j(X). Thus by 1.3.13 (3), π = Φ|A is a ∗-homomorphism from A to R, with respect to the new product on R. Since π extends the identity map on j(X), it clearly also maps into B. The ﬁnal point is that the natural unital completely isometric surjection R → I(X) guaranteed by 4.2.5, is a ∗-homomorphism by 1.3.10. Hence it is clear that (B, j), as a C ∗ -extension of X, is X-equivalent to (Ce∗ (X), i). Putting these facts together, we see that (Ce∗ (X), i) has the announced universal property. 4.3.4 (C ∗ -envelopes and the Shilov boundary) We use the notation (Ce∗ (X), i) for any C ∗ -envelope of a unital operator space X. Suppose that X is a uniform algebra, or more generally a unital function space. From the universal property of the last theorem, Ce∗ (X) is a homomorphic image of a commutative unital C ∗ -algebra, and hence Ce∗ (X) is commutative. Thus Ce∗ (X) = C(Ω) for some compact space Ω. The universal property of the Arveson–Hamana theorem, together with basics of the duality between compact spaces and commutative unital C ∗ -algebras (e.g. see A.5.4), translates to show that Ω satisﬁes the universal property of Theorem 4.1.5. Hence in this case, Ce∗ (X) coincides with the continuous functions on the usual Shilov boundary ∂X of X. This suggests that in the general case of a unital operator space X, the C ∗ -algebra Ce∗ (X) should be regarded as a noncommutative Shilov boundary. We deﬁne the C ∗ -envelope of a nonunital operator algebra A to be a pair (B, i), where B is the C ∗ -subalgebra generated by the copy i(A) of A inside a C ∗ -envelope (Ce∗ (A1 ), i) of the unitization A1 of A (see 2.1.11). Proposition 4.3.5 Let A be an operator algebra, and let (Ce∗ (A), i) be a C ∗ envelope of A. Then i is a homomorphism, and Ce∗ (A) has the following universal property: Given any C ∗ -cover (B, j) of A, there exists a (necessarily unique and surjective) ∗-homomorphism π : B → Ce∗ (A) such that π ◦ j = i. Proof The result reduces to 4.3.1 if A is unital. Assume that A is nonunital and let (B, j) be a C ∗ -cover of A. Then B is nonunital as well, by 2.1.8, and j extends by 2.1.15 to a completely isometric unital homomorphism j 0 : A1 → B 1 , whose range generates B 1 as a C ∗ -algebra. Thus by the Arveson–Hamana theorem 4.3.1, there is a surjective ∗-homomorphism ρ : B 1 → Ce∗ (A1 ) such that ρ◦j 0 = i, where i : A1 → Ce∗ (A1 ) is the canonical embedding. Let π be ρ restricted to B, then π is a ∗-homomorphism with π(j(a)) = ρ(j 0 (a)) = j(a) ∈ Ce∗ (A), for all a ∈ A. Thus π maps B into Ce∗ (A), and the above shows that π ◦ j = i. The last identity forces i to be a homomorphism. 2

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4.3.6 (Properties of the C ∗ -envelope) If A is an approximately unital operator algebra, or unital operator space, then its C ∗ -envelope (Ce∗ (A), i) is easily seen to be both a rigid and an essential extension of A, in the sense of 4.2.3. One need only to recall that Ce∗ (A) is a subspace of the injective envelope (using 4.3.3, and also 4.2.8 (2) in the operator algebra case), and that the injective envelope has these properties (see Lemma 4.2.4). For example, to see that Ce∗ (A) is essential, we take a complete contraction u : Ce∗ (A) → B(H), which restricts to a complete isometry on the copy of A. Extend u to a complete contraction I(A) → B(H) by 1.2.10, and then use the ‘essential’ property of I(A). Although we will not use this, it follows from the last paragraph in 4.2.7 that for a unital operator space X, Ce∗ (X) and Ce∗ (X + X ) coincide. Also, if u : X → Y is a surjective unital linear complete isometry onto another unital operator space, then one may ‘extend’ u to a ∗-isomorphism between any C ∗ envelopes (Ce∗ (X1 ), j1 ) and (Ce∗ (X2 ), j2 ). Indeed a routine ‘diagram chase’ shows that (Ce∗ (X2 ), j2 ◦ u) is a C ∗ -envelope for X1 . 4.3.7 (Examples) In the rest of this section, we look at several examples of the C ∗ -envelope or noncommutative Shilov boundary. (1) The ﬁrst example to consider is T n , the upper triangular n × n matrices, sitting inside Mn . Since Mn is simple (i.e. has no nontrivial two-sided ideals), it follows from the fourth remark in 4.3.2 that Mn is the C ∗ -envelope of T n . (2) More generally, we consider the C ∗ -envelope of a unital subspace X of Mn . The C ∗ -subalgebra of Mn generated by X is ﬁnite-dimensional, and hence is ∗-isomorphic to a ﬁnite direct sum B of full ‘matrix blocks’ Mnk . Some of these blocks are redundant. That is, if p is the central projection in B corresponding to the identity matrix of this block, then the map x → x(1B − p) is completely isometric on X. If one eliminates such blocks, then the remaining direct sum of blocks is the C ∗ -envelope of X. The next example we consider is the ‘noncommutative version’ of Example 4.1.9 (1). In the latter, we showed that the classical Shilov boundary of 1n is the (n−1)-torus T(n−1) . Here we consider 1n with its natural operator space structure (that is, its Max structure). Let C ∗ (Fn−1 ) be the full C ∗ -algebra of the free group on n− 1 generators u1 , . . . , un−1 , and consider the map i : 1n → C ∗ (Fn−1 ), which takes e1 to u0 = 1 and ek to uk−1 , for k = 2, . . . , n. We will show that i is the ‘Shilov representation’ of Max(1n ). Proposition 4.3.8 (1) Max(1n ) is a unital operator space (with unit e1 ). Indeed the above map i : Max(1n ) → C ∗ (Fn−1 ) is a unital complete isometry, whose range generates C ∗ (Fn−1 ). (2) Ce∗ (Max(1n )) = C ∗ (Fn−1 ). Proof Note that {uk : k ≤ n} is a set of contractions, so that the map i above is clearly a contraction. Therefore i is a complete contraction by (1.12). Conversely, suppose that Max(1n ) ⊂ B(H) completely isometrically, with ek

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corresponding to contractions Tk . For each k, form the canonical ‘unitary dilation’ Uk ∈ M2 (B(H)), whose 1-1-corner is Tk . Let Vj−1 = Uj U1−1 , these are also unitary. By the universal property of C ∗ (Fn−1 ), there exists a unique ∗-homomorphism π : C ∗ (Fn−1 ) → M2 (B(H)) such that π(uk ) = Vk for all k = 0, 1, . . . , n − 1. If Φ is the projection from M2 (B(H)) onto its 1-1-corner, then Φ(π(·)U1 ) is a complete contraction extending i−1 . Thus i−1 is a complete contraction, and i is a complete isometry. Clearly Ran(i) is generating. To prove (2), we let (A, j) be a C ∗ -envelope of Max(1n ). By 4.3.1, there exists a surjective unital ∗-homomorphism π : C ∗ (Fn−1 ) → A such that π ◦ i = j. Let Vk = π(uk ) for k = 0, 1, . . . , n − 1. Thus the Vk are unitaries which generate A. Suppose that C ∗ (Fn−1 ) is represented as a unital ∗-subalgebra of B(H). By 1.2.10 we may extend the map i ◦ j −1 to a completely contractive map u : A → B(H). Since u is unital, it is completely positive by 1.3.3. Note that Vk = π(i(ek+1 )) = j(ek+1 ), so that u(Vk ) = i(ek+1 ) = uk . Therefore u(Vk∗ Vk ) = u(1) = IH = u∗k uk = u(Vk )∗ u(Vk ), and similarly u(Vk Vk∗ ) = u(Vk )u(Vk )∗ . Thus by 1.3.11 we have that u(Vk a) = u(Vk )u(a) and u(Vk∗ a) = u(Vk )∗ u(a) = u(Vk∗ )u(a),

a ∈ A.

Thus u is a ∗-homomorphism. Hence u maps into C ∗ (Fn−1 ), and is clearly an inverse for π. Thus π is one-to-one. 2 4.3.9 (Noncommutative Dirichlet and logmodular algebras) Suppose that A is a closed unital-subalgebra of a unital C ∗ -algebra B. We say that A is a Dirichlet algebra if A+A is norm dense in B. We say that A is left convexly approximating in modulus if every positive b ∈ B is a uniform limit of terms of the form n ∗ k=1 ak ak for ak ∈ A. Here the n are varying too. The word ‘left’ here refers to the preference of products a∗ a as opposed to aa∗ ; thus the reader may guess the meaning of right convexly approximating in modulus. We say that A has factorization (resp. is logmodular) if every element b ∈ B such that b ≥ 1 for some > 0, is of the form (resp. is a uniform limit of terms of the form) a∗ a where a ∈ A−1 . A logmodular algebra is both left and right convexly approximating in modulus. To see this, ﬁrst approximate a positive b ∈ B with b + n1 . To illustrate these notions, we recall that the disc algebra A(D) is a Dirichlet subalgebra of C(T), and that any Dirichlet uniform algebra is logmodular (see [167]). The algebra of n × n upper triangular matrices in Mn is Dirichlet, and is known to have factorization (this is the Choleski factorization). Thus it is logmodular. The Hardy space H ∞ (D), or any of its usual uniform algebra generalizations, or more generally still Arveson’s noncommutative H ∞ spaces, are known to have factorization, and are therefore logmodular. We will say a little more about these spaces in the Notes section for 4.3. For the reader’s convenience we will simply recall the deﬁnition: a noncommutative H ∞ is a w∗ -closed unitalsubalgebra A of a von Neumann algebra B, such that B has a faithful normal tracial state τ , A+A is w∗ -dense in B, and such that the unique faithful normal

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conditional expectation Φ of B onto ∆(A) for which τ ◦ Φ = τ , is multiplicative on A. A simple example is the upper triangular n × n matrices inside B = Mn . Our point is simply that the examples above satisfy the hypotheses of the next two results. Both results generalize useful facts well known for logmodular uniform algebras. The second result is a generalization of the important ‘uniqueness of representing measure’ for point evaluations on logmodular algebras. Proposition 4.3.10 Suppose that A is a unital-subalgebra of a unital C ∗ -algebra B, which is either Dirichlet, or is left or right convexly approximating in modulus. Then B = Ce∗ (A). Proof In the case that A is Dirichlet, the canonical ∗-epimorphism from B to Ce∗ (A) is by 1.3.6 an isometry, and is therefore one-to-one. Suppose that A is left convexly approximating in modulus (the ‘right’ case is similar). We apply the fourth remark in 4.3.2. Assume that I ⊂ B is a closed two-sided ideal such that the canonical map Q : A → B/I factoring through qI the canonical maps A → B → B/I is a complete isometry. It suﬃces to check that I = (0). Fix b ∈ I with b ≥ 0. Then b is a limit of terms of the form n ∗ qI (b) is the limit of k ak , for ak ∈ A. Since k=1 a qnI is a ∗-homomorphism, n ∗ ∗ q (a ) q (a ) = Q(a ) Q(a ). Also, b is a limit of terms terms I k I k k k k=1 k=1 n k=1 a∗k ak . However the last quantity is, by 1.2.5, the square of the norm of the column [ak ] in Cn (A). Since Q is a complete isometry, this norm coincides with the square of the norm of the column [Q(ak )] in Cn (B/I). We have n n b = lim a∗k ak = lim Q(ak )∗ Q(ak ) = qI (b) = 0. k=1

k=1

Thus b = 0, which implies that I = (0).

2

Theorem 4.3.11 Suppose that A is either a Dirichlet or logmodular unitalsubalgebra of a unital C ∗ -algebra B. (1) Any unital completely contractive (resp. completely isometric) homomorphism π : A → B(H) has a unique extension to a completely positive and completely contractive (resp. and completely isometric) map from B into B(H). (2) Every ∗-representation of B is a boundary representation for A (see 4.1.11). Proof Assume that π : A → B(H) is a unital completely contractive homomorphism. The claim regarding the existence of a completely contractive, and hence completely positive, extension of π is clear, by 1.2.10 and 1.3.3. The completely isometric statement is obtained by applying 4.3.10, and the fact that Ce∗ (A) is an essential extension of A (see 4.3.6). The claim regarding uniqueness is clear in the Dirichlet algebra case, by 1.3.6. Thus we now assume that A is logmodular and shall prove uniqueness. We suppose that Φ and Ψ are two completely positive extensions of π to all of B. Let ζ be a unit vector in H, and suppose that a, b ∈ A with ba = 1. Then

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1 = ζ, ζ = π(ba)ζ, ζ = Φ(b)Ψ(a)ζ, ζ = Ψ(a)ζ, Φ(b)∗ ζ. By the Cauchy–Schwarz inequality, we have 1 ≤ Ψ(a)∗ Ψ(a)ζ, ζ Φ(b)Φ(b)∗ ζ, ζ. By the Kadison–Schwarz inequality (see 1.3.9), we have 1 ≤ Ψ(a∗ a)ζ, ζ Φ(bb∗ )ζ, ζ. Since A is logmodular, this yields 1 ≤ Ψ(x)ζ, ζ Φ(x−1 )ζ, ζ for all strictly positive x ∈ B. Writing x = eh for h ∈ Bsa , we may then replace h with th for real t, to obtain 1 ≤ Ψ(eth )ζ, ζ Φ(e−th )ζ, ζ. Let f (t) = g(t)k(t), where g(t) = Ψ(eth )ζ, ζ and k(t) = Φ(e−th )ζ, ζ. By the above, f has a local minimum at t = 0. Thus if f (0) exists, then f (0) = 0. By the undergraduate ‘product rule’, f (0) equals ) 1 * ) 1 * lim k(t) Ψ (eth −1) ζ, ζ + g(t) Φ (e−th −1) ζ, ζ = Ψ(h)ζ, ζ−Φ(h)ζ, ζ. t→0 t t Thus Ψ(h)ζ, ζ = Φ(h)ζ, ζ. Hence Ψ(h) = Φ(h) for all selfadjoint h ∈ B. Thus Ψ = Φ. This proves (1), and (2) follows from (1) and the deﬁnitions. 2 4.4 THE INJECTIVE ENVELOPE, THE TRIPLE ENVELOPE, AND TROS This section is a little more technical (and terse) than some of the others in this chapter. The reader should feel free to skim through these results: they will be used to give quick proofs of a few results, and then after this the injective envelope will be rarely mentioned again. 4.4.1 (TROs and their morphisms) We recall that a ternary ring of operators or TRO is a closed linear subspace Z of B(K, H) (or of a C ∗ -algebra) satisfying ZZ Z ⊂ Z. For x, y, z ∈ Z, we sometimes write xy ∗ z as [x, y, z], this is the triple product on Z. The basic example of a TRO is pA(1 − p), for a C ∗ -algebra A and a projection p in A (or in M (A)). A subtriple of a TRO Z is a closed subspace Y of Z satisfying Y Y Y ⊂ Y . A triple morphism between TROs is a linear map which respects the triple product: thus T ([x, y, z]) = [T x, T y, T z]. These are the natural morphisms between TROs, and we note that they naturally arise when restricting a ∗-homomorphism from a C ∗ -algebra A to a subtriple of A. However there is another important class of maps from any TRO Z to itself. Recall from Example 3.1.2 (6) that a TRO Z is a bimodule over the C ∗ -algebras ZZ and Z Z. Thus we may consider, for example, the bounded right module maps on Z. We will not use this in this chapter, but by a result of Lin (see 8.1.16 (3)), these

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module maps are in a canonical bijective correspondence with the left multipliers (in the usual sense of 2.6) of the C ∗ -algebra ZZ . Indeed in Section 8.3 we will study TROs and their morphisms in much more detail. Some authors substitute the word ‘ternary’ for ‘triple’ in the deﬁnitions above, because of the potential confusion with the JB ∗ -triple literature, where ‘triple’ has a diﬀerent connotation. We allow ourselves the use of the word ‘triple’ here because we shall not discuss JB ∗ -triples in this text, and hence there should be little danger of confusion. 4.4.2 (An important construction) The following notation will appear frequently in the next several pages. We ﬁx an operator space X ⊂ B(H). The construction and results which follow are not altered essentially if we take X to be a subspace of B(K, H), or of an injective C ∗ -algebra say, but for simplicity we take B(H) here. Consider the Paulsen system S(X) ⊂ M2 (B(H)) in the usual way (see 1.3.14). By 4.2.7, there is a completely positive idempotent map Φ on M2 (B(H)) whose range is an injective envelope I(S(X)) of S(X). Also, I(S(X)) is a unital C ∗ -algebra in a new product ◦Φ (see (4.2)). Write p and q for the canonical projections IH ⊕ 0 and 0 ⊕ IH . Since Φ(p) = p and Φ(q) = q, it follows from 2.6.16 that Φ is ‘corner-preserving’, and that its ‘1-2-corner’ P is an idempotent map on B(H). By deﬁnition of the product ◦Φ , it is clear that p and q are complementary projections in the C ∗ -algebra I(S(X)). With respect to these projections, I(S(X)) may be viewed as consisting of 2 × 2 matrices, in the usual way (see 2.6.14). Let Ikl (X), or simply Ikl , denote its ‘k-l-corner’, for k, l = 1, 2. Thus I11 is the unital C ∗ -algebra pI(S(X))p, I22 is (1 − p)I(S(X))(1 − p), and I12 = pI(S(X))(1 − p) = Ran(P ). Note that by 1.3.12, Φ(pa) = pΦ(a) = pa for any a ∈ I(S(X)). Thus pa belongs to I(S(X)), as does ap by the same reasoning. Also, multiplication by p or 1 − p in I(S(X)) is the same in the product of M2 (B(H)) or in the new product ◦Φ . We write J for the canonical map from X into I12 (X); we shall see shortly that (I12 (X), J) is an injective envelope of X. We have the following diagram: C X I(X) = I12 (X) I11 (X) X → S(X) = → I(S(X)) = . X C I(X) = I21 (X) I22 (X) We temporarily write Z for I12 . According to Example 3.1.2 (5), we see that the ‘corner’ Z of I(S(X)) is an operator I11 -I22 -bimodule. Similarly, by 4.4.1, Z is a TRO with triple product [x, y, z] deﬁned from the product of the C ∗ -algebra I(S(X)). With this product we have ZZ Z ⊂ Z, ZZ ⊂ I11 , and Z Z ⊂ I22 . Note that in terms of the product in B(H), we have [x, y, z] = P (xy ∗ z),

x, y, z ∈ Z.

(4.6)

This follows from the deﬁnition of the new product on I(S(X)) (as given by (4.2)), and the deﬁnition of P as the 1-2-corner of Φ. Theorem 4.4.3 (Hamana–Ruan) If X is an operator space, let A be the injective C ∗ -algebra I(S(X)) considered above, and let p and 1 − p be the two

Some ‘extremal theory’

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complementary diagonal projections in A mentioned above. Then pA(1 − p) is an injective envelope of X. Proof We suppose that X ⊂ B(H), and we use the notation established above. By 1.2.11, it is easy to see that Z = pA(1 − p) is injective. Let v : Z → Z be a completely contractive map extending the identity map on J(X). By 4.2.4, we need to show that v = IZ . By Paulsen’s lemma 1.3.15, v gives rise to a canonically associated map on S(Z), the latter viewed as a subset of A in the obvious way. Since A is injective, we may extend further to a complete contraction Ψ from A to itself. Note that the restriction of Ψ to S(X) is the identity map. By the rigidity property of A (see 4.2.4), both Ψ and v are the identity map. 2 Corollary 4.4.4 (Hamana–Ruan) An operator space X is injective if and only if X ∼ = pA(1 − p) completely isometrically, for a projection p in an injective C ∗ -algebra A. Proof If X is injective, then the 1-2-corner of the injective C ∗ -algebra I(S(X)) above, is simply X itself. Indeed if (Y, i) is any injective envelope of an injective space X, then i is necessarily surjective. The converse is obvious. 2 4.4.5 (Remarks) (1) We saw in 4.4.2 that for any operator space X, the space Z = I12 (X) may be viewed as a TRO. If X happened originally to be a TRO inside B(K, H), and if one traces through the construction in 4.4.2, using B(H ⊕ K) in place of M2 (B(H)), one ﬁnds that the ‘triple product’ [x, y, z] in X coincides with its ‘triple product’ in Z. This follows from (4.6). (2) Although the C ∗ -algebras I11 , I22 , I(S(X)) in 4.4.2 seem to depend on a particular embedding X ⊂ B(H), in fact up to appropriate isomorphism they do not. Indeed if u : X → B(K) is a complete isometry, and if Y = u(X), then it is easy to see that u ‘extends’ to a ∗-isomorphism θ between I(S(X)) and I(S(Y )) (for example, see the proof of 4.4.6 below). This ∗-isomorphism restricts to a triple isomorphism between I12 (X) and I12 (Y ), and also to ∗-isomorphisms between I11 (X) and I11 (Y ), and between I22 (X) and I22 (Y ). Corollary 4.4.6 (Hamana–Kirchberg–Ruan) A surjective complete isometry between TROs is a triple morphism. Proof Such a surjective complete isometry u : X → Y gives rise, by 1.3.15, to a canonical complete order isomorphism between the operator systems S(X) and S(Y ). This isomorphism extends, by 4.2.5, to a completely isometric unital surjection θ between I(S(X)) and I(S(Y )). We assume that these latter C ∗ -algebras have been chosen as in 4.4.2 and Remark 4.4.5 (1), so that the triple products of X and Y coincide with those on I12 (X) and I12 (Y ) respectively. By 1.3.10, θ is a ∗-isomorphism. Since θ(1 ⊕ 0) = 1 ⊕ 0, θ is ‘corner-preserving’ as in 2.6.16. Since the 1-2-corner θ12 of θ is the restriction of θ to a subtriple, it is a triple 2 morphism. However, θ12 agrees with u on the copy of X.

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4.4.7 (TROs and the triple envelope) From 4.4.6, it follows that an operator space can have at most one ‘triple product’ with respect to which it is completely isometrically ‘triple isomorphic’ to a TRO. From this fact and Theorem 4.4.3, it follows that any injective envelope of an operator space X has a unique triple product with respect to which ‘it is a TRO’. Moreover, given any two injective envelopes, the map given by 4.2.5 is necessarily a triple morphism. In the sequel we can therefore regard I(X) as a TRO without any ambiguity. We now turn to the triple envelope T (X) of an operator space X. This is the generalization to nonunital operator spaces of the C ∗ -envelope or ‘noncommutative Shilov boundary’ (see Section 4.3). In Section 8.3 we will revisit this topic in more detail, and we shall see there that the triple envelope has most of the properties of the injective envelope. Also it has an additional universal property that is very useful, namely, T (X) is the smallest TRO or C ∗ -module containing X (see 8.3.9 for a precise statement). Since it is much smaller, it is often much more tractable than I(X). Most occurrences of the injective envelope in the next couple of sections could be replaced by the triple envelope, and in some places this would be preferable. We will not do this here however, mainly for pedagogical reasons, namely to avoid using basic facts about C ∗ -modules that will be fully developed in Chapter 8. For now, we simply give one construction of T (X), and make some deductions. We view I(X) as a TRO as above, and we deﬁne T (X) to be the smallest subtriple of I(X) containing X. We claim that T (X) = Span{x1 x∗2 x3 x∗4 · · · x2n+1 : n ≥ 0, x1 , x2 , . . . , x2n+1 ∈ X}.

(4.7)

Note that the ‘odd products’ appearing in (4.7), are formed by repetitions of the ‘triple product’ of I(X). To prove (4.7), note that any subtriple of I(X) containing X must contain such ‘odd products’, and hence also the closure of their span. Conversely, if u, v, w are three such ‘odd products’, then uv ∗ w is another ‘odd product’. Thus the right side of (4.7) is indeed a subtriple of I(X) containing X. By the discussion at the end of the second last paragraph, we see that as a triple system, T (X) does not really depend on the particular injective envelope I(X) chosen above. If X is a C ∗ -algebra or TRO, then by Remark 4.4.5 (1) and (4.7), it follows that X is its own triple envelope. On the other hand, if X is a unital operator space, then we saw in 4.2.8 that I(X) may be taken to be a unital C ∗ -algebra, containing X as a unital subspace. By adding ‘1s’ to the ‘odd products’ in (4.7) it is clear that T (X) is simply the C ∗ -subalgebra of I(X) generated by i(X). By the ﬁrst few lines of 4.3.3 we conclude that Ce∗ (X) is a triple envelope of X. 4.4.8 (Shilov inner product) Regard I(X) as the 1-2-corner of the C ∗ -algebra I(S(X)) as in 4.4.2, T (X) ⊂ I(X), and deﬁne B = T (X) T (X). By our convention in 1.1.2, this is a closed set, indeed this is a C ∗ -subalgebra of I22 (X), hence of I(S(X)). Then the triple envelope T (X) is a right operator B-module, and moreover we may deﬁne a B-valued ‘inner product’ y|z = y ∗ z, for y, z ∈ T (X).

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The ensuing map X × X → T (X) T (X) ⊂ I22 (X) is sometimes referred to as a Shilov inner product on X. Theorem 4.4.9 (Youngson) Let P be a completely contractive idempotent map on a TRO Z. For x, y, z ∈ Z we have P (P (x)P (y)∗ P (z)) = P (xP (y)∗ P (z)) = P (P (x)y ∗ P (z)) = P (P (x)P (y)∗ z). Furthermore Ran(P ) is completely isometrically isomorphic to a TRO. Proof Suppose that Y is the range of P , and regard P as valued in Y . Let i : Y → Z be the inclusion. Using 1.3.15, we have associated unital complete contractions P˜ and ˜i between the Paulsen systems S(Z) and S(Y ), such that P˜ ◦ ˜i = I on S(Y ). By deﬁnition of injectivity, we may extend these to complete contractions P : I(S(Z)) → I(S(Y )) and i : I(S(Y )) → I(S(Z)). Since P ◦ i restricted to S(Y ) is the identity map on S(Y ), by the rigidity property of the injective envelope we have that P ◦ i is the identity map on I(S(Y )). We will assume that I(S(Z)) has been written as a 2×2 matrix C ∗ -algebra as in (1), 4.4.2, and 4.4.3, so that the 1-2-corner of this C ∗ -algebra is an injective envelope I(Z) of Z, and so that the ‘triple product’ on Z agrees with the one on I(Z). Deﬁne Q = i ◦ P , this is a completely positive unital idempotent map on I(S(Z)). Also, all of our maps ﬁx the diagonal projections p = 1 ⊕ 0 and q = 0 ⊕ 1, and are therefore ‘corner-preserving’ as in 2.6.16. Let c be the canonical map taking Z into the 1-2-corner of I(S(Z)). We have Q(c(z)) = i (P (c(z))) = i (P˜ (c(z))) = c(i(P (z))),

z ∈ Z.

(4.8)

By 1.3.13, the range of Q is a C ∗ -algebra with product given by Q(Q(x)Q(y)) = Q(Q(x)y) = Q(xQ(y))

(4.9)

for x, y ∈ I(S(Z)). The products here are the product of the C ∗ -algebra I(S(Z)). As in the discussion in 4.4.2, the projections p, q allow us to write Ran(Q) as a 2 × 2 matrix C ∗ -algebra, Q being corner-preserving. By the reasoning in 4.4.2, the 1-2-corner W of this C ∗ -algebra may be regarded as a TRO, with ‘triple product’ given by the formula [x, y, z] = u(xy ∗ z), for x, y, z ∈ W , where u is the 1-2-corner of Q. By (4.8), the restriction of u to the copy of Z in the 1-2-corner, is i ◦ P . Thus it follows that this triple product, restricted to the copy of Y , is simply P (xy ∗ z), for x, y, z ∈ Y . Thus Y is completely isometrically isomorphic to a subtriple of W . Using (4.9), it is easy to see that Q(Q(x)Q(y)Q(z)) equals Q(Q(Q(x)Q(y))Q(z)) = Q(Q(xQ(y))Q(z)) = Q(xQ(y)Q(z)), for x, y, z ∈ I(S(Z)). Similarly, Q(Q(x)Q(y)Q(z)) = Q(Q(x)yQ(z)) = Q(Q(x)Q(y)z). The desired formulae follow from these last relations upon choosing the x, y ∗ , z from the copy of Z inside I(S(Z)), and using equation (4.8). 2

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Corollary 4.4.10 Suppose that P is a completely contractive idempotent map from a TRO Z onto a subspace Y for which Y Y Y ⊂ Y . Then P is a ‘conditional expectation’ in the sense that P (xy ∗ z) = xy ∗ P (z),

P (xz ∗ y) = xP (z)∗ y, P (zy ∗ x) = P (z)y ∗ x

for all x, y ∈ Y and z ∈ Z. 4.4.11 (An essential ideal) We deﬁne two C ∗ -algebras C = C(X) = I(X)I(X) and D = D(X) = I(X) I(X), in the notation in 4.4.2. The products here are in the C ∗ -algebra I(S(X)), as in 4.4.8. These are C ∗ -subalgebras of I11 (X) and I22 (X) respectively. Facts in the last paragraph of 4.4.2 show that the subset C(X) I(X) L= I(X) D(X) of the C ∗ -algebra I(S(X)), is an ideal. In fact L, is an essential ideal (see 2.6.10). To see this, suppose that K is an ideal in I(S(X)) with K ∩ L = (0). Let π : I(S(X)) → I(S(X))/K be the canonical ∗-epimorphism. Since π|L is one-toone, it is completely isometric by 1.2.4. By 2.6.15, it is clear that π maps each of the four corners into a matching corner of I(S(X))/K. Let πij be the i-j-corner map of π. Since π is a unital ∗-homomorphism, we have π21 (z) = π12 (z ∗ )∗ , and π11 and π22 are unital maps between the diagonal corners. Let Φ (resp. Φij ) be the restriction of π (resp. πij ) to S(X) (resp. to the appropriate corner of S(X)). Then Φii is the identity map of C, and Φ12 = Φ21 is completely isometric (since π|L is). Thus by 1.3.15, Φ is a complete isometry. Since I(S(X)) is an essential extension of S(X), π is a complete isometry as well, and hence K = (0). Hence L is an essential ideal by 2.6.10. Proposition 4.4.12 Let X be an operator space, and suppose that a ∈ I11 (X), or that a is a right module map on the triple envelope T (X) of X (with respect to the module action discussed in 4.4.8). If ax = 0 for all x ∈ X, then a = 0. Proof If a is a right module map on T (X), and if a(X) = 0, then we clearly have a(x1 x∗2 x3 x∗4 · · · x2n+1 ) = a(x1 )x∗2 x3 x∗4 · · · x2n+1 = 0, for x1 , x2 , . . . , x2n+1 in X. Thus a = 0, by (4.7). For the other assertion, we may assume that a ≤ 1. By replacing a by a ∗ a, we may also suppose that 0 ≤ a ≤ 1, where this last ‘1’ is the identity of I11 . Deﬁne u(z) = (1 − a)z, for z ∈ I(X). Since u(x) = x for x ∈ X, we see that u is the identity map, by rigidity. Thus aI(X) = 0, and so (a ⊕ 0)L = 0, where L is as in 4.4.11. By the last line of 4.4.11, and the ﬁrst deﬁnition in 2.6.10, it follows that a ⊕ 0 = 0, so a = 0. 2 Proposition 4.4.13 Suppose that X is an operator space with an injective envelope which is a C ∗ -algebra B. Then M2 (B) is an injective envelope for the Paulsen system S(X). Moreover, we may take I11 (X) = I22 (X) = B.

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Proof By 4.2.8 (3), B is unital, with identity 1 say. The C ∗ -algebra M2 (B) is injective by 4.2.10, and has as subsystems S(X) ⊂ S(B) ⊂ M2 (B), where we have identiﬁed the diagonal idempotents in S(X) and S(B) with 1. To see the ﬁrst assertion, it suﬃces to show that the identity map is the only S(X)-projection Ψ : M2 (B) → M2 (B). Such a Ψ ﬁxes the diagonal idempotents of S(X), hence is ‘corner-preserving’: indeed as in 2.6.16, we may write v1 u Ψ = . u v2 Now u|X = IX , so by the rigidity property of the injective envelope, u = IB . Thus Ψ is a S(B)-projection. Now one sees that Ψ ﬁxes 1 ⊗ M2 , which is a unital C ∗ -algebra. Hence, by 1.3.12, Ψ is a M2 -bimodule map. Thus v1 = v2 = u = IB , and so Ψ is the identity map. The last assertion is clear from the variant of the construction in 4.4.2 with B(H) replaced by B (if necessary use also Remark 4.4.5 (2)). 2 4.5 THE MULTIPLIER ALGEBRA OF AN OPERATOR SPACE We concentrate on ‘left multipliers’ in this section; the ‘right-handed’ variant is left to the reader, and the associated ‘left adjointable multipliers’ will be more fully developed in Section 8.4. We will also begin to see the applications of these fruitful concepts. The left multiplier algebra Ml (X) of an operator space X, (resp. the left adjointable multiplier algebra Al (X)) turns out to be an abstract operator algebra (resp. C ∗ -algebra) whose elements are maps on X. To illustrate the versatility of this construction, we point out: (1) If A is an approximately unital operator algebra, then Ml (A) is the usual left multiplier algebra LM (A) considered in Section 2.6. If A is a C ∗ -algebra, then Al (A) = M (A). (2) If E is a Banach space, then Ml (Min(E)) coincides with the classical function multiplier algebra M(E) considered in Section 3.7, whereas Al (Min(E)) is the classical ‘centralizer algebra’ considered there. (3) If Z is a right Hilbert C ∗ -module, then we shall show in 8.4.2 that Ml (Z) and Al (Z) are respectively the algebras of bounded right module maps, and adjointable maps, on Z. Thus although Ml (X) and Al (X) are deﬁned purely in terms of the matrix norms and vector space structure on X, they often encode ‘operator algebra structure’. Operator spaces X with trivial (one-dimensional) multiplier algebras are exactly the spaces lacking ‘operator algebraic structure’ in a sense which one can make precise. 4.5.1 (The left multiplier algebra of an operator space) For an operator space X, we deﬁne a left multiplier of X to be a linear map u : X → X such that

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there exists a Hilbert space H, an S ∈ B(H), and a linear complete isometry σ : X → B(H) with σ(ux) = Sσ(x) for all x ∈ X. We deﬁne the multiplier norm of u, to be the inﬁmum of S over all possible H, S, σ as above. We deﬁne Ml (X) to be the set of left multipliers of X. Notice that we may replace the B(H) in the deﬁnition of Ml (X) by an arbitrary C ∗ -algebra, without any real change in meaning. Or, we may replace B(H) by B(K, H), where K is another Hilbert space. We leave it to the reader to check that this allowance does not enlarge the set Ml (X) or change the multiplier norm. The ‘multiplier norm’ clearly satisﬁes all the properties of a norm, except for the triangle inequality. In fact from the next theorem, the triangle inequality property is easily seen (using the fact that the set of maps u characterized in that theorem is clearly convex). Indeed from this theorem it is easily seen that Ml (X) is a normed algebra, and a unital subalgebra of CB(X), but with a possibly larger norm. These last facts will be reproved later in the proof of 4.5.5 too. Notice also that if we ﬁx such σ : X → B(K, H), and deﬁne Mσl (X) to be the set of S ∈ B(K) such that Sσ(X) ⊂ σ(X), then Mσl (X) is an operator algebra. Although we shall not use this explicitly, later we shall see that there is a ‘universal embedding’ σ0 such that Ml (X) = Mσl 0 (X). Indeed this also follows from Theorem 4.5.5 and its proof. For any linear map u : X → X, we deﬁne τu : C2 (X) → C2 (X) to be the map u ⊕ IX ; that is: " # x u(x) = , x, y ∈ X. τu y y ‘Left multipliers’ are best viewed as a sequence of equivalent deﬁnitions, as in the following theorem. To explain the notation in this result: in (iii)–(v), we are viewing X ⊂ T (X) ⊂ I(X) ⊂ I(S(X)) as in 4.4.2 and 4.4.8. Here T (X) is the triple envelope (see 4.4.7). Thus I11 (X) is the C ∗ -algebra deﬁned in 4.4.2, and the inner product in (v) is the one mentioned in 4.4.8. The matrices in (v) are indexed on rows by i, and on columns by j. Thus the inequality in (v) takes place in the C ∗ -algebra Mn (B), where B = T (X) T (X) (see 4.4.8). Theorem 4.5.2 Let u : X → X be a linear map. The following are equivalent: (i) u is a left multiplier of X with multiplier norm ≤ 1. (ii) τu is completely contractive. (iii) There exists a (necessarily unique, by 4.4.12) a ∈ I11 (X) of norm ≤ 1, such that u(x) = ax for all x ∈ X. (iv) u is the restriction to X of a (necessarily unique, by 4.4.12) contractive right module map a on T (X) with a(X) ⊂ X. (v) [u(xi )|u(xj )] ≤ [xi |xj ], for all n ∈ N and x1 , . . . , xn ∈ X. The inﬁmum in the deﬁnition of the ‘multiplier norm’ (see 4.5.1) is achieved. Proof Assume (i) and let σ, S, H be chosen as in the deﬁnition of Ml (X) above. Then for x, y ∈ X, we have

Some ‘extremal theory’ 169 " # x σ(x) x σ(ux) S 0 . τu ≤ max{S, 1} = = y σ(y) y σ(y) 0 I Thus τu ≤ max{S, 1}. A similar argument shows that τu cb ≤ max{S, 1}. Then (ii) follows by the deﬁnition of the ‘multiplier norm’. We now prove that (ii) implies (iii). We give Paulsen’s short proof of this result found by Blecher, Eﬀros, and Zarikian [57]. Another short proof will be sketched in 8.4.5. Using notation in 4.4.2, let     I11 I(X) I11 C1 0 X I11 I(X)  . S =  0 C1 X  ⊂  I11 I(X) I(X) I22 X X C1 To explain this notation, if we let M be the last space, then M should be viewed as a corner, and ∗-subalgebra, of M2 (I(S(X))) in the obvious way. Thus M is an injective C ∗ -algebra. We assume that τu cb ≤ 1 and deﬁne a map on S by     λ 0 x1 λ 0 u(x1 ) µ x2  . Φ :  0 µ x2  −→  0 y1∗ y2∗ ν u(y1 )∗ y2∗ ν Let Dn be the diagonal ∗-subalgebra of Mn . Since τu is a completely contractive left D2 -module map on C2 (X), Φ is completely positive, by the discussion in 3.6.1. By Arveson’s extension theorem 1.3.5, we may extend this map to a completely positive map Φ : M → M . Since Φ ﬁxes the diagonal D3 , fact 2.6.17 tells us that Φ is ‘corner-preserving’. Write Φij for the ‘i-j-corner’ of Φ, here i, j = 1, 2, 3. View the lower 2 × 2 corner of Φ as a map Ψ on I(S(X)). The restriction of Ψ to S(X) is the identity map, thus by rigidity Ψ is the identity map on I(S(X)). Hence Φ ﬁxes the C ∗ -subalgebra   C 0 0  0 I11 I(X)  0 I(X) I22 of M . By 1.3.12, Φ is a bimodule map over this subalgebra. Thus Φ(wz) = Φ(w)z, where w is the matrix with a 1 in the 1-2-corner and zeroes elsewhere, and z is the matrix with an element from X in the 2-3-corner and zeroes elsewhere. We immediately obtain (iii) by inspection of the relation Φ(wz) = Φ(w)z, and the fact that Φ extends Φ . The desired element a in I11 (X), is Φ12 (1). That (iii) implies (i) is clear from the deﬁnitions in 4.5.1, if we regard X as a subspace of the C ∗ -algebra I(S(X)), as discussed in 4.4.2, and taking S in 4.5.1 to be a ⊕ 0 ∈ I(S(X)). This also shows the ﬁnal assertion of the theorem. We do not need this, but (iv) implies (i) easily via the theory of C ∗ -modules. In the rest of the proof we set Y = T (X), viewed as a subtriple of I(X), with X identiﬁed as a subspace of Y . We let B be the C ∗ -algebra Y Y , and we denote the canonical B-valued inner product in (v) as y ∗ z, for y, z ∈ Y . We

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recall that products like these, and the ones met below, make sense and may be interpreted, for example, as products in the C ∗ -algebra I(S(X)). To show that (iii) implies (iv), let a be as in (iii). Using (4.7) we clearly have aXX XX · · · X = u(X)X XX · · · X ⊂ Y, and aY ⊂ Y . Also, axy ∗ z = (ax)(y ∗ z) for x, y, z ∈ Y . Thus the map z → az is a contractive right B-module map on Y , whose restriction to X is u. If (iv) holds, then a well-known result of Paschke implies that a(z)∗ a(z) ≤ z ∗ z,

z ∈ Y.

(4.10) − 12

We sketch the quick argument for (4.10) here. Set bn = (z ∗ z + n1 ) , the inverse taken in the unitization of B (which in this case may be thought of as the span of B and the identity of I22 (X)). Then (zbn )∗ zbn = bn z ∗ zbn ≤ 1 (by basic spectral theory for the selfadjoint z ∗ z). Thus zbn ≤ 1, so that a(zbn ) ≤ 1. We deduce that a(zbn )∗ a(zbn ) ≤ 1. However a is a B 1 -module map clearly, and it follows that a(z)∗ a(z) ≤ z ∗ z + n1 . Letting n → ∞ yields (4.10). For b1 , . . . , bn ∈ B and x1 , . . . , xn ∈ X, set z = i xi bi . By (4.10), b∗i u(xi )∗ u(xj )bj = a(z)∗ a(z) ≤ z ∗ z = b∗i x∗i xj bj . i,j

i,j

A well-known positivity criterion (see [407, IV.3.2] or 8.1.12) yields (v). Let [xij ], [yij ] ∈ Mn (X), and let k ≤ n. If (v) holds, then we have ∗ ∗ ykj ] ≤ [x∗ki xkj + yki ykj ] . [u(xki )∗ u(xkj ) + yki

Summing over k, and taking the norm, we obtain xij ∗ ∗ ∗ ∗ . u(xki ) u(xkj ) + yki ykj ≤ xki xkj + yki ykj = yij k

k

This is equivalent to saying that τu cb ≤ 1. Hence (v) implies (ii).

2

4.5.3 (Matrix norms on Ml (X)) For u = [uij ] ∈ Mn (Ml (X)), write Lu for the operator on Cn (X) given by the usual formula for the product of a matrix and a column. This is a left multiplier on Cn (X) (for if aij is associated to uij as in Theorem 4.5.2 (iii), then a variant of the proof that (iii) implies (i) in 4.5.2 shows that Lu ∈ Ml (Cn (X))). We deﬁne ‘multiplier matrix norms’ on Ml (X), by assigning to u = [uij ] above, the multiplier norm of Lu (that is, its ‘norm’ in Ml (Cn (X))). In the proof of 4.5.5, we show that these quantities are well deﬁned, and are norms deﬁning an operator algebra structure on Ml (X). Moreover Mn (Ml (X)) ∼ = Ml (Cn (X))

as operator algebras,

(4.11)

via the map u → Lu above. Indeed, note that if (4.11) holds isometrically for every n ∈ N, then it holds completely isometrically. This may be seen by composing the isometric isomorphisms Mm (Mn (Ml (X))) ∼ = Mmn (Ml (X)) ∼ = Ml (Cmn (X)) ∼ = Mm (Ml (Cn (X))).

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4.5.4 (A matricial generalization of Theorem 4.5.2) Theorem 4.5.2 has a variant characterizing ‘matrices of multipliers’. For example, if u : Cn (X) → Cn (X) is a linear map, then the following are equivalent: (i) u is a left multiplier of Cn (X) with multiplier norm ≤ 1, (ii) u ⊕ IX : Cn+1 (X) → Cn+1 (X) is completely contractive, (iii) there exists an a ∈ Ball(Mn (I11 (X))) such that u(x) = ax for all x ∈ Cn (X). To prove this, one may mimic the proof of 4.5.2. We give further hints for the diﬃcult implication that (ii) implies (iii) here; one follows the proof above closely, replacing the space M in that proof by the following space:   Mn (I11 ) Cn (I11 ) Cn (I(X))  Rn (I11 ) I11 I(X)  , Rn (I(X) ) I(X) I22 which may be regarded as a unital ∗-subalgebra of M2n (I(S(X))). The last few steps are modiﬁed by considering Φ applied to the sum over j of products of a matrix with an ej in the 1-2-corner and zeroes elsewhere, and the matrix with an element xj from X in the 2-3-corner and zeroes elsewhere. Here (ej ) is the canonical basis of Cn . The desired element a is the matrix with jth column Φ12 (ej ). Hence a ≤ Φ12 cb [e1 · · · en ] ≤ 1. Theorem 4.5.5 If X is an operator space, then the ‘multiplier norms’ deﬁned in 4.5.1 and 4.5.3 are norms. With these matrix norms Ml (X) is an operator algebra, and X is a left operator Ml (X)-module. Moreover, the inclusion map from Ml (X) to CB(X) is a one-to-one completely contractive homomorphism. Proof Again we use notation from 4.4.2, as well as the completely isometric embeddings X ⊂ I(X) ⊂ I(S(X)). We saw there that I(X) is a left operator I11 (X)-module. We deﬁne IMl (X) = {a ∈ I11 (X) : aX ⊂ X}. This is clearly an operator algebra, a subalgebra of the C ∗ -algebra I11 (X). Moreover, X is a left operator IMl (X)-module (as can be seen by using 3.1.2 (2) and 3.1.12, if necessary). The equivalence of (i) and (iii) in 4.5.2, shows that Ml (X) is a subalgebra of CB(X), and yields an isometric isomorphism θ : IMl (X) → Ml (X). Namely θ(a)(x) = ax for any x ∈ X. Indeed θ is a complete isometry, using 4.5.4. Therefore, the ‘multiplier matrix norms’ are norms, and indeed Ml (X) ∼ = IMl (X)

as operator algebras.

(4.12)

Since every operator module is a matrix normed module (see 3.1.5 (1)), the canonical embedding from Ml (X) to CB(X) is completely contractive, and it is one-to-one by 4.4.12. 2

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4.5.6 (Right multipliers) There are deﬁnitions and results analoguous to the above, but with ‘left’ replaced by ‘right’. Thus we may regard Mr (X) ⊂ CB(X). The latter inclusion is as sets (not necessarily isometrically), and we need to put the reverse of the usual composition multiplication on this subset of CB(X). The reader can see the reason for this by considering, for example, the ‘right-hand version’ of (4.12), namely that Mr (X) ∼ = {b ∈ I22 (X) : Xb ⊂ X}. From these descriptions it is clear that as operators on X, any v ∈ Mr (X) commutes with every u ∈ Ml (X). 4.5.7 (Adjointable multipliers) We write Al (X) for the C ∗ -algebra ∆(Ml (X)) (see 2.1.2 for this notation). Similarly, Ar (X) = ∆(Mr (X)). The operators in Al (X) are called left adjointable multipliers, for reasons that will be clear in Section 8.4, where we will give a more thorough development of A l (X), and its links to C ∗ -modules. For now we give just one quick result about Al (X): Proposition 4.5.8 Let X be an operator space. (1) A linear map u : X → X is in Al (X) if and only if there exist a Hilbert space H, an S ∈ B(H), and a linear complete isometry σ : X → B(H) with σ(ux) = Sσ(x) for all x ∈ X, and such that also S ∗ σ(X) ⊂ σ(X). (2) Al (X) ∼ = {a ∈ I11 (X) : aX ⊂ X and a∗ X ⊂ X} as C ∗ -algebras. (3) If u, S and σ are as in (1), then the involution u∗ in Al (X) is the map x → σ −1 (S ∗ σ(x)) on X. (4) The canonical inclusion map from Al (X) into CB(X) (or into B(X)) is an isometric homomorphism. Proof Assertion (2) is clear from (4.12), since the algebras in (2) are the diagonal C ∗ -algebras of the algebras in (4.12). Thus if u ∈ Al (X), then we have a corresponding a in the set on the right in (2) such that u(x) = ax for all x ∈ X. If we regard I(S(X)) as a C ∗ -subalgebra of some B(H), we see that the required condition in (1) holds. Conversely, suppose that S, H, σ are as in (1). If Mσl (X) is as in 4.5.1, then the restriction to ∆(Mσl (X)) of the canonical contractive homomorphism from Mσl (X) to Ml (X), is a ∗-homomorphism into Al (X), by the last paragraph of 2.1.2. Since S ∈ ∆(Mσl (X)), u ∈ Al (X). This completes the proof of (1). Note that S ∗ ∈ ∆(Mσl (X)) is taken by this ∗-homomorphism to u∗ ∈ Al (X). This gives (3). Finally, (4) follows from the above and from A.5.9. 2 4.5.9 (Restrictions of multipliers) It follows immediately from the deﬁnitions 4.5.1, that if u ∈ Ml (X), and if Z is a closed subspace of X such that u(Z) ⊂ Z, then u|Z ∈ Ml (Z). Moreover, the multiplier norm of u|Z is smaller than the multiplier norm of u. A similar argument using 4.5.8 shows that if u ∈ Al (X), and if u(Z) ⊂ Z and u∗ (Z) ⊂ Z, then u|Z ∈ Al (Z). 4.5.10 (The Banach space case) If E is a Banach space, then M(E) = Ml (Min(E)) and Z(E) = Al (Min(E)) (see 3.7.1 for this notation). We will prove this momentarily.

(4.13)

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If X is any operator space with an injective envelope (B, j) such that B is a unital C ∗ -algebra, then putting (4.12) together with 4.4.13 yields the fact that Ml (X) ∼ = {a ∈ B : aj(X) ⊂ j(X)} as operator algebras. Using this fact, together with the fact that I(Min(E)) is a commutative C ∗ -algebra C(Ω) (see 4.2.11), we conclude that Ml (Min(E)) ∼ = {a ∈ C(Ω) : aj(E) ⊂ j(E)}. This shows that Ml (Min(E)) is a minimal operator space, and that the inclusion of Ml (Min(E)) in M(E) is a contraction (by the implication (iv) ⇒ (i) in 3.7.2). Conversely, by 3.7.2 again and deﬁnition 4.5.1, we also have a contractive inclusion of M(E) in Ml (Min(E)). Thus Ml (Min(E)) = M(E). Also, Al (Min(E)) = Z(E), since these are the ‘diagonals’ of the previous two equal algebras. Proposition 4.5.11 Let A be an approximately unital operator algebra. Then Ml (A) = CBA (A) = LM (A), and Mr (A) = RM (A), as operator algebras. Proof Identify LM (A) = CBA (A) (see Section 2.6). Let λ : LM (A) → CB(A) be the canonical completely isometric embedding. From the characterization of LM (A) in 2.6.2 (2), and the deﬁnition in 4.5.1, we see that λ(u) ∈ Ball(Ml (A)), for all u ∈ Ball(LM (A)). Thus λ, viewed as a map into Ml (A), is contractive. A similar argument with matrices, using Corollary 2.6.6, shows that λ is completely contractive. Since the inclusion Ml (A) ⊂ CB(A) is a complete contraction, by 4.5.5, we see that λ : LM (A) → Ml (A) is a complete isometry. Similarly, we have a completely isometric homomorphism ρ : RM (A) → Mr (A). If u ∈ Ml (A), and b ∈ A, then using 4.5.6 we see that u(ab) = u(ρ(b)a) = ρ(b)(u(a)) = u(a)b, so 2 that u ∈ CBA (A). Thus λ is surjective. Proposition 4.5.12 If ν : X → Y is a linear surjective complete isometry between operator spaces, then the map taking u to νuν −1 is a completely isometric isomorphism from Ml (X) to Ml (Y ). Proof The isometry follows from the deﬁnition in 4.5.1: if u = σ −1 (Sσ(·)), for σ and S as in 4.5.1, then ν(u(ν −1 (y))) = (σ ◦ ν −1 )−1 (S(σ ◦ ν −1 )(y)). Thus νuν −1 Ml (Y ) ≤ uMl (X) , and the other direction follows by symmetry. The complete isometry is similar, and is left as an exercise using (4.11). 2 The last result is an almost tautological ‘Banach–Stone type’ result for operator spaces. In fact many theorems of this type may be deduced from it. As an example, we give the following ‘Banach–Stone theorem’ for operator algebras: Theorem 4.5.13 Let v : A → B be a completely isometric linear surjection between approximately unital operator algebras. Then there exists a completely isometric homomorphism θ from A onto B, and a unitary U with U and U −1 in M (B), such that v = θ(·)U . Proof If A and B are unital, then LM (A) = A and LM (B) = B. Hence by 4.5.11 and 4.5.12, there is a completely isometric isomorphism θ : A → B such that vλ(a)v −1 = λ(θ(a)), for any a ∈ A. Here λ denotes the left regular representation either on A or B. This gives v(a) = θ(a)v(1 A ). Hence if we set U = v(1A ), then there exists an a0 ∈ Ball(A) with 1B = v(a0 ) = θ(a0 )U , so that

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The multiplier algebra of an operator space

U ∗ = θ(a0 ) ∈ B, by an early fact in A.1.1. By 2.1.3, v(1A ) is an isometry, and it lies in ∆(B). By symmetry v(1A ) must also be a coisometry, and therefore a unitary, in ∆(B). One way to see this is to consider v op : Aop → B op as in 1.2.25, and use the fact that ∆(B)op = ∆(B op ). In the general case, clearly v ∗∗ : A∗∗ → B ∗∗ is a completely isometric linear surjection. By the ﬁrst part, there is a unital completely isometric isomorphism θ : A∗∗ → B ∗∗ , and a unitary U ∈ ∆(B ∗∗ ) such that v(a) = θ(a)U for a ∈ A. Inside B ∗∗ , we have that BU −1 = θ(A) is an operator algebra with cai. We have BU −1 BU −1 = BU −1 , by A.6.2. Thus BU −1 B = B. Also, BBU −1 = BU −1 , by A.6.2 again. Thus by 2.5.10, BU −1 = B. Thus BU = B and θ(A) = B. Hence U, U −1 ∈ RM (B). A similar argument applied to U −1 B, shows that U, U −1 ∈ LM (B). So U, U −1 ∈ M (B). 2 The asymmetry in the statement of 4.5.13 is easily removed by setting θ equal to U ∗ θ(·)U . Then θ is also an isomorphism of A onto B, and θ(·)U = U θ (·). 4.5.14 (One-sided M -projections) If X is an operator space, then a linear idempotent P : X → X is said to be a left M-projection if the map P (x) σP (x) = x − P (x) is a complete isometry from X → C2 (X). A similar deﬁnition (involving R2 (X)) pertains to right M-projections. These projections are connected to the noncommutative variant on M -ideals studied in Section 4.8. Here we will prove just one result concerning them: Theorem 4.5.15 If P is an idempotent linear map on an operator space X, then the following are equivalent: (i) (ii) (iii) (iv)

P is a left M -projection. The map τP introduced above 4.5.2 is completely contractive. P is a (selfadjoint) projection in the C ∗ -algebra A (X). There exist a completely isometric embedding σ : X → B(H), and a projection e ∈ B(H), such that σ(P x) = eσ(x) for all x ∈ X. (v) There exists a completely isometric embedding σ : X → B(H) such that σ(x)∗ σ(y) = 0,

x ∈ P (X), y ∈ (I − P )(X).

Proof To show that (i) implies (iv), suppose that X ⊂ B(H), and view C2 (X) as a subset of M2 (X) ⊂ M2 (B(H)) in the usual way. Then take σ = σP (the map in 4.5.14), regarded as a map X → M2 (B(H)), and set 1 0 e= . 0 0 That (iv) implies (v) is clear, since e(1 − e) = 0.

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175

Assume (v). If x ∈ X, then by the C ∗ -identity, x2 equals σ(x)2 = σ(P x) + σ(x − P x)2 = σ(P x)∗ σ(P x) + σ(x − P x)∗ σ(x − P x). But the latter is the norm of σP (x) in C2 (X). Thus σP is an isometry. A similar argument shows that it is a complete isometry, which shows (i). To prove the equivalence of (ii) and (iii), note that by 4.5.2, (ii) holds if and only if P ∈ Ball(Ml (X)). Since P = P 2 , and since Ml (X) is an operator algebra, this occurs when P = P ∗ (see 2.1.3), in which case P ∈ ∆(Ml (X)) = A (X). That (iv) implies (iii) follows by 4.5.8 (1) and (3). Conversely, assume (iii). By 4.5.2, there exists an a ∈ Ball(I11 ) such that P x = ax for all x ∈ X. Since P is an idempotent map, we have (a2 − a)X = 0, which implies that a2 = a by 4.4.12. As in the last paragraph, a is a (selfadjoint) projection. Now one may 2 deduce (iv) by regarding I(S(X)) as a C ∗ -subalgebra of some B(H). We refer the reader to [73], for example, for explicit calculation of the multiplier algebras for some other classes of operator spaces. 4.6 MULTIPLIERS AND THE ‘CHARACTERIZATION THEOREMS’ It is very simple to deduce the BRS and CES theorems from 4.5.2. In fact we give a more general result which has such theorems as immediate consequences. 4.6.1 (Oplications) If X and Y are operator spaces, then a (left) oplication of Y on X is a completely contractive bilinear map m : Y × X → X (see 1.5.4), such that there is an element e ∈ Ball(Y ) (or more generally, a net of elements (et )t in Ball(Y )) such that m(e, x) = x (resp. m(et , x) → x) for all x ∈ X. The word ‘oplication’ is intended to be a contraction of the phrase ‘operator multiplication’. The element e is called a left identity for the oplication. Theorem 4.6.2 (Oplication theorem) Let m : Y × X → X be an oplication. (1) There exists a (necessarily unique) linear complete contraction θ from Y into Ml (X) such that θ(y)(x) = m(y, x) for all y ∈ Y, x ∈ X. If e is a left identity for the oplication, then θ(e) = IX . (2) If Y is an algebra (resp. C ∗ -algebra), and if m is a module action, then θ is a homomorphism into Ml (X) (resp. ∗-homomorphism into Al (X)). (3) If Y is an operator system, whose unit is a left identity for m, then θ maps into Al (X), and is completely positive. Proof We assume the presence of a left identity e, and leave the case involving the net as an exercise (in which case one needs to use limits in the formulae below). Let θ : Y → B(X) be the map associated with m. Clearly θ(e) = IX . If τ is as in 4.5.1, if m2 is as in 1.5.4, and if y ∈ Ball(Y ) and x, x ∈ X, then # " # " m(y, x) x y 0 x 0 x τθ(y) ≤ = , . = m2 x x 0 e x 0 x Thus τθ(y) is contractive, and an analoguous argument with matrices shows that it is completely contractive. By 4.5.2, we deduce that θ is a contractive map into

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Multipliers and the ‘characterization theorems’

Ml (X). Given [yij ] ∈ Ball(Mn (Y )), we consider [θ(yij )] as a map v on Cn (X), via the usual formula for the product of a matrix and a column. Arguing as above, but now using larger matrices, we see that v ∈ Ball(Ml (Cn (X))). Thus θ is a complete contraction into Ml (X). This proves (1). Since θ(y)(x) = m(y, x) for all y ∈ Y, x ∈ X, the ﬁrst assertion in (2) is quite clear. The second one follows from the last paragraph of 2.1.2, and the fact that 2 Al (X) = ∆(Ml (X)). In a similar way, (3) follows from 1.3.8 and 1.3.3. 4.6.3 The BRS theorem 2.3.2 follows immediately from the last result. For if A is an algebra satisfying the BRS hypotheses, then 4.6.2 provides a completely contractive homomorphism θ from A into the operator algebra Ml (A). That θ is a complete isometry is obvious: for example, θ(a) ≥ a1 = a for a ∈ A. Indeed one sees easily from this approach, that the ‘associativity’ condition for the algebra is not needed in the BRS characterization (this was proved in the original paper [69] using the injective envelope). One way to see this is to consider the product on A both as a left and as a right oplication; and then the ‘automatic associativity’ follows from 4.6.2 (1) and the last assertion in 4.5.6. Corollary 4.6.4 Let A be a unital Banach algebra, and let A1 and A2 be operator space structures on A (keeping the same norm). Suppose that the canonical map A1 ⊗h A2 → A2 , induced by the product on A, is completely contractive. Then there is a third operator space structure A3 on A between A1 and A2 , such that A3 is completely isometrically isomorphic to a unital operator algebra. Proof By the oplication theorem 4.6.2, there is a unital completely contractive homomorphism θ : A1 → Ml (A2 ) with θ(a)(b) = ab for all a, b ∈ A. On the other hand, we have a canonical complete contraction Ml (A2 ) → CB(A2 ) by 4.5.5, and a canonical complete contraction CB(A2 ) → A2 given by ‘evaluation at 1A ’. Thus we have a sequence of completely contractive maps θ

A1 −→ Ml (A2 ) −→ CB(A2 ) −→ A2 which compose to the identity map IA . Thus θ is an isometric homomorphism. Since Ml (A2 ) is an operator algebra, we may deﬁne an operator algebra A3 to be A with the matrix norms pulled back via θ from Ml (A2 ). The other assertions are now evident from the above. 2 4.6.5 (The commutative case) The last result is general enough to explain many of the results found in the 1970s, distinguishing general operator algebras amongst the Banach algebras. As an example, we show how it gives Theorem 3.7.9. Thus consider a unital Banach algebra A such that the product on A is contractive as a map from A ⊗g2 A to A. We remarked in 3.7.7 that this is equivalent to it being completely contractive as a map from Max(A) ⊗ h Min(A) to Min(A). Thus we deduce from 4.6.4 that A is isometrically isomorphic to an operator algebra. Indeed, the proof of 4.6.4 shows that A is isomorphic to a subalgebra of Ml (Min(A)). However from (4.13), Ml (Min(A)) is a uniform algebra, which implies that A is a uniform algebra.

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4.6.6 (Applications to operator modules) The reader may want to glance at Section 3.1 to recall the deﬁnition of an operator module. We will assume that all operator bimodules are nondegenerate in this section. Any operator space X is an operator Ml (X)-Mr (X)-bimodule, and is thus also an operator Al (X)-Ar (X)-bimodule. This follows, for example, from the technique in the proof in 4.5.5 showing that X is an operator Ml (X)-module. We call these bimodule structures the extremal operator module actions on X. Conversely, if A is a unital Banach algebra which is also an operator space, then by the oplication theorem, the left operator module actions of A on X are in oneto-one correspondence with the completely contractive unital homomorphisms from A into Ml (X). If A is a C ∗ -algebra then we may replace Ml (X) by Al (X) here. This generalizes the phenomena in Section 3.7. We may now give a generalization of the Christensen–Eﬀros–Sinclair theorem 3.3.1; an abstract characterization of nondegenerate operator bimodules: Theorem 4.6.7 Suppose that X is an operator space, that A and B are operator spaces which are also approximately unital Banach algebras, and that X is a nondegenerate left A-module and a nondegenerate right B-module. Then X is an operator A-B-bimodule if and only if axn ≤ an xn ,

and

xbn ≤ xn bn ,

for every n ∈ N, a ∈ Mn (A), b ∈ Mn (B) and x ∈ Mn (X). In particular, the centered equations imply that a(xb) = (ax)b for all a ∈ A, x ∈ X, b ∈ B. Proof The centered equation here is simply saying that X is both a left and a right h-module in the sense of 3.1.3; and then the one direction follows from a remark in 3.1.3. Conversely, if the centered equation holds, then the two module actions are oplications on X. By the oplication theorem, the two actions are ‘prolongations’ (in the sense of 3.1.12) of the extremal Ml (X)-Mr (X)-bimodule actions (see 4.6.6). Since X is an operator Ml (X)-Mr (X)-bimodule, and since prolongations of operator modules remain operator modules (see 3.1.12), we deduce that X is an operator A-B-bimodule. Since these extremal actions commute, so do the original module actions. This proves the last assertion. 2 4.6.8 (Examples) The following examples illustrate the main point in 4.6.6. (1) Let H c be a column Hilbert space (see 1.2.23). We leave it as an exercise that Ml (H c ) = B(H) and Mr (H c ) = C (see also 8.4.2 for a more general result). Thus if A is a unital Banach algebra which is also an operator space, the left operator module actions of A on H c are exactly in correspondence with the completely contractive unital homomorphisms θ : A → B(H). Indeed we saw this already in Proposition 3.1.7. On the other hand, the right operator module actions are all ‘scalar’: in correspondence with the characters of A (see A.4.2). (2) Consider the possible left module actions on X = 1n . We equip 1n with its natural (maximal) operator space structure, so that X is the operator space dual of ∞ n (see (1.30)). There are plenty of Banach module actions of a

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Banach algebra A on X, corresponding to contractive unital homomorphisms from A to B(1n ). Similarly, there are plenty of ‘matrix normed module actions’ ∗ (see 3.1.4) on X. For example, since X = (∞ n ) we have from Example 3.1.5 1 (2) that Max(n ) is canonically a matrix normed ∞ n -module. Considering the into B(X), A.5.9 shows that there is an corresponding homomorphism from ∞ n isometric copy of the Banach algebra ∞ inside B(X) or CB(X). One might n ﬁrst guess that this copy equals Al (X). In fact this is false, and indeed we shall show that Ml (Max(1n )) = C. Equivalently, by the main point in 4.6.6, there are no ‘nonscalar’ (see the last line of (1) above) operator module actions on X = Max(1n ). We recall that in 4.3.8 we showed that X is a unital operator space, and that Ce∗ (X) = C ∗ (Fn−1 ). By the last paragraph of 4.4.7, the latter is also a triple envelope of X. Hence, by 4.5.2 (iv), any u ∈ Ml (X) is necessarily the restriction to X of a right module map on C ∗ (Fn−1 ). Since the latter is a unital C ∗ -algebra, any right module map on it is given by left multiplication by a ﬁxed a ∈ C ∗ (Fn−1 ). Recall from 4.3.8 that X is identiﬁed with the span of 1 and the generators u1 , . . . , un−1 of C ∗ (Fn−1 ). Since u(X) ⊂ X, we have that a = a1 ∈ X, and auj belongs to X for any j. Hence we must have a ∈ C1, and therefore u ∈ CIX . Hence Ml (Max(1n )) = C as promised. 4.6.9 (CES-representations of operator modules) If X is an operator bimodule over algebras A and B as in 4.6.7), then one may extract from the proof of 4.6.7 an explicit CES-representation for X (see 3.3.3). Indeed let J be the canonical completely isometric embedding from X into I(S(X)) (see 4.4.2). By the oplication theorem 4.6.2 and (4.12), there exist completely contractive homomorphisms θ : A → I11 (X) and π : B → I22 (X), with J(ax) = θ(a)J(x) and J(xb) = J(x)π(b),

a ∈ A, b ∈ B, x ∈ X.

Thus if we represent I(S(X)) faithfully and nondegenerately on a Hilbert space L, and if we regard J, θ and π as valued in B(L), then (J, θ, π) is an explicit CES-representation of the operator A-B-bimodule X. We next improve on the extension theorem from Section 3.6. Theorem 4.6.10 Let X be a nondegenerate operator A-B-bimodule over unital C ∗ -algebras A and B. Then X is an injective operator space if and only if X has the following extension property: For every nondegenerate operator A-B-bimodule W , closed A-B-submodule Y of W , and completely contractive A-B-bimodule map u : Y → X, there exists a completely contractive A-B-bimodule map u ˆ : W → X extending u. Proof Suppose that X satisﬁes the above extension property. By the CES theorem 3.3.1, X may be realized as an A-B-submodule of some B(K, H), where H is a Hilbert A-module and K is a Hilbert B-module. By the hypothesis applied to the inclusion map X ⊂ B(K, H), there is a completely contractive idempotent map from B(K, H) onto X. Since B(K, H) is injective, so is X.

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If X is injective, then by the proof of 4.4.4, I12 (X) = X. Represent the C ∗ -algebra I(S(X)) faithfully and nondegenerately on a Hilbert space. The two diagonal projections 1 ⊕ 0 and 0 ⊕ 1 determine a splitting of the Hilbert space as H ⊕ K, say (e.g. see 2.6.14). Hence I11 (X) and I22 (X) are unital ∗-subalgebras of B(H) and B(K) respectively, so that we have B(H) B(K, H) I (X) X ⊂ = B(H ⊕ K). I(S(X)) = 11 X I22 (X) B(H, K) B(K) By injectivity, there is a completely positive idempotent map Φ from B(H ⊕ K) onto I(S(X)). By 2.6.16, Φ decomposes as a 2 × 2 matrix of maps. Let P be the ‘1-2-corner’ of Φ, then P : B(K, H) → X is a completely contractive idempotent map onto X. By 1.3.12, Φ is a ‘conditional expectation’ onto I(S(X)). It follows that P is a left I11 -module map. However if X is an operator A-B-bimodule, then as in 4.6.9 there is a unital completely contractive homomorphism θ from A to I11 ⊂ B(H) implementing the left module action of A on X. Hence H is a Hilbert A-module, with A-action given by θ. Since θ maps into I11 , we see that P is a left A-module map onto X. Similarly, P is a right B-module map onto X. We saw in 3.6.4 that B(K, H) is injective as an operator A-B-bimodule. It is now easy to see that so is X. 2 4.6.11 (Bimodule injective envelopes) In the following discussion, A and B are unital C ∗ -algebras. Suppose that X is an operator A-B-bimodule. We saw in 4.4.2 that the injective envelope I(X) = I12 (X) considered above, is an operator module over I11 . Hence, for example via the maps θ and π in 4.6.9, I(X) is an operator A-B-bimodule. By 4.6.10, it follows that I(X) is injective in the category of operator A-B-bimodules and completely bounded bimodule maps (see 3.6.4). In fact, I(X) is also an ‘injective envelope’ of X in the latter category. To be more precise: one may adapt all the deﬁnitions in 4.2.3 to the A-B-bimodule situation, simply by changing the word ‘map’ to ‘A-B-bimodule map’, ‘operator space’ to ‘operator A-B-bimodule’, and ‘subspace’ to ‘A-B-submodule’. Thus we have notions of ‘A-B-rigid extension’, and so on. What we said in the last paragraph clearly implies that the operator space injective envelope I(X) is also an A-B-injective extension of X. Since I(X) is rigid, it is an A-B-rigid extension of X. A trivial modiﬁcation of the proof of 4.2.5 shows that there is an essentially unique A-B-rigid injective A-B-extension of X. Conversely, putting the last two sentences together, an obvious diagram chase shows that any A-B-rigid injective extension of X, is also rigid as an operator space extension, and is hence an injective envelope of X in the old sense (of 4.2.3). Such observations are often useful, as applications such as the following show: Corollary 4.6.12 If I(X) is an injective envelope of X, then for any cardinals I and J, MI,J (I(X)) is an injective envelope of MI,J (X), and of KI,J (X). Proof Note that MI,J (X), KI,J (X), and MI,J (I(X)) are operator bimodules ∞ over ∞ I and J , by 3.1.14 (3). Also MI,J (I(X)) is an injective operator space, by

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the proof in the ﬁrst paragraph of 4.2.10. By 4.6.10, it is also injective in the cate∞ gory of operator ∞ I -J -bimodules. Suppose that u : MI,J (I(X)) → MI,J (I(X)) ∞ is a completely contractive ∞ I -J -bimodule map extending the identity map on the image of KI,J (X). It is fairly obvious that this bimodule condition implies that u is of the form u([zpq ]) = [upq (zpq )], for maps upq : I(X) → I(X). Clearly upq is a completely contractive map extending the identity map on i(X). By rigidity, upq is the identity map, for all p, q. Hence u is the identity map. Thus ∞ MI,J (I(X)) is an injective ∞ I -J -rigid extension of KI,J (X), and MI,J (X). By the observation above 4.6.12, we obtain the desired result. 2 4.7 MULTIPLIERS AND DUALITY Operator space multipliers yield deep facts about w ∗ -topologies. For example: Theorem 4.7.1 (Magajna) If X is a dual operator space, then any operator in Ml (X) is w∗ -continuous. Proof This proof is again rather technical. As in the proof of 2.7.13, there is a canonical w∗ -continuous projection q : X ∗∗ → X, which induces a completely isometric w∗ -homeomorphism v : X ∗∗ /Ker(q) → X (using A.2.5 (3) if necessary). If u ∈ Ml (X), then u∗∗ ∈ B(X ∗∗ ). We claim that: q(u∗∗ (η)) = u q(η),

η ∈ X ∗∗ .

(4.14)

Assuming (4.14), it follows that u∗∗ induces a map u˙ in B(X ∗∗ /Ker(q)), namely u( ˙ η) ˙ = (u∗∗ (η))˙, where η˙ is the equivalence class of η ∈ X ∗∗ in the quotient. ˙ by basic Banach space duality principles Since u∗∗ is w∗ -continuous, so is u, (e.g. A.2.4). Using (4.14) it is easy to see that u˙ = v −1 uv. Since u, ˙ v, and v −1 ∗ are w -continuous, so is u, and thus the theorem is proved. In order to prove (4.14), we will use the notation in 4.4.2, and we will be silently using the principles in A.2.3. We let A be the C ∗ -algebra I(S(X)), and let p be the projection in A corresponding to the identity of I11 . We regard A ⊂ A∗∗ , then p may be viewed as a projection in A∗∗ , and A∗∗ has four corners with respect to p, as in 2.6.14. Let c11 and c12 be, respectively, the canonical embeddings of I11 = pAp and I(X) = pA(1 − p) in A. We will write C11 = c∗∗ 11 ∗∗ ∗∗ and C12 = c∗∗ into A∗∗ . We view X as a subspace 12 , these embed I11 and I(X) of pA(1 − p) ⊂ pA∗∗ (1 − p) ⊂ A∗∗ , and thus we may identify X ∗∗ with the w∗ -closure of X in the corner pA∗∗ (1 − p). Via this embedding and 1.3.15, we view S(X ∗∗ ) ⊂ A∗∗ . Also by 1.3.15, q induces a canonical complete contraction Q : S(X ∗∗ ) → S(X) ⊂ A, with Q|S (X) = I|S (X) . Since A is injective, we may extend Q to a complete contraction R : A∗∗ → A. Since R|S (X) = I|S (X) , it follows by rigidity (see 4.2.3) that R|A = IA . By 1.3.12, R is an A-bimodule map. Let a ∈ I11 correspond to u, as in 4.5.2 (iii). Then C11 (a) = c∗∗ 11 (a) = c11 (a). Similarly C12 (z) = c12 (z) if z ∈ I(X). Since R is an A-bimodule map, we have

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R(C11 (a) C12 (η)) = c11 (a) R(C12 (η)) = c11 (a) c12 (q(η)) = c12 (uq(η)),

(4.15)

for η ∈ X ∗∗ ⊂ I(X)∗∗ . On the other hand, we claim that C11 (a) C12 (η) = C12 (u∗∗ (η)),

η ∈ X ∗∗ .

(4.16)

Indeed since the product in a W ∗ -algebra is separately w∗ -continuous, both sides of (4.16) may be viewed as w ∗ -continuous functions taking η ∈ X ∗∗ into the W ∗ algebra A∗∗ . Since A is a subalgebra of A∗∗ , if z ∈ X ⊂ I(X) then C11 (a) C12 (z) = c11 (a) c12 (z) = c12 (az) = c12 (uz) = C12 (u∗∗ (z)). Thus (4.16) follows by a routine w ∗ -density argument. By (4.15) and (4.16), 2 c12 (uq(η)) = R(C12 (u∗∗ (η))) = c12 (q(u∗∗ (η))). This proves (4.14). Corollary 4.7.2 If B is an operator algebra which is also a dual operator space, then the product on B is separately w ∗ -continuous. Proof If a ∈ B, then the map b → ab on A is clearly a left multiplier, and therefore is w∗ -continuous by 4.7.1. Similarly the product is w ∗ -continuous in the ﬁrst variable. 2 Next we turn to a general functional analytic result: Lemma 4.7.3 (1) Let T : E → F be a one-to-one linear map between Banach spaces, and suppose that F is a dual Banach space. The following are equivalent: (i) E is a dual Banach space and T is w ∗ -continuous, (ii) T (Ball(E)) is w∗ -compact. (2) Let X and Y be operator spaces, with Y a dual operator space, and let u : X → Y be a one-to-one linear map such that un (Ball(Mn (X))) is w∗ compact for every positive integer n. Then the predual of X given in the proof of (1), is an operator space predual of X, and u is w ∗ -continuous. Proof (1) That (i) implies (ii) is clear. Given (ii), we see by the Principle of Uniform Boundedness that T (Ball(E)), and therefore also T , is bounded. We may assume that T is a contraction. Suppose that Z is the predual of F , viewed as a subset of F ∗ , and let W = T ∗ (Z), a linear subspace of E ∗ . The canonical map j : E → W ∗ is one-to-one and contractive, and j(x), T ∗ (z) = T ∗ (z), x = T (x), z. We will show that j is an isometric isomorphism. Given ψ ∈ Ball(W ∗ ), deﬁne a ˜ map ψ(z) = ψ(T ∗ (z)), for z ∈ Z. Then ψ˜ ∈ Ball(Z ∗ ) = Ball(F ). If we can show ˜ that ψ = T (x), for some x ∈ Ball(E), then we are done, for in this case it is clear that ψ = j(x). So suppose, by way of contradiction, that ψ˜ ∈ / T (Ball(E)). By assumption, T (Ball(E)) is w∗ -closed, and so by the Hahn–Banach theorem there

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˜ exists z ∈ Z such that ψ(z) > 1 and |T (x), z| ≤ 1 for all x ∈ Ball(E). The latter condition implies that T ∗(z) ≤ 1, whereas the former condition implies the contradictory assertion that ψ(T ∗ (z)) > 1. That T is w∗ -continuous with respect to the predual W is now clear. (2) We will use 1.6.4 to check that the predual from (1), is an operator space predual. Let xλ = [xλij ] be a net in Ball(Mn (X)), and x = [xij ] ∈ Mn (X), with < xλij , w > → < xij , w > for all i, j and w ∈ W . Equivalently, u(xλij ) → u(xij ) in the w∗ -topology of Y . By 1.6.3 (1), the matrices [u(xλij )] converge in the w∗ topology to [u(xij )] in Mn (Y ). By hypothesis, [u(xij )] ∈ un (Ball(Mn (X))), so that x ∈ Ball(Mn (X)), and we are done. 2 Theorem 4.7.4 Let X be a dual operator space. (1) Ml (X) is a dual operator algebra, and Al (X) is a W ∗ -algebra. (2) A bounded net (at )t in Ml (X) or Al (X) converges in the w∗ -topology to an element a in the same space, if and only if at (x) → a(x) in the w∗ -topology in X for all x ∈ X. Proof For the ﬁrst assertion in (1), it suﬃces by Theorem 2.7.9 to show that Ml (X) is a dual operator space. To do this we shall use 4.7.3 applied to the canonical embedding u : Ml (X) → CB(X), and basic facts about CB(X) (see 1.6.1). We will show that if (at )t is a net in Ball(Mn (Ml (X))) converging in the w∗ -topology in Mn (CB(X)) to a, then a ∈ Ball(Mn (Ml (X))). Let µ be the isomorphism in (4.11). We will test if µ(a) ∈ Ball(Ml (Cn (X))) using 4.5.2 (ii). Choosing x, y in Mm (Cn (X)), we need to check that µ(a)m (x) ≤ x . (4.17) y y We do have

µ(at )m (x) ≤ x . y y

(4.18)

Let z (resp. zt ) be the matrix on the left side of (4.17) (resp. (4.18)). These matrices are in M2m,m (Cn (X)) = M2mn,m (X), and zt → z in the w∗ -topology in M2m,m (X), by 1.6.3 (1). Hence (4.17) follows from (4.18). Thus a ∈ Ball(Ml (X)). We have now proved that Ml (X) is a dual operator space. Thus Ml (X) is a dual operator algebra, by Theorem 2.7.9. That is, there is a w ∗ -homeomorphic completely isometric isomorphism of Ml (X) onto a w∗ -closed unital-subalgebra B of some B(H). Clearly ∆(B) = B ∩ B ∗ is a von Neumann algebra by 2.1.2; hence Al (X) ∼ = ∆(B) is a W ∗ -algebra. Assertion (2) follows from (1), and the deﬁnition of the w ∗ -topologies concerned (see the proof of 4.7.3). 2 Lemma 4.7.5 Any dual operator space X is a normal dual bimodule, and hence is also a dual operator bimodule (see 3.8.3), over Ml (X) and Mr (X) (and also over Al (X) and Ar (X)).

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Proof The fact that X is an operator bimodule follows from 4.6.6. The other facts follow immediately from 4.7.4 (2) and 4.7.1. 2 Theorem 4.7.6 Let X and M be dual operator spaces, and let m : M × X → X be an oplication. Suppose that m is w ∗ -continuous in the ﬁrst variable. Then we may conclude: (1) m is separately w∗ -continuous. (2) There exist Hilbert spaces H, K, a unital w ∗ -continuous completely contractive map π : M → B(K), and a w∗ -continuous complete isometry v from X into B(H, K), with v(m(b, x)) = π(b)v(x) for all x ∈ X, b ∈ M . (3) If M is also a W ∗ -algebra (resp. algebra) and if m is a module action, then π in (2) may be taken to be a ∗-homomorphism (resp. homomorphism). Proof First apply Theorem 3.8.3 to the Ml (X) action on X (after appealing to 4.7.5). This provides the Hilbert spaces H, K, and the map v. It also provides a w∗ -continuous completely contractive homomorphism ρ : Ml (X) → B(K), with ρ(u)v(x) = v(ux) for all x ∈ X, u ∈ Ml (X). Next apply the oplication theorem 4.6.2, to get a unital completely contractive θ : M → Ml (X) such that θ(b)x = m(b, x) for all x ∈ X, b ∈ M . It is elementary to check that θ is w ∗ continuous using, ﬁrst, 4.7.4 (2), second, the fact that m is w ∗ -continuous in the ﬁrst variable, and third, the Krein–Smulian theorem A.2.5 (2). Let π = ρ ◦ θ, and now (2) is clear. If b ∈ M , and xt → x in the w∗ -topology in X, then π(b)v(xt ) → π(b)v(x) in the w∗ -topology in B(H, K). This follows since v is a w ∗ -w∗ -homeomorphism, and operator multiplication is separately w ∗ -continuous. Thus m(b, xt ) → m(b, x), which gives (1). Adding the ideas in 4.6.2 (2), and 1.2.4, gives (3). 2 The last result generalizes Theorem 3.8.3, and allows us to characterize dual operator modules over algebras which need not be operator algebras: Corollary 4.7.7 Suppose that X is a left operator module over a unital Banach algebra and operator space A, that A and X are dual operator spaces, and that the module action is w∗ -continuous in the ﬁrst variable. Then X is a dual operator module in the sense of 3.8.1. 4.8 NONCOMMUTATIVE M -IDEALS The theory of M -ideals of Banach spaces emerged in the early 1970s with the paper [4] of Alfsen and Eﬀros. It is an important tool in functional analysis and approximation theory (see [195] for an extensive survey of results). A one-sided variant of this classical theory was recently developed in [57,440,71,73,442], and is intimately connected with the operator space multipliers met in the last sections. Indeed, we already met in 4.5.14 the ‘left M -projections’: these are exactly the orthogonal projections in the C ∗ -algebra Al (X). If X is a dual operator space, then we saw that Al (X) is a von Neumann algebra. Applying basic von Neumann algebra theory then leads to a cogent ‘noncommutative’ generalization of the

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classical theory of M -ideals. The aim of this short section is to give some of the ﬂavor of this subject, and to hopefully act as a conduit to the more satisfying account in the cited papers. En route, we will also compute the various M -ideals of operator algebras. 4.8.1 (M -ideals: classical and one-sided) We begin with some deﬁnitions. In 3.7.4 we deﬁned M -projections. A subspace Y of a Banach space E is called an M -summand of E if Y is the range of an M -projection. A subspace Y of E is called an M -ideal in E if Y ⊥⊥ is an M -summand in E ∗∗ . Of course Y ⊥⊥ is just the w∗ -closure of Y in E ∗∗ , and may be identiﬁed with Y ∗∗ (see A.2.3). If X is an operator space, then a map P on X is a complete M -projection if the ampliﬁcation Pn is an M -projection on Mn (X) for every n ∈ N. This immediately leads, by straightforward analogy with the last paragraph, to the notion of complete M -summand and complete M -ideal. In 4.5.14 we deﬁned left M -projections. By analogy to the above, a right M -summand of X is the range of a left M -projection on X. We say that Y is a right M -ideal if Y ⊥⊥ is a right M -summand in X ∗∗ . Similar deﬁnitions pertain to the ‘other-handed’ case, i.e., right M -projections, left M -ideals, and so on. We shall see that all these deﬁnitions are interconnected in a tidy way. 4.8.2 (Transferring the classical theory) Few of the proofs of results in the extensive classical M -ideal theory transfer verbatim to the one-sided case; most require new, noncommutative, arguments. We will see an example of the latter in Theorem 4.8.6. As an illustration of an ‘uncomplicated transfer’, we mention the fact that for any one-sided M -summand J of an operator space X, there exists a unique contractive linear idempotent of X onto J, namely the onesided M -projection P deﬁning J. To prove this, note that it suﬃces to show that Ker(P ) ⊂ Ker(Q) for any contractive idempotent Q onto Ran(P ), for then Q(x) = Q(x − (x − P (x))) = P (x) for every x ∈ X. If y ∈ Ker(P ), let z = Q(y). For t > 0, we have tz ≤ t2 z2 + y2 , (t + 1)z = Q(tz + y) ≤ tz + y = y the second equality because the map σP in 4.5.14 is an isometry. Squaring, we obtain (2t + 1)z2 ≤ y2 . Since t > 0 was arbitrary, z = 0. 4.8.3 (Examples) We showed in 3.7.4 that the M -projections on a Banach space E are exactly the projections in the centralizer algebra Z(E) discussed in Section 3.7. We showed in (4.13) that Z(E) = Al (Min(E)). Thus, by 4.5.15, the right M -projections on Min(E) are exactly the classical M -projections on E. Therefore, left M -summands (resp. left M -ideals) of Min(E) are exactly the same as the classical M -summands (resp. M -ideals) of E. Similar assertions hold for the ‘other-handed’ variants. Hence classical M -ideals and summands are particular examples within the one-sided M -ideal theory. We see next that complete M -ideals and summands (see 4.8.1) are also particular examples within the one-sided theory.

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Proposition 4.8.4 Suppose that X is an operator space. (1) A projection P : X → X is a complete M -projection if and only if it is both a left M -projection and a right M -projection. (2) A subspace Y of X is a complete M -summand if and only if it is both a left M -summand and a right M -summand. (3) A subspace Y of X is a complete M -ideal if and only if it is both a left M -ideal and a right M -ideal. Proof (1) If P is both a left and right M -projection, then by ﬁrst applying the map σP from 4.5.14, and then the ‘row variant’ of σP , we obtain P (x) P 2 (x) 0 P (x) (I − P )P (x) = = x = x − P (x) P (I − P )(x) (I − P )2 (x) 0 x − P (x) , which is simply max {P (x) , x − P (x)}, for x ∈ X. Similar identities hold for matrices, and so P is a complete M -projection. Conversely, if P is a complete M -projection, then the mapping u : X −→ X ⊕∞ X ⊂ M2 (X) : x −→ (P x) ⊕ (x − P x) is completely isometric. It follows that if u2,1 is as in 1.2.1, then " # P (x) P (x) = u2,1 = max {P x , x − P x} = x . x − P (x) x − P (x) Again it is easy to generalize this to matrices, so that P is a left M -projection. A similar argument shows that P is a right M -projection. One direction of (2) follows from (1), the other follows from (1) together with the ‘uniqueness’ assertion proven in 4.8.2. Now (3) is also clear. 2 Theorem 4.8.5 Let A be an approximately unital operator algebra. (1) The M -ideals in A are the complete M -ideals. These are exactly the twosided ideals in A which possess a two-sided contractive approximate identity. (2) The M -summands in A are the complete M -summands. These are exactly the principal ideals Ae for a central projection e ∈ M (A). (3) The right M -ideals in A are exactly the right ideals in A which possess a left contractive approximate identity. (4) The right M -summands in A are exactly the principal right ideals eA for a projection e ∈ LM (A). Proof (4) By 4.5.15 together with 4.5.11, the left M -projections on A are exactly the projections e in LM (A) (or equivalently in ∆(LM (A))). w∗

(3) If J is a right M -ideal of A, then J ∗∗ = J ⊥⊥ = J is a right M summand. Hence by (4), J ⊥⊥ = eA∗∗ for a projection e ∈ A∗∗ . Considered as subsets of A∗∗ , we have JA ⊂ J ∗∗ . But also JA ⊂ A. Thus JA ⊂ J ∗∗ ∩ A, and the latter space (by A.2.3 (4)) equals J. So J is a right ideal of A. Since

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A∗∗ is unital, we must have e ∈ J ∗∗ . Therefore e is a left identity for J ∗∗ . By Proposition 2.5.8, J has a left contractive approximate identity. Conversely, if J is a closed right ideal of A with a contractive left approximate identity, then by Proposition 2.5.8 and 2.5.2, we have that J ∗∗ is a subalgebra of A∗∗ , and J ∗∗ has a left identity e of norm 1. Note that e is a projection in A∗∗ . Moreover, J ∗∗ A∗∗ ⊂ J ∗∗ by a routine separate w∗ -continuity argument similar to those encountered in Section 2.5. We obtain J ∗∗ = eJ ∗∗ ⊂ eA∗∗ ⊂ J ∗∗ . Thus J ∗∗ = eA∗∗ . Since e is a projection in A∗∗ , we see by (4) that J ∗∗ is a right M -summand of A∗∗ . Hence J is a right M -ideal of A. (2) If e is a central projection in M (A), then Ae is a complete M -summand by (4), the ‘left-handed’ version of (4), and 4.8.4 (2). Thus it is an M -summand. Conversely, suppose that P is an M -projection on A. We may assume that A is unital (the general case follows easily from this case by considering the M projection P ∗∗ on A∗∗ , and using a deﬁnition of M (A) from Section 2.6). We set z = P (1). As in the third centered equation in the proof of 3.7.3, we have P ∗ (ϕ) + (I − P )∗ (ϕ) ≤ 1, for any ϕ ∈ A∗ of norm 1. If ϕ is a state, we deduce that 1 = ϕ(1) = P ∗ (ϕ)(1) + (I − P )∗ (ϕ)(1) ≤ P ∗ (ϕ) + (I − P )∗ (ϕ) = 1. If 0 ≤ a ≤ c and 0 ≤ b ≤ d, and a + b = c + d, then a = c and b = d. Thus ϕ(z) = P ∗ (ϕ)(1) = P ∗ (ϕ) ≥ 0.

(4.19)

Thus z is Hermitian in A (see A.4.2). Claim 1: z 2 = z, or equivalently by A.4.2, ϕ(z 2 − z) = 0.

(4.20)

for all states ϕ on A. To prove this we will need Claim 2: If ϕ is a state on A with ϕ(z) = 1, then (4.20) holds. Claim 2 follows by spectral theory: regard ϕ|C ∗ (1,z) as a probability measure µ on [−1, 1]. Since ϕ(z) = 1, µ is the point mass δ1 . To see Claim 1, we may assume that ϕ(z) and ϕ(1 − z) are nonzero (for if ϕ(z) = 0, for example, then ϕ(1 − z) = 1, so that by Claim 2 for 1 − z, we have ϕ((1 − z)2 ) = ϕ(1 − z), which is equivalent to (4.20)). From (4.19) we see ﬁrst that ψ = P ∗ (ϕ)/P ∗ (ϕ) is a state of A, and second that ψ(z) =

ϕ(P (1)) ϕ(P (z)) = = 1. P ∗ (ϕ) P ∗ (ϕ)

By Claim 2, we have ψ(z 2 ) = ψ(z). Equivalently, ϕ(P (z 2 )) = ϕ(P (z)) = ϕ(z). Also, ϕ((I − P )((1 − z)2 )) = ϕ(1 − z) (replacing P by I − P and z = P (1) by (I − P )(1)). A line of algebra shows that ϕ(z 2 − z) = 0, establishing Claim 1.

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By 2.1.3, z is a (selfadjoint) projection in A. If ϕ is any state with P ∗ (ϕ) = 0, and if ψ is as deﬁned above, extend ψ to a state ψ˜ on some C ∗ -cover of A. By the Cauchy–Schwarz inequality we obtain, for any a ∈ A, that ˜ ∗ ) ψ(1 − z) = 0. |ψ(a(1 − z))|2 ≤ ψ(aa Thus ϕ(P (a(1 − z))) = 0. Since this holds for an arbitrary state, A.4.2 implies that P (a(1 − z)) = 0. Thus A(1 − z) ⊂ (I − P )(A). Symmetrical arguments show that (1 − z)A ⊂ (I − P )(A), Az ⊂ P (A), and zA ⊂ P (A). If a ∈ A, then a = az + a(1 − z), so that P (a) = az. Similarly, P (a) = za. Now (2) is clear. (1) This is similar to the proof of (3), but uses (2) instead of (4). 2 The following is a sample result from the theory of one-sided M -ideals: Theorem 4.8.6 Let X be an operator space (resp. dual operator space) and suppose that {Jλ : λ ∈ Λ} is a family of right M -ideals (resp. right M -summands) of X. Then the closure of Span{∪λ∈Λ Jλ } in the norm (resp. w∗ -) topology is a right M -ideal (resp. right M -summand) of X. Proof (Sketch) By basic functional analysis, (∪λ∈Λ Jλ )⊥⊥ is the w∗ -closure of the span of ∪λ∈Λ Jλ⊥⊥ . Thus it suﬃces to prove the ‘respectively’ assertion, and we may assume henceforth that X is a dual operator space. For the remainder of the proof, we rely heavily on the fact that A l (X) is a W ∗ -algebra (see 4.7.4), and on facts about projections in von Neumann algebras. We claim ﬁrst that if P, Q are left M -projections on X, then the w ∗ -closure of P (X) + Q(X) is the right M -summand (P ∨ Q)(X), where P ∨ Q is the usual ‘join’ of projections in the von Neumann algebra Al (X). Since (P ∨ Q)P = P , and similarly for Q, it follows that P (X)+ Q(X) ⊂ (P ∨Q)(X). Now (P ∨Q)(X) is w∗ -closed, since P ∨ Q is a w∗ -continuous projection (see 4.7.1), and so P (X) + Q(X)

w∗

⊂ (P ∨ Q)(X).

To see the reverse inclusion, we will use the formula p ∨ q = w∗ − lim (p + q)1/n , n→∞

valid for any two projections p, q in a von Neumann algebra. This formula follows easily from e.g. [214, Lemma 5.1.5]. It follows from this and from Theorem 4.7.4 (2), that if x ∈ (P ∨ Q)(X), then x = (P ∨ Q)(x) = w∗ − lim (P + Q)1/n (x). n→∞

(4.21)

By the spectral theorem, for each ﬁxed n ∈ N, there exist polynomials f k without constant terms, such that fk (P +Q) → (P +Q)1/n . Hence (P +Q)1/n (x) lies in the norm closure of P (X)+Q(X). Thus from (4.21) we see that x ∈ P (X) + Q(X) We have now proved the above claim.

w∗

.

188

Notes and historical remarks

Finally, suppose that {Jλ : λ ∈ Λ} is a family of right M -summands as above. Suppose that {Pλ } is the associated family of left M -projections, and set P = ∨λ∈Λ Pλ . We claim that the right M -summand P (X) equals the w ∗ closure W of Span{∪λ∈Λ Jλ }. Indeed, the argument at the beginning of the last paragraph shows immediately that P (X) contains W . For any ﬁnite subset ∆ of Λ, deﬁne P∆ = ∨λ∈∆ Pλ . Clearly {P∆ } is an increasing net in the W ∗ -algebra Al (X), and P∆ → P in the w∗ -topology. Thus, using Theorem 4.7.4 (2) again, it follows that P∆ (x) → P (x) in the w∗ -topology of X, for any x ∈ X. By the 2 previous paragraph, P∆ (x) lies in W . Thus P (x) ∈ W . Hence W = P (X). Combining facts from the last two results, we have: Corollary 4.8.7 Let A be an approximately operator algebra. The closed span in A of any family of right ideals of A, each possessing a left contractive approximate identity for itself, is a right ideal also possessing a left contractive approximate identity. It is interesting that the last result fails badly for Banach algebras: indeed it is easy to ﬁnd ﬁnite-dimensional counterexamples. In 8.5.19 (resp. 8.5.16) we will show that the right M -ideals (resp. right M summands) in a right Hilbert C ∗ -module are exactly the closed right submodules (resp. orthogonally complemented right submodules). Results in the noncommutative M -ideal theory may then be seen to be generalizations of some aspects of the theory of C ∗ -modules. See [56] for more on this perspective. 4.9 NOTES AND HISTORICAL REMARKS 4.1: Again the story begins with Arveson, and his introduction of noncommutative generalizations of the Choquet and Shilov boundaries of unital operator spaces in the foundational papers [21,22]. Arveson built up a formidable array of machinery, including the theory of boundary representations, multivariable dilation theory, and much more. Remark 4.1.13 (2) is from [21]. Proposition 4.1.12 is a reﬁnement due mostly to Muhly and Solel, of an advance made in [21]. It is adapted from [283] and [128]. Putting 4.1.12 together with ideas of Woronowicz [435] permits a construction of the noncommutative Shilov boundary diﬀerent from the one presented in Section 4.3. Namely, any completely isometric unital representation π of a unital operator algebra B has a dilation ρ which is a completely isometric boundary representation. It is easy to see that the C ∗ -algebra generated by ρ(B) is a C ∗ -envelope of B. This is presented in [128] from the perspective of Agler’s model theoretic approach [1], and a simpliﬁcation of the argument appears in [24]. Kirchberg has a related approach we believe (but we are not sure if this is in print). There are many equivalent deﬁnitions of Choquet boundary points in the literature. Indeed the Choquet boundary, the classical Shilov boundary, and the boundary theorem are a crucial part of the theory of function spaces and uniform algebras. For example, see the texts [167,400]. The Shilov boundary of a nonunital

Some ‘extremal theory’

189

function space always exists too, in the form of a line bundle—see [53, Section 4] and the notes for 8.3. 4.2: Injective envelopes of unital operator spaces were ﬁrst considered by Hamana in [184] in the late 1970s; many of these proofs are valid for general operator spaces. He also studied injective envelopes of operator systems in several of his papers (see reference list), and gave various excellent applications. Ruan studied the injective envelope of a general operator space in [371]. Hamana claims some of the same results at a similar date; he refers to some conference talks which he gave, and in [189, 193] he refers to [188] which we have not seen. Some of this was published in [190, 191], and ﬁnally in [193]. We have followed the presentation from [189] more closely than that of [371] (although they are similar). Injective envelopes have been used to study nonselfadjoint operator algebras for a long time (e.g. see [69, 40]). The fact that an injective (envelope of a) C ∗ -algebra (or approximately unital operator algebra) is a unital C ∗ -algebra, we could only ﬁnd in [65]. The result 4.2.9 appears in [61]. In the C ∗ -algebra case this result also follows from 1.3.12. If we drop the hypothesis in 4.2.9 that B = P (A) is a subalgebra of A (we still require P (1A ) = 1A ), then an identical proof shows that P (a1 P (a2 )) = P (P (a1 )a2 ) = P (P (a1 )P (a2 )) for all a1 , a2 ∈ A. Such results fail for general Banach algebras. For work on ‘separable injectivity’ see [367], for example. In [365], Robertson classiﬁes injective Hilbertian spaces. 4.3: The result 4.3.1 was proved by Arveson in several important cases in [21, 22]. The general case of 4.3.1, the name ‘C ∗ -envelope’, and 4.3.5 and 4.3.6 in the unital case, are from [184]. The general case of 4.3.5 was observed in [72]. The observation in Example 4.3.7 (2) has been put to eﬀect by Zarikian, who has developed software to compute C ∗ -envelopes, multipliers, and so on, for subspaces of Mn (see [441]). Proposition 4.3.8 appears in [310, 445, 446]; the proof of (2) given here is due to Pisier. The C ∗ -envelope is generally considerably smaller than the injective envelope, and is therefore often more tractable. Some interesting C ∗ -envelopes have been computed in [21, 22, 276, 284], for example, and in the literature on nonselfadjoint direct limit algebras. A fruitful and important concept is that of a Shilov module, or Shilov representation, of an operator algebra A (e.g. see [127, 281]). This is a Hilbert A-module which is an A-submodule of a Hilbert Ce∗ (A)-module. Every boundary representation is a Shilov representation, but not vice versa. Algebras with factorization have been studied by many authors, e.g. see [20, 108,340,349]. The term convexly approximating in modulus we have seen used for function algebras in [127]. For more on Arveson’s noncommutative H ∞ algebras (also known as ﬁnite maximal subdiagonal algebras) see, for example, [20, 275, 270, 339]. Items 4.3.10 and 4.3.11 are from [61]. In the commutative case, the important results about generalized H ∞ algebras are known to hinge on the commutative version of 4.3.11. E.g. see [258] for a discussion of this point. 4.4: For 4.4.3 and 4.4.4, see the Hamana and Ruan references in the Notes to 4.2. Result 4.4.6 appears in Ruan’s thesis [369], but was noticed independently by Hamana (see Notes for 4.2), and Kirchberg in (we believe) the 1980s. The remark

190

Notes and historical remarks

in the last paragraph of 4.4.7 is from [445]. Youngson’s theorem is from [439]. The proof here is diﬀerent, but it is not better than the original, which used a useful fact due to Eﬀros and Størmer [151]. There are related results at the Banach space level (e.g. see [151, 164, 379], and references therein). Corollary 4.4.10 is also true with the word ‘completely’ removed [141]. The triple envelope is due to Hamana (see 8.3, and references in the Notes to 8.3 for more historical details). The triple envelope, or noncommutative Shilov boundary, is generally much smaller than I(X); for example if X is a subspace of a C ∗ -algebra A, then T (X) may be realized as a subtriple of A∗∗ (see [58]). The other unattributed results in this section are from [68], except for the observations 4.4.13, and part of 4.4.12 (from [53]). The use of essential ideals in 4.4.11 and 4.4.12, was suggested by [163] (see also the proof of [194, Lemma 3.2]). It is shown in [68] that I11 (X) = M (C(X)) = LM (C(X)) = RM (C(X)), and similarly for I22 (X) and D(X) (in the notation of 4.4.2 and 4.4.11). Neal and Russo have recently given remarkable intrinsic characterizations, up to complete isometry, of one-sided ideals of C ∗ -algebras, TROs, and C ∗ -algebras as operator spaces with certain aﬃne geometric properties. See [289, 290]. 4.5: The multipliers discussed in this section have a complicated history, in part because three of the papers concerned did not appear in the journals they were ﬁrst submitted to. Kirchberg (e.g. see [229–231]), and K. H. Werner [425], had considered multipliers of special classes of operator spaces or operator systems. Around 1998, W. Werner considered one-sided multipliers of nonunital operator systems (c.f. [427]). Using the injective envelope, he proved the analogue of some parts of 4.5.2, 4.5.8, and 4.5.9 for such multipliers. Hamana in [187], and Frank and Paulsen in [163], considered respectively the multiplier algebra M (A), and LM (A) and ‘local multiplier algebras’, in terms of the injective envelope. These authors considered the case when A is a C ∗ -algebra. In early 1999, independently to Werner, and inspired by the characterizations in Section 3.7 of Chapter 3, Blecher introduced multipliers of general operator spaces [53], in order to improve and unify the analoguous noncommutative characterization theorems. (We learned of Werner’s work on operator system multipliers in the middle of 2000, but believed it to be only distantly related to the general operator space setting. In fact there is a direct connection. The point is that an operator space X may be embedded in the selfadjoint subspace of the Paulsen system with zero main diagonal entries. The latter nonunital operator system falls within the framework of his ‘nonunital system’ theory [426]. By making appropriate choices, and applying Werner’s original theorem to certain 4 × 4 matrices, one may recover Ml (X) and some of the implications in 4.5.2. These arguments may be found in the more recent paper [428].) The paper [53] introduced the conceptual framework that Section 4.5 and 4.6.1–4.6.8 fall within. This paper was written with an emphasis on the triple envelope (since this may be viewed as the ‘noncommutative Shilov boundary’), as opposed to the much larger injective envelope. For those familiar with C ∗ modules this is a natural approach (see also [56] for a recent survey from the

Some ‘extremal theory’

191

perspective of C ∗ -modules of some topics in the second half of the present chapter). After the main conceptual advances in [53] were made, work also began on the tidy alternative approach based directly on the injective envelope presented in [68]. About a year later, the basic list of characterizations of operator space multipliers was completed in [57]. The route which we have followed in these sections is a combination of the approaches of the last three papers mentioned. The equivalence of (i), (iv), and (v) in Theorem 4.5.2 is from [53], the equivalence with (iii) is from [68], whereas the equivalence with (ii) was proved in [57] (although the later short proof presented here comes from [314]). In fact (ii) was inspired by Werner’s theorem discussed above, and is a noncommutative analogue of the pretty norm formula in 3.7.3 (iii). Because of the simplicity of 4.5.2 (ii), it is easy to see that the class of left multipliers is closed with respect to most of the ‘usual constructions’, for example, quotients, tensor products, duality, interpolation, ampliﬁcation (e.g. see [57, 73]). Formula (4.11) is from [53]. As C ∗ -algebras, Mn (Al (X)) ∼ = Al (Cn (X)), since Mn (∆(Ml (X))) = ∆(Mn (Ml (X))) ∼ = ∆(Ml (Cn (X))) = Al (Cn (X)), the ‘∼ =’ here following from 2.1.2 and (4.11). If X is a dual operator space then the reader may ﬁnd in [73] the relation MI (Ml (X)) ∼ = Ml (MI,J (X)) as dual operator algebras, for any cardinals I, J. A similar relation holds with Ml replaced by Al . These are valid for any operator space if I, J are ﬁnite. The generalization in 4.5.4 of Paulsen’s trick is from [440]. The statement in Theorem 4.5.5 is from [53], but the proof here, centered on the relation (4.12), is from [68]. Similarly, for 4.5.7–4.5.13: these appear in [53], but many of the proofs here are from [68]. Also, 4.5.11 is closely related to [163]. Indeed some proofs in [68] were inspired by some of Frank and Paulsen’s techniques in [163], and conversely [68] shows that some of those authors results on multipliers of C ∗ algebras in terms of the injective envelope, ﬁt harmoniously within the injective envelope approach to multipliers on general operator spaces. One may recover, for example, the fact that LM (A) ⊂ I(A) by combining 4.5.11 with (4.12). The deﬁnition of a left M -projection P in 4.5.14 may be restated informally as the statement that X is a ‘column sum’ of P (X) and (I − P )(X). Items 4.5.14 and 4.5.15 are from [57, 440]. See [53, 56, 57, 73], for example, for further theory of one-sided multipliers and adjointable maps of operator spaces, and computation of these multiplier algebras for certain classes of operator spaces. The ﬁrst ‘noncommutative Banach–Stone theorem’, for surjective isometries between C ∗ -algebras, is due to Kadison [212]. A good survey of isometries with more recent proofs may be found in [158]. Arveson in [21, 22] proved various theorems of this type for unital completely isometric maps between certain operator algebras. Once Hamana had established the existence of the C ∗ -envelope in general, it was fairly obvious that Arveson’s techniques showed that unital surjective completely isometric maps between general unital operator algebras are multiplicative (see [40, 144]). Isometries between general nonselfadjoint operator algebras were studied in [196, 278, 14], for example. We do not have the

192

Notes and historical remarks

space to list all the papers on the subject of isometries between particular classes of operator algebras. There are also ‘Banach–Stone theorems’ for algebras with one-sided identity, such as ideals in C ∗ -algebras [55]; and further generalizations may be found in [219]. The case of nonsurjective complete isometries was discussed in [58, 59, 61]. See also [10, 96, 86], for example. Kirchberg has some deep characterizations of certain classes of maps in his papers cited here. Junge, Ruan, and Sherman have recently characterized complete isometries between noncommutative Lp spaces (see [210] and references therein). The Banach–Stone theorem 4.5.13 may also be proved by extending u to a surjective complete isometry on a containing C ∗ -algebra (e.g. the injective or C ∗ - envelope), and then using 4.4.6 or a Banach–Stone theorem for C ∗ -algebras. See, for example, 8.3.13. This is related to the topic of ‘recovering’ the product on a unital operator algebra from its operator space structure, which is discussed fully in Section 6 of [56]. Indeed, Theorem 4.5.13 shows that an approximately unital operator algebra can have only one operator algebra product (compatible with the operator space structure and the cai). If X is a unital operator space then Ml (X) ⊂ X completely isometrically, via the map T → T (1). This follows easily from the fact that, as operator algebras, Ml (X) ∼ = {a ∈ Ce∗ (X) : aX ⊂ X}, which in turn follows from the ﬁrst fact in the second paragraph of 4.5.10. 4.6: The results in the ﬁrst two thirds of this section are due to Blecher [53] (although admittedly some of the proofs in early versions of that paper were too complicated). As we said in the Notes to 4.5, operator space multipliers were introduced in order to establish a ‘better route’ to the BRS and CES theorems; thus it was not surprising that 4.5.2 (ii) yields these theorems (as was observed by Blecher–Eﬀros–Zarikian, and Paulsen, independently). Indeed 4.5.2 (ii) yields a characterization of operator algebras with a one-sided contractive approximate identity (see [54]). Some of these ideas were taken further with the introduction in [218, 219] of quasimultipliers, a variant of the notion of one-sided multiplier. Perhaps the main result in [218] is a bijective correspondence between operator algebra products on an operator space X, and contractive quasimultipliers. Some other applications of the techniques in this section to operator modules may be found in [53], for example. Results 4.6.9–4.6.11 are from [68], and 4.6.12 is from [57] (but see 3.11 in [186]). See [163] for the case of 4.6.10 when X is a C ∗ -algebra. In connection with 4.6.10, many papers studying injectivity in some other categories of operator modules are referenced in the Notes to 3.6; and we also note that the analogue of 4.2.4 holds for operator bimodules, with most parts of the proof unchanged (see [68, 217]). However, we would like to mention here the surprising and useful results of Frank and Paulsen [163]. They study a variant of injectivity where one extends completely bounded module maps, but do not insist that the extension has the same norm. One of their results implies that if A is a C ∗ -algebra, and if u is any completely bounded A-module map on I(A) whose restriction to A is IA , then u is the identity map (cf. 4.2.3). This has some lovely consequences,

Some ‘extremal theory’

193

such as the fact that if there exists a completely bounded A-module projection from B(H) onto a ∗-subalgebra A, then A is injective in the usual sense. Finally, we prove two characterizations of uniform algebras. First, we give the original proof from [40] of Corollary 3.7.10. If A is a unital operator algebra and also a minimal operator space, then it is completely isometrically isomorphic by 4.2.8 to a subalgebra of the C ∗ -algebra I(Min(A)). However, we saw in 4.2.11 that I(Min(A)) is linearly completely isometric, and hence ∗-isomorphic by 4.5.13, for example, to a C(K)-space. This gives the result. One may also characterize uniform algebras as the unital operator algebras for which the multiplication viewed as a map on A ⊗ A is completely contractive with respect to the minimal tensor product (see 1.5.1). Indeed together with Zeibig, we noted that since the minimal tensor product is ‘injective’ (see 1.5.1) we may extend the product to a completely contractive map M : I(A) ⊗min I(A) → I(A). However I(A) is a C ∗ -algebra as we saw in 4.2.8. By the rigidity property of the injective envelope, it follows that M (1, x) = M (x, 1) = x for all x ∈ I(A). Since the Haagerup tensor product dominates the minimal tensor product (see 1.5.13), we have that M satisﬁes the conditions of the BRS theorem (except associativity). As is pointed out in 4.6.3, the BRS theorem does not require the associativity hypothesis, so that M is an operator algebra product on I(A). By the Banach–Stone theorem for operator algebras, M is the usual product on I(A). Next we point out that since ⊗min is ‘commutative’ (see 1.5.1), applying the last part of the above argument to the reversed product on I(A), we see that the reversed product is the usual product. Thus I(A) is a commutative C ∗ -algebra. 4.7: Theorem 4.7.1 was proved in the Al (X) case, and 4.7.2 in the unital case, by Blecher–Eﬀros–Zarikian and Blecher [57, 54]. The general cases are new results from a joint paper in preparation with Magajna. We thank Magajna for permission to include these here. All the results in this section are from these papers, to which we refer for further applications, and other facts about w ∗ topologies on operator spaces and algebras, such as the facts which follow: Note that Al (X) need not even be an AW ∗ -algebra if X is an operator space which is a dual Banach space. However in the latter case, it is true that any T ∈ Al (X) is w∗ -continuous, and if A is a unital operator algebra with a Banach space predual then ∆(A) is a W ∗ -algebra. It is worth pointing out that although multipliers work very well with respect to operator space duality, the closely related triple and injective envelopes do not. Indeed if A is a dual operator algebra, then Ce∗ (A) and I(A) need not be W ∗ -algebras. For example, take A = U(X) (see 2.2.10), where X is the span of {1, x, x2 } in C([0, 1]), or the span of the two generators of the free group F2 inside the reduced group C ∗ -algebra. In the former case, the C ∗ -envelope is M2 (C([0, 1])), and so the injective envelope is M2 (I(C([0, 1]))). However, it is well known folklore that I(C([0, 1])) is the Dixmier algebra, a commutative AW ∗ -algebra which is not a W ∗ -algebra (see 3.9.7 in [320]). Similar arguments pertain in the free group case (see [185, Corollary 3.8]). 4.8: See [195] and [31] for surveys of the classical theory of M -ideals, and references to the extensive literature. Eﬀros and Ruan introduced and studied

194

Notes and historical remarks

complete M -ideals of operator spaces in [148]. Assertions (1) and (2) of 4.8.5 in the case of C ∗ -algebras are from [4,3,393], the general case of (1) and (2) is from [144], which relied on results from [393]. The argument given here for these items is adapted from [437], which is a simpliﬁcation of the original argument from [393]. Arias and Rosenthal showed in [17] that complete M -ideals in operator algebras (and more general operator spaces) have in fact a quite strong kind of approximate identity. M -ideals in nonselfadjoint operator algebras have also been considered, for example, in [10, 115, 144, 343]. One nice application of M ideal theory is the following observation: if I is a closed two-sided ideal in an operator algebra, and if both A and I have contractive approximate identities, then for each a ∈ A the distance from a to I is achieved. This follows from 4.8.5 (1), and the ‘proximinality’ property that every M -ideal possesses [195]. The other results in this section are from [57, 73, 440]. See also [71, 442]. One-sided M -ideals were intended as a generalization of classical M -ideals to ‘noncommutative functional analysis’, where one might expect to have left and right ‘ideals’. What we call ‘one-sided M -ideals’ here were called ‘complete onesided M -ideals’ in [57]. One-sided M -ideals and summands are stable under quotients, tensors, interpolation, duality, and so on. The reason for this is usually that multipliers work well under these constructions, as observed earlier. Thus, for example, using interpolation one can show that in contrast to the classical case, Lp spaces and their noncommutative variants can possess nontrivial onesided M -ideals. Or as another example, one can show that one-sided M -ideals in a dual operator space X satisfy a Kaplansky density theorem. It is not hard to see that a left M -ideal in a left operator module is necessarily a submodule. See [73] for these, and very many other such results. Some other applications of ‘one-sided M -structure’ may be found in [54, 55], for example. For references on noncommutative convexity and ‘matricial extreme points’, see for example [2,152,153,156,157,223,266,422,429], and references therein. Noncommutative Choquet theory will play an exciting role in the future. A sample of other interesting related topics which we had no space to include in this chapter: the question of the uniqueness of operator algebra structures on a given Banach algebra, and the companion question of when contractive maps are automatically completely contractive (e.g. see [21, 109, 282, 337], and references therein); questions involving algebraic or linear isomorphism, or peturbations, of nonselfadjoint algebras (e.g. see [342] or Chapter 18 of [108]); ‘approximately ﬁnite operator algebras’ and related questions about K-theory (e.g. see [341, 353]); Popescu’s noncommutative disc algebra and H ∞ (not connected to Arveson’s subdiagonal algebras discussed earlier) and dilation theory (see [347, 348], and references therein); the Muhly–Solel work on tensor algebras, correspondences and dilations (see [284–287], and references therein); and the operator space aspects of ‘interpolation’ as studied by Agler, Cole, McCarthy, McCullouch, Paulsen, Wermer, and others (see [276] and references therein). Some results of the ﬂavor in this chapter, but for algebras with a one-sided cai, or no kind of identity, may be found in [55, 60, 218, 219].

5 Completely isomorphic theory of operator algebras

5.1 HOMOMORPHISMS OF OPERATOR ALGEBRAS In the ﬁrst four chapters, we studied operator algebras up to completely isometric isomorphism. There, the most important mappings between operator algebras (the morphisms in the language of categories) were the completely contractive homomorphisms. In this chapter we turn to the study of operator algebras ‘up to isomorphism’ or ‘up to complete isomorphism’. Here we must deal with homomorphisms which are merely bounded or completely bounded. Theorem 5.1.2 below, which is the main result of Section 5.1, clariﬁes the relationship between completely bounded and completely contractive representations. 5.1.1 (Similarities) Let A be an operator algebra, let H be a Hilbert space and let π : A → B(H) be a bounded homomorphism. If S : H → H is any invertible bounded operator, then the mapping πS : A → B(H) deﬁned by letting πS (a) = S −1 π(a)S for any a ∈ A, is a bounded homomorphism. Moreover, we clearly have πS ≤ S −1 Sπ. If further π is completely bounded, then πS is completely bounded as well, with πS cb ≤ S −1 Sπcb. We say that two representations π, ρ : A → B(H) are similar to each other if there exists S as above such that ρ = πS (in which case π = ρS −1 ). Theorem 5.1.2 (Paulsen) Let A be an operator algebra and let π : A → B(H) be a completely bounded representation. Then there exists a bounded invertible operator S : H → H such that πS : A → B(H) is completely contractive. If further A is unital and π is unital, then this may be achieved with S satisfying S −1 S = πcb . Proof We ﬁrst assume that A is unital and π : A → B(H) is unital. By the representation theorem for completely bounded maps (see 1.2.8) we may ﬁnd a Hilbert space K, a unital completely contractive representation θ : A → B(K) and linear maps V : H → K and W : K → H such that V W ≤ πcb and π(a) = W θ(a)V for any a ∈ A. We introduce two (closed) subspaces F ⊂ E ⊂ K deﬁned by E = [θ(A)V (H)] and F = E ∩ Ker(W ). We claim that these two spaces are θ(A)-invariant. Indeed this is clear for E. To prove it for F , ﬁx x in F , and consider a sequence (xi )i≥1 in the linear span of {θ(a)V ζ : a ∈ A, ζ ∈ H}

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Homomorphisms of operator algebras

converging to x ∈ F . For each may ﬁnd ﬁnite families (aik )k in A and i ≥i 1, we i i (ζk )k in H such that xi = k θ(ak )V ζk . By assumption W x = 0, and so lim π(aik )ζki = lim W θ(aik )V ζki = 0. i

i

k

k

If a ∈ A then W θ(a)x is the limit of W θ(a)xi . For each i ≥ 1, W θ(a)xi = W θ(a)θ(aik )V ζki = W θ(aaik )V ζki k

=

π(aaik )ζki

= π(a)

k

k

π(aik )ζki

k

because π is a homomorphism. In the limit we have that W θ(a)x = 0, that is, θ(A)x belongs to F . Since E and F are θ(A)-invariant, by 3.2.2 the space L = E F is semiinvariant in the sense of 3.2.1. Let p be the projection from E onto L. By 3.2.2 θ induces a completely contractive homomorphism θ˜: A → B(L) satisfying ˜ θ(a)p = pθ(a),

a ∈ A.

(5.1)

Since θ is unital, the range of V is contained in E. Let T = pV : H → L. Since W V = W θ(1)V = π(1) = IH and W (IE − p) = 0, we have for ζ ∈ H that ζ = W V ζ = W pV ζ + (IE − p)V ζ = W T ζ. Thus W|L ◦T = IH . Since W|L is one-to-one, we deduce that T is an isomorphism, with T −1 = W|L . Clearly T ≤ V and T −1 ≤ W , and hence T T −1 is less than or equal to πcb . We claim that ˜ = π(a), T −1 θ(a)T

a ∈ A.

(5.2)

Indeed, for a ∈ A and ζ ∈ H, we have ˜ ˜ ζ = T −1 θ(a)pV ζ = T −1 pθ(a)V ζ T −1 θ(a)T by (5.1). Hence using the fact that T −1 p is equal to W|E , we have ˜ ζ = W θ(a)V ζ = π(a)ζ. T −1 θ(a)T Since H and L are isomorphic (via T ), they are actually isometrically isomorphic, and hence there exists a unitary U : H → L. We let S = U −1 T : H → H; this is an isomorphism and SS −1 = T T −1 ≤ πcb . By (5.2), we have ˜ ˜ S for any a ∈ A. Thus πS −1 : a → U ∗ θ(a)U is a complete π(a) = S −1 U ∗ θ(a)U contraction. This concludes the unital case. Assume now that A is unital, but π(1) = IH . Then q = π(1) is an idempotent. Since every idempotent operator is similar to an orthogonal projection (this is

Completely isomorphic theory of operator algebras

197

an easy exercise), we may assume without loss of generality that q is selfadjoint. Let K = Ran(q). We have π = qπ(·)q, and in particular, K is π(A)-invariant. Moreover the induced ‘restriction’ of π to K is unital, and so by the ﬁrst part of the proof, there is an isomorphism S : K → K such that the mapping πS is completely contractive from A into B(K). Then if S = S ⊕ I : H → H, then πS : A → B(H) is a complete contraction. Finally if A is nonunital, we can reduce to the unital case by means of the unitization A1 (see 2.1.11). Indeed if π : A → B(H) is any completely bounded homomorphism, and if π ˜ : A1 → B(H) ˜ is a completely is the extension of π obtained by letting π ˜ (1) = IH , then π bounded homomorphism as well. Hence π ˜ is similar to a complete contraction and this clearly implies that π itself is similar to a complete contraction. 2 5.1.3 (Similarity problems) Paulsen’s theorem 5.1.2 is the key to many important similarity problems. We do not intend to discuss such problems in the book, and refer the reader to [335], which is devoted to this topic, to [314], and to the Notes to Section 5.1. We merely mention that most similarity problems reduce to the following question: for a ﬁxed operator algebra A, is every bounded representation π : A → B(H) similar to a contractive one? It clearly follows from Theorem 5.1.2 that this property holds provided that every bounded homomorphism on A is automatically completely bounded. We note for later use that nuclear C ∗ -algebras satisfy the above property. Indeed, if A is a nuclear C ∗ -algebra and π : A → B(H) is a bounded homomorphism, then π is completely bounded and πcb ≤ π2 . This result was proved by Bunce [78] and Christensen [91] independently (see also [335, Chapter 7] for a proof). This applies in particular when A is a commutative C ∗ -algebra. 5.1.4 (Isomorphisms of operator algebras) Let A be a Banach space which is also an algebra. We say that A is isomorphic to an operator algebra if there is an operator algebra B ⊂ B(H), and a bounded invertible homomorphism ρ from A onto B. In that case, ρ−1 : B → A also is a bounded homomorphism, and A is a C-Banach algebra (see A.4.1) with C ≤ ρ2 ρ−1 . If ρ is an isometry we say that A is isometrically isomorphic to an operator algebra. These classes have various stability properties. Assume that A is isomorphic (resp. isometrically isomorphic) to an operator algebra. Clearly the same holds true for any subalgebra of A. On the other hand, if I is any index set, then the algebra ∞ I (A) with coordinatewise product is isomorphic (resp. isometrically isomorphic) to an operator algebra. Lastly, if J ⊂ A is any closed two-sided ideal, then the quotient algebra A/J is isomorphic (resp. isometrically isomorphic) to an operator algebra. Indeed this follows from the isometric version of Proposition 2.3.4. 5.1.5 (A counterexample) It is far from true that every Banach algebra is isomorphic to an operator algebra. One way to see this is to recall that operator algebras are Arens regular (see 2.5.2 and 2.5.4), and to observe that Arens regularity is preserved under isomorphism. Thus a Banach algebra which is not Arens regular cannot be isomorphic to an operator algebra. We refer the reader

198

Homomorphisms of operator algebras

to the Notes to Section 2.5 for further discussion of Arens regularity. In particular A = 1Z , equipped with convolution product, is not Arens regular, and hence it is not isomorphic to an operator algebra. We will next provide a direct proof of this last fact without using Arens regularity. Assume to the contrary that there exists, for some Hilbert space H, a bounded homomorphism ρ : A → B(H) whose range B = ρ(A) is closed. Since A is unital, replacing H by [BH] if necessary, we may assume that ρ(1) = IH . Let (en )n denote the canonical basis of A = 1Z , and let T = ρ(e1 ) ∈ B(H). The element e1 is invertible and en1 = en for any n ∈ Z. Since ρ is unital, this implies that T is invertible and that T n = ρ(en ) for any n ∈ Z. Thus we have T n ≤ ρ for any n ∈ Z. By Nagy’s similarity theorem [403], this implies that T is similar to a unitary operator in B(H), that is, there is an invertible bounded operator S : H → H such that S −1 T S is a unitary. Applying von Neumann’s inequality for unitaries (the simplest case), we deduce that T is polynomially bounded. That is, there is a constant K ≥ 1 such that f (T )≤ K sup{|f (z)| : z ∈ C, |z| ≤ 1}, for any polynomial f . Since f (e1 ) = ρ−1 f (T ) and en1 = en for any n, we derive an estimate

|αk | ≤ C sup αk z k : z ∈ T , k≥0

k≥0

for any ﬁnite family (αk )k≥0 of complex numbers. This estimate implies that the Fourier series of any f ∈ C(T) is absolutely convergent, a contradiction. 5.1.6 (Complete isomorphisms of operator algebras) Let A be an operator space which is also an algebra. We say that A is completely isomorphic to an operator algebra if there is an operator algebra B ⊂ B(H), and a completely bounded invertible homomorphism ρ : A → B such that ρ−1 is completely bounded. The case when ρ is a complete isometry corresponds to A being an abstract operator algebra in the sense of Section 2.1. Stability properties mentioned in 5.1.4 hold as well in the completely bounded setting. Namely if A is completely isomorphic to an operator algebra then the same holds for subalgebras of A, for any direct sum ∞ I (A) (I being an index set), and for quotients of A. The latter result uses the full statement of Proposition 2.3.4. An operator space structure (in the sense of 1.2.2) on an algebra A for which A is completely isomorphic to an operator algebra, will be called an operator algebra structure on A. Recall that if A already has a norm, we demand that the operator space structure is compatible with this one. Proposition 5.1.7 Let A be a commutative C ∗ -algebra. Then an operator space structure on A is an operator algebra structure (for the usual product) if and only if it is completely isomorphic to the minimal one. Proof Min(A) is the canonical structure of A (see 1.2.3) and our statement says that up to complete isomorphism, this is actually the only one. Assume that A is equipped with a operator space structure for which it is completely isomorphic to an operator algebra B. We let ρ : A → B ⊂ B(H) be the corresponding completely bounded invertible homomorphism. By the last paragraph

Completely isomorphic theory of operator algebras

199

of 5.1.3, ρ is completely bounded on Min(A). Hence ρ : Min(A) → B is a complete isomorphism by (1.10), which concludes the proof. 2 5.1.8 (A necessary condition) Let m : A ⊗ A → A be the linear mapping induced by the multiplication on an algebra A. If A is an operator algebra, then m extends to a completely contractive map on A ⊗h A (see 2.3.1). Hence if A is completely isomorphic to an operator algebra, the mapping m extends to a completely bounded map m : A ⊗h A → A. The key result for the study of operator algebras up to complete isomorphism will be Theorem 5.2.1, which asserts that the converse is true. In the next statement, we give a simple application of the above necessary condition. Proposition 5.1.9 Let X be an operator space and let CB(X) have its canonical matrix norms (1.6). Then CB(X) is completely isomorphic to an operator algebra if and only if X is completely isomorphic to a column Hilbert space. Proof The ‘if’ part follows from (1.14). Conversely, if CB(X) is completely isomorphic to an operator algebra, then we have a completely bounded multiplication m : CB(X) ⊗h CB(X) → CB(X) (see 5.1.8). By (1.35) we may regard X ⊗min X ∗ , and hence X ⊗ X ∗ , as a subspace of CB(X). Then we have x, x ∈ X, ϕ, ϕ ∈ X ∗ . m x ⊗ ϕ, x ⊗ ϕ = ϕ, x x ⊗ ϕ , Fix x ∈ X and ϕ ∈ X ∗ with x = ϕ = 1. For any x1 , . . . , xn in X and ϕ1 , . . . , ϕn ∈ X ∗ , we have n n n = m x ⊗ ϕk , xk ⊗ ϕ , ϕk , xk = ϕk , xk x ⊗ ϕ k=1

k=1

CB(X)

k=1

CB(X)

and by (1.40) the latter quantity is dominated by m [x ⊗ ϕ1 · · · x ⊗ ϕn ]Rn (CB(X)) [x1 ⊗ ϕ · · · xn ⊗ ϕ ]t Cn (CB(X)) . Since the embedding ϕ → x ⊗ϕ is a complete isometry from X ∗ to the ‘subspace’ X ⊗min X ∗ of CB(X), the norm of [x ⊗ ϕ1 · · · x ⊗ ϕn ] in Rn (CB(X)) is equal to the norm of [ϕ1 . . . ϕn ] in Rn (X ∗ ). Likewise the norm of [x1 ⊗ ϕ · · · xn ⊗ ϕ ]t in Cn (CB(X)) is equal to the norm of [x1 · · · xn ]t in Cn (X). Thus we have proved n ϕk , xk ≤ m [ϕ1 · · · ϕn ]Rn (X ∗ ) [x1 · · · xn ]t Cn (X) . k=1

Thus the duality pairing extends to a (completely) bounded functional X ∗ ⊗h X −→ C. By the CSPS theorem for bilinear forms (see 1.5.7 and 1.5.8), there exist a Hilbert space H and completely bounded maps v : X → H c and w : H c → X ∗∗ such that wv : X → X ∗∗ is the canonical embedding iX . Since iX is a complete isometry, x = wn vn (x) ≤ wcb vn (x) for any n ≥ 1 and any x ∈ Mn (X). Thus v induces a complete isomorphism between X and the range of v. 2

200

Completely bounded characterizations

5.1.10 (Interpolation) Let (A0 , A1 ) be a compatible couple of Banach algebras and for any θ ∈ (0, 1), let Aθ = [A0 , A1 ]θ denote the Banach algebra obtained from A0 and A1 by the complex interpolation method (2.3.6). Using functorial properties of interpolation, one can easily deduce from Proposition 2.3.7 that: (1) If A0 , A1 are isomorphic to operator algebras, then Aθ is isomorphic to an operator algebra. (2) If A0 and A1 are also operator spaces and if A0 , A1 are completely isomorphic to operator algebras, then Aθ equipped with its canonical operator space structure (see 1.2.30) is completely isomorphic to an operator algebra. 5.2 COMPLETELY BOUNDED CHARACTERIZATIONS The following theorem, which is the main result of this section, is a completely isomorphic counterpart of the BRS theorem (see 2.3.2). Theorem 5.2.1 Let A be an operator space which is also an algebra, and let m : A ⊗ A → A denote the multiplication on A. Then A is completely isomorphic to an operator algebra if and only if m extends to a completely bounded map m : A ⊗h A → A. The proof of this will be completed in 5.2.9 below. We divide it into several intermediate steps which provide interesting related results and generalizations. 5.2.2 (Polynomials in noncommuting variables) For any integer n ≥ 1, we let P n be the algebra of all polynomials in n2 noncommuting variables without constant terms. These variables will be denoted by Xij (1 ≤ i, j ≤ n). Let A be an algebra and assume that A is equipped with two operator space structures, which we denote by A1 and A2 . For any F in P n and any [aij ] in the algebra Mn (A), we let F [aij ] ∈ A be deﬁned by substituting the aij ’s in for the variables. Next, if N ≥ 1 is an integer, if F = [Fkl ]1≤k,l≤N is any element of MN (P n ) (each entry Fkl being in P n ), and if a = [aij ]1≤i,j≤n is in Mn (A), we deﬁne F (a) = Fkl [aij ] ∈ MN (A). Then given any positive number δ > 0 and F ∈ MN (P n ) as above, we set F A1 ,A2 ,δ = sup F (a)MN (A2 ) : a ∈ Mn (A), aMn (A1 ) ≤ δ . This deﬁnition is not changed if the supremum is taken over all a ∈ Mn (A) with aMn (A1 ) < δ. If A1 = A2 then F A1 ,A2 ,δ will be denoted by F A,δ , and we will use the symbol A for A1 and A2 . Moreover we let F A = F A,1 . Lemma 5.2.3 Let B be an operator algebra, let I be an index set, and let C be a subalgebra of ∞ I (B). Suppose that J ⊂ C is a closed two-sided ideal, and consider the operator algebra A = C/J. Then for any n, N ≥ 1 and any F ∈ M N (P n ), we have F A ≤ F B .

Completely isomorphic theory of operator algebras

201

∞ Proof Let c ∈ Ball(Mn (C)). Since Mn ∞ I (B) = I Mn (B) isometrically (see 1.2.17), we may write c = (bλ )λ∈I with bλ ∈ Mn (B) and bλ ≤ 1. Since the multiplication on ∞ I (B) is obtained by taking the multiplication of B coordinatewise, we see that F (c) = F (bλ ) λ for any F ∈ P n . Thus if F ∈ MN (P n ) we ∞ also have F (c) = F (bλ ) λ and since MN ∞ I (B) = I MN (B) isometrically, we obtain that F (c) = supλ {F (bλ )}. Hence F (c) ≤ F B . Taking the supremum over c ∈ Ball(Mn (C)), we deduce that F C ≤ F B . Fix a ∈ Mn (C/I), with a < 1. Let q : C → C/I = A be the canonical quotient map. By deﬁnition of the quotient (see 1.2.14), we may ﬁnd c ∈ M n (C) such that c < 1 and qn (c) = a. Since q is a homomorphism, qn (F (c)) = F (a) for any F ∈ MN (P n ). Thus F (a) ≤ F (c), from which we deduce that 2 F A ≤ F C . The result therefore follows by the ﬁrst part of this proof. 5.2.4 (Homogeneous polynomials) r For any integer r ≥ 1, we consider the set Λr = {1, . . . , n} × {1, . . . , n} . For any α = ((i1 , j1 ), . . . , (ir , jr )) ∈ Λr , we set αp = (ip , jp ). We say that a polynomial F ∈ P n is homogeneous of degree r if it lies in the linear span of {Xα1 · · · Xαr : α ∈ Λr }. We say that a matrix valued polynomial F = [Fkl ] ∈ MN (P n ) is homogeneous of degree r if any F ∈ MN (P n ) admits each entry Fkl is homogeneous of degree r. Clearly a unique decomposition as a ﬁnite sum F = r≥1 Fr , where Fr ∈ MN (P n ) is homogeneous of degree r. If B is an operator algebra then for any b ∈ M n (B) and any complex number z we have Fr (zb) = z r F (b). Thus Fr (b) =

1 2π

!

2π

e−irt F (eit b) dt .

0

This implies that Fr B ≤ F B for any r ≥ 1. Proposition 5.2.5 Let B be an operator algebra and let A be an algebra equipped with two operator space structures denoted by A1 and A2 . (1) Let M, δ > 0. The following assertions are equivalent: (i) For any integers n, N ≥ 1 and any F ∈ MN (P n ), we have F A1 ,A2 ,δ ≤ M F B .

(5.3)

(ii) There exist an index set I, a subalgebra C ⊂ ∞ I (B), a closed two-sided ideal J ⊂ C, and an algebra isomorphism ρ : A → C/J such that ρ : A1 −→ C/Jcb ≤ δ −1

and

ρ−1 : C/J −→ A2 cb ≤ M.

(2) Let M, δ > 0, and assume that for any n, N ≥ 1, (5.3) is satisﬁed for any matrix valued homogeneous polynomial F ∈ MN (P n ). Then there exists an isomorphism ρ : A → C/J as in (ii) with ρ : A1 −→ C/Jcb ≤ 2δ −1

and

ρ−1 : C/J −→ A2 cb ≤ M.

202

Completely bounded characterizations

Proof We will ﬁrst deduce (2) from (1). Assuming (1), let F = r≥1 Fr be an arbitrary element of MN (P n ), with Fr homogeneous of degree r (see the discussion in 5.2.4). Since Fr (a) = 2−r Fr (2a) for any a ∈ Mn (A) and any r ≥ 1, we have Fr A1 ,A2 ,δ/2 ≤ 2−r Fr A1 ,A2 ,δ for any r ≥ 1. Thus Fr A1 ,A2 ,δ/2 ≤ 2−r Fr A1 ,A2 ,δ . F A1 ,A2 ,δ/2 ≤ r≥1

r≥1

Thus if (5.3) holds for any homogeneous polynomial, we conclude by 5.2.4 that 2−r M Fr B ≤ M F B . F A1 ,A2 ,δ/2 ≤ r≥1

This shows (5.3) with δ replaced by δ/2. Hence (2) follows from (1). Turning to (1), we now prove the easy fact that (ii) implies (i). Assume (ii), and consider F ∈ MN (P n ) and a ∈ Mn (A), with aMn (A1 ) ≤ δ. Then ensures that F ρn (a)Mn (C/J) ≤ 1, and so Lemma 5.2.3 (ρn (a)) ≤ F B . Since ρ is a homomorphism, we have F ρn (a) = ρN F (a) . Hence F (a)MN (A2 ) ≤ ρ−1 : C/J −→ A2 cb F (ρn (a)) ≤ M F B . Taking the supremum over a, we obtain (5.3). We now assume (i) and shall prove (ii), which is the main implication. Let Σ1 = {a ∈ K(A1 ) : a ≤ δ}, let Σ2 be the closed unit ball of K(B), and let I = Σ2 Σ1 be the set of all functions from Σ1 into Σ2 . If a (resp. b) is an element of Σ1 (resp. Σ2 ), we denote its entries by aij (resp. bij ), i, j ≥ 1. It will be convenient to regard ∞ I (B) as the space of bounded functions from I into B. For any a ∈ Σ1 and any integers i, j ≥ 1, we deﬁne such a function fija : I → B by letting fija (λ) = λ(a)ij . We let V be the subalgebra of ∞ I (B) generated by all the fija , with a ∈ Σ1 and i, j ≥ 1. Let C = V . Our goal is to show that we can deﬁne a completely bounded homomorphism q : C → A2 such that q(fija ) = aij ,

a ∈ Σ1 , i, j ≥ 1.

(5.4)

To this end, let N ≥ 1 be an integer and let v = [vkl ] ∈ MN (V ). According to the deﬁnition of V , there exist pairwise distinct a(1), . . . , a(r) in Σ1 , and an integer m ≥ 1, such that each vkl lies in the algebra generated by the ﬁnite set a(p) fij : 1 ≤ p ≤ r, 1 ≤ i, j ≤ m . pq Let n = mr. Here it is convenient to denote the generators of P n as Xij , for 1 ≤ i, j ≤ n and 1 ≤ p, q ≤ m. Thus we regard Mn (P n ) as consisting of r2 ‘block matrices’ each of size m × m. With this notation we see that there exist polynomials Fkl ∈ P n only depending on the variables   11 [Xij ]   11 rr .. [Xij ] ⊕ · · · ⊕ [Xij ] , . rr [Xij ]

Completely isomorphic theory of operator algebras a(1) a(r) such that vkl = Fkl [fij ] ⊕ · · · ⊕ [fij ] . For 1 ≤ k, l ≤ N and λ ∈ I, vkl (λ) = Fkl λ(a(1))ij ⊕ · · · ⊕ λ(a(r))ij .

203

(5.5)

The reader may verify this algebraic relation by substituting in an uncomplicated sample polynomial v. We let F = [Fkl ] ∈ MN (P n ), and we observe that

F B = sup F [β(1)] ⊕ · · · ⊕ [β(r)] MN (B) , where the supremum runs over all β(1), . . . , β(r) ∈ Mn (B) such that [β(1)] ⊕ · · · ⊕ [β(r)] = sup β(p)Mm (B) ≤ 1. M (B) n

p

We claim that v = F B . ∞ Indeed the isometric identiﬁcation MN ∞ I (B) = I MN (B) yields

v = sup v(λ)MN (B) : λ ∈ I

= sup F [λ(a(1))ij ] ⊕ · · · ⊕ [λ(a(r))ij ] MN (B) : λ ∈ I

(5.6)

by (5.5). However the last quantity also equals

sup F [b(1)ij ] ⊕ · · · ⊕ [b(r)ij ] MN (B) : b(1), · · · , b(r) ∈ Ball(K(B)) . The claim then follows by the preceding calculation. For any 1 ≤ p ≤ r, let α(p) ∈ Mm (A1 ) be the truncation of a(p) deﬁned by α(p)ij = a(p)ij for 1 ≤ i, j ≤ m. Then α(1) ⊕ · · · ⊕ α(r) ≤ sup a(p) ≤ δ, Mn (A1 ) p

hence it follows from our assumption (5.3) and from (5.6) that ≤ M v. F [a(1)ij ] ⊕ · · · ⊕ [a(r)ij ] MN (A2 )

This shows that (5.4) deﬁnes a bounded homomorphism q from V into A2 , and that passing to the closure, we have q : C −→ A2 ≤ M. cb Let n ≥ 1 be an integer and let a ∈ Mn (A1 ), with a = δ. We regard a as an element of Σ1 in the obvious way, and we let v = [fija ] ∈ Mn (V ). Then v ≤ 1 and qn (v) = a. This shows that q is onto and that if we let J = Ker(q), the corresponding inverse mapping ρ from A into C/J is completely bounded on A1 , 2 with ρ : A1 → C/Jcb ≤ δ −1 . This shows (ii).

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Completely bounded characterizations

5.2.6 Let X be an operator space, let H be a Hilbert space, and let v be a completely bounded map from X into B(H). For any r ≥ 1, we let v • · · · • v : X ⊗ · · · ⊗ X −→ B(H) be the linear mapping on the r-fold tensor product X ⊗· · ·⊗X taking x1 ⊗· · ·⊗xr to v(x1 ) · · · v(xr ) for any x1 , . . . , xr ∈ X. By 1.5.8, this mapping extends to a completely bounded map on X ⊗h · · · ⊗h X, with v • · · · • vcb ≤ vrcb. Lemma 5.2.7 Let X be an operator space, let r, N ≥ 1 be two integers, and let z ∈ MN (X ⊗ · · · ⊗ X), the tensor product having r factors. Then

zMN (X⊗h ···⊗h X) = sup (IMN ⊗ (v • · · · • v))(z)M (B(H)) , N

where the supremum runs over all separable Hilbert spaces H and all completely contractive maps v : X → B(H). Proof That the left-hand side is greater than the right was observed in 5.2.6. Conversely, let z ∈ MN (X ⊗ · · · ⊗ X), and let j : X ⊗h · · · ⊗h X → B(H0 ) be a completely isometric embedding of the r-fold Haagerup tensor product of X, for some Hilbert space H0 . Let Hr = H0 . By the CSPS theorem (see 1.5.7 and 1.5.8), there exist Hilbert spaces H1 , . . . , Hr−1 and completely contractive maps vk : X → B(Hk , Hk−1 ), 1 ≤ k ≤ r, such that j(x1 ⊗ · · · ⊗ xr ) = v1 (x1 ) · · · vr (xr ) for any x1 , . . . , xr ∈ X. Let H = Hr ⊕ Hr−1 ⊕ · · · ⊕ H0 be the Hilbertian direct sum of the Hk ’s, and let v : X → B(H) be deﬁned by letting v(x) ζr , . . . , ζ0 = 0, vr (x)(ζr ), vr−1 (x)(ζr−1 ), . . . , v1 (x)(ζ1 ) for any x ∈ X and ζk ∈ Hk . It is plain that v is completely contractive. Moreover for any x1 , . . . , xr ∈ X and any ζ ∈ Hr = H0 , we have v(x1 ) · · · v(xr ) ζ, 0, · · · , 0 = 0, . . . , 0, v1 (x1 ) · · · vr (xr )(ζ) = 0, . . . , 0, j(x1 ⊗ · · · ⊗ xr )(ζ) Since j is a complete isometry, we deduce that v • · · · • v is a complete isometry on the r-fold Haagerup tensor product of X, and hence zh = (IMN ⊗ (v • · · · • v))(z)MN (B(H)) . Let (pt )t ⊂ B(H) be a bounded net of ﬁnite rank projections converging to the identity in the strong operator topology. Then for any T1 , . . . , Tr ∈ B(H), the net of operators pt T1 pt · · · pt Tr pt converges to T1 · · · Tr in the WOT. Thus letting vt = pt v(·)pt for any t, we see that

zh = sup (IMN ⊗ (vt • · · · • vt ))(z)MN (B(H)) . t

Since each vt is completely contractive and acts on a ﬁnite-dimensional (hence separable) Hilbert space, this yields the result. 2

Completely isomorphic theory of operator algebras

205

Theorem 5.2.8 Let A be an algebra, with r-fold multiplication mapping denoted by mr : A ⊗ · · · ⊗ A → A for any r ≥ 1. As usual, we let m = m2 . We assume that A is equipped with two operator space structures denoted by A1 and A2 . (1) Assume that there exist two constants C ≥ 1 and K > 0 such that mr : A1 ⊗h · · · ⊗h A1 −→ A2 ≤ CK r−1

(5.7)

cb

for any r ≥ 1. Then there exist an operator algebra D and an algebra isomorphism ρ : A → D such that ρ : A1 −→ Dcb ≤ 2K

and

ρ−1 : D −→ A2 cb ≤ CK −1 .

(2) If the identity mapping IA is completely bounded from A1 into A2 , and if m extends to a completely bounded map m : A1 ⊗h A2 → A2 , then there exists an algebra isomorphism ρ : A → D from A onto some operator algebra D, such that ρ : A1 → D and ρ−1 : D → A2 are completely bounded. Proof (1) Assume (5.7). By 5.2.5 (2), it suﬃces to show that (5.3) holds for homogeneous polynomials with B = B(2 ), δ = K −1 , and M = CK −1 . Let n, N, r ≥ 1 and let F ∈ MN (P n ) be a homogeneous polynomial of degree r. Using notation from 5.2.4 and the identiﬁcation MN (P n ) = MN ⊗ P n , we write λα ⊗ Xα1 · · · Xαr , with λα ∈ MN . F = α∈Λr

Let a = [aij ] ∈ Mn (A1 ) with a ≤ δ = K −1 , and let a = Ka. Then F (a) = λα ⊗ aα1 · · · aαr α∈Λr

= IMN ⊗ mr λα ⊗ aα1 ⊗ · · · ⊗ aαr . α∈Λr

Hence by (5.7), we have F (a)MN (A2 ) ≤ CK r−1 λα ⊗ aα1 ⊗ · · · ⊗ aαr α∈Λr

MN (A1 ⊗h ···⊗h A1 )

≤ CK −1 λα ⊗ a α1 ⊗ · · · ⊗ a αr α∈Λr

MN (A1 ⊗h ···⊗h A1 )

.

It follows from Lemma 5.2.7 that the norm on the right is equal to

sup F [v(aij ] MN (B(2 )) v : A1 → B(2 ), vcb ≤ 1 . This quantity is less than or equal to F B(2 ) , since a Mn (A1 ) ≤ 1. Thus F (a)MN (A2 ) ≤ CK −1 F B(2 ) . This completes the proof of (1).

206

Completely bounded characterizations

(2) If C = IA : A1 → A2 cb and K = m : A1 ⊗h A2 → A2 cb are both ﬁnite, then by induction and the ‘functoriality’ of the Haagerup tensor product, we see that mr : A1 ⊗h · · · ⊗h A1 ⊗h A2 → A2 cb ≤ K r−1 for any r ≥ 2. Hence (5.7) holds for any r ≥ 1. Thus (2) follows from (1). 2 5.2.9 (Proof of Theorem 5.2.1) The ‘only if’ part was observed in 5.1.8. The ‘if’ part is a special case of 5.2.8 (2), obtained by taking A = A1 = A2 . In fact, it follows from Theorem 5.2.8 (1) that if m : A ⊗h A → A is completely bounded and m = 0, then there exists an operator algebra B and a completely bounded homomorphism ρ : A → B such that ρcb ≤ 2mcb and ρ−1 cb ≤ m−1 cb . 5.2.10 (A completely isometric characterization) Let A be an operator space which is also an algebra. Applying 5.2.5 (1) with A = A1 = A2 we ﬁnd that if B is an operator algebra, then A is completely isomorphic to a quotient of a subalgebra of some direct sum ∞ I (B) if and only if there exist constants M, δ > 0 such that F A,δ ≤ M F B for any integers n, N ≥ 1 and any F ∈ MN (P n ). Moreover A is (completely isometrically isomorphic to) a quotient of a subalgebra of a direct sum ∞ I (B) if and only if this holds with M = δ = 1, that is: F A ≤ F B ,

n, N ≥ 1, F ∈ MN (P n ).

(5.8)

Thus A is an operator algebra if and only if (5.8) holds true with B = B( 2 ). Of course this may be seen as a ‘von Neumann’s inequality’ characterization of operator algebras. 5.2.11 (Isomorphic characterizations) Let P[Z1 , . . . , Zn ] denote the algebra of all polynomials in n noncommuting variables Z1 , . . . , Zn . We may obviously regard P[Z1 , . . . , Zn ] as a subalgebra of P n by identifying Zk with Xkk for any 1 ≤ k ≤ n. Let A be an algebra which is also a Banach space, and let δ > 0 be a positive number. Restricting the deﬁnitions from 5.2.2 to diagonal matrices, we deﬁne for any F ∈ P[Z1 , . . . , Zn ]: F A,δ = sup F (a1 , . . . , an )A : ak ∈ A, sup ak ≤ δ . k

Arguing as in the proofs of 5.2.3 and 5.2.5, we ﬁnd that if B is an operator algebra, then A is isomorphic to a quotient of a subalgebra of some direct sum ∞ I (B) if and only if there exist two constants M, δ > 0 such that F A,δ ≤ M F B,1,

n ≥ 1, F ∈ P[Z1 , . . . , Zn ].

(5.9)

In particular (5.9) holds with B = B(2 ) if and only if A is isomorphic to an operator algebra. Moreover we see that (5.9) holds with M = δ = 1 (resp. and with B = B(2 )) if and only if A is isometrically isomorphic to a quotient of a subalgebra of some direct sum ∞ I (B) (resp. to an operator algebra). We give two more results concerning the isomorphic theory which are straightforward applications of our operator space techniques.

Completely isomorphic theory of operator algebras

207

Corollary 5.2.12 (Varopoulos) Let A be a Banach space which is also an algebra. Then A is isomorphic to an operator algebra if and only if there is a constant K > 0 such that for any ϕ ∈ Ball(A∗ ) and any r ≥ 2, there exist Hilbert spaces C = H0 , H1 , . . . , Hr−1 , Hr = C and bounded linear maps vk : A → B(Hk , Hk−1 ), 1 ≤ k ≤ r, such that v1 · · · vr ≤ K r and ϕ, a1 a2 · · · ar = v1 (a1 ) v2 (a2 ) · · · vr (ar ),

a1 , . . . , ar ∈ A.

Proof Indeed according to (1.10) and (1.12), A is isomorphic to an operator algebra if and only if it satisﬁes the conclusion of 5.2.8 (1), with A1 = Max(A) and A2 = Min(A). By the latter proposition together with (1.9), this holds true if and only if, for some constant K, we have ϕ ◦ mr : Max(A) ⊗h · · · ⊗h Max(A) −→ C ≤ K r cb for any r ≥ 2 and ϕ ∈ Ball(A∗ ). The result therefore follows by applying the CSPS theorem (see 1.5.8) to ϕ ◦ mr . 2 Corollary 5.2.13 (Tonge) Let A be a Banach space which is also an algebra and assume that the multiplication m : A ⊗ A → A is bounded from A ⊗ g2 A into A (see (A.6)). Then A is isomorphic to an operator algebra. Proof According to (1.10) and (1.46), our assumption is equivalent to the map m : Max(A) ⊗h Min(A) → Min(A) being completely bounded. Hence the result 2 follows by applying 5.2.8 (2), with A1 = Max(A) and with A2 = Min(A). 5.2.14 As a variant of the above statement, we note that if A is a Banach space which is also an algebra, then Min(A) is completely isomorphic to an operator algebra if and only if its multiplication m is bounded from A ⊗γ2 A into A (see (A.5) for a deﬁnition). Indeed by (1.44), the boundedness of m : A ⊗γ2 A → A is equivalent to the complete boundedness of m : Min(A) ⊗h Min(A) → Min(A). Hence the result follows from 5.2.1. See also 5.4.11 below for a related result. 5.2.15 (Multiplication and interpolation) As in 5.1.10 (2), let (A0 , A1 ) be a compatible couple of Banach algebras with operator space structures, and assume that A0 and A1 are both completely isomorphic to operator algebras. For any 0 ≤ θ ≤ 1, let mθ : Aθ ⊗h Aθ → Aθ be the product map on Aθ . Then θ mθ cb ≤ m0 1−θ cb m1 cb ,

θ ∈ (0, 1).

To see this, let n ≥ 1 be any integer and let σn,θ : Mn (Aθ ) × Mn (Aθ ) −→ Mn (Aθ ) be the multiplication mapping on n × n matrices, regarded as a bilinear map. Then by (1.24) and the interpolation theorem for multilinear maps [33, Section 4.4], we have σn,θ ≤ σn,θ 1−θ σn,θ θ . Passing to the limit when n → ∞ yields the result.

208

Completely bounded characterizations

Theorem 5.2.16 Let A be an algebra which is also a dual operator space and assume that A is completely isomorphic to an operator algebra. If the product on A is separately w∗ -continuous, then there exist a dual operator algebra B and a w∗ -continuous invertible homomorphism ρ : A → B such that ρ and ρ−1 are completely bounded. Proof This is a variant of the proof of Theorem 2.7.9. By hypothesis there is an operator algebra C and a complete algebra isomorphism σ : C → A. By 2.5.4, C is Arens regular, and hence A is Arens regular as well. Passing to second duals, σ ∗∗ : C ∗∗ → A∗∗ is a complete isomorphism and also a homomorphism (the latter by simple facts in Section 2.5). By assumption, A = X ∗ completely isometrically for some operator space X. Let Q : A∗∗ → A be the adjoint mapping of the canonical embedding iX : X → X ∗∗ . Then Lemma 2.7.12 ensures that Q is a homomorphism. Thus the mapping Qσ ∗∗ : C ∗∗ → A is a w∗ -continuous homomorphism. Consequently, its kernel J is a w ∗ -closed ideal of C ∗∗ , and hence the resulting quotient algebra B = C ∗∗ /J is a dual operator algebra by 2.7.3 and 2.7.11. The mapping τ : B → A induced by Qσ ∗∗ is a w∗ -continuous isomorphism. Since Q is a complete quotient map, it is clear that τ and its inverse are completely bounded, which concludes the proof. 2 We end this section with a few words on operator modules. In Chapter 3, we only considered the ‘completely isometric’ version of the theory. As mentioned in the introduction to that chapter one may also deﬁne a ‘completely isomorphic’ version of operator bimodules (and matrix normed bimodules). In this direction, the appropriate version of the ‘CES theorem’ 3.3.1 is the following result. Theorem 5.2.17 Let A and B be operator algebras, and suppose that X is an A-B-bimodule. The module actions on X extend to completely bounded maps A ⊗h X −→ X

and

X ⊗h B −→ X

if and only if there exists a Hilbert space H, a completely bounded linear map Φ : X → B(H), and completely bounded homomorphisms θ : A → B(H) and π : B → B(H) which all are complete isomorphisms onto their respective ranges, and such that θ(a)Φ(x) = Φ(ax)

and

Φ(x)π(b) = Φ(xb),

for all a ∈ A, b ∈ B and x ∈ X. Proof The ‘if’ part is easy, as in 3.3.1; we only prove the more diﬃcult ‘only if’ implication. We let u : A ⊗h X → X and v : X ⊗h B → X denote the mappings induced by the module actions. Consider the algebra D of matrices a x 0 b for a ∈ A, b ∈ B, x ∈ X, as discussed in 3.3.4. One may give the algebra D any operator space structure which retains the original matrix norms on the three

Completely isomorphic theory of operator algebras

209

nonzero corners, but for speciﬁcity we will use the structure A ⊕∞ X ⊕∞ B. It is a simple matter to check that the multiplication on D is completely bounded. This follows because, for example, a1 a2 a1 x2 + x1 b2 a1 x1 a2 x2 = 0 b1 0 b2 0 b1 b2 = max{a1 a2 , a1 x2 + x1 b2 , b1 b2 }, which is clearly dominated by 2 max{a1 , vcbx1 , b1} max{a2 , ucbx2 , b2} a1 x1 a2 x2 , ≤ K 0 b1 0 b2 for the constant K = 2 max{1, vcb} max{1, ucb}. Thus we may appeal to Theorem 5.2.1 to obtain a complete isomorphism ρ of D onto some concrete operator subalgebra of B(H) say. Letting " # " # " # a0 00 0x θ(a) = ρ ; π(b) = ρ ; Φ(x) = ρ , 00 0b 00 we obtain the desired result.

2

5.2.18 (Dual modules) Assume that A, B are dual operator algebras, that X is a dual operator space and that the module actions A × X → X and X × B → X are both completely bounded and separately w ∗ -continuous. Then the conclusions of Theorem 5.2.17 hold with the additional property that the mappings θ, π and Φ are w∗ -continuous. Indeed with these assumptions, the algebra D in the above proof is a dual operator space and the product on D is separately w∗ -continuous. Hence Theorem 5.2.16 ensures that the homomorphism ρ : D → B(H) in the proof of 5.2.17 can be taken to be w ∗ -continuous. Then the mappings θ, π and Φ deﬁned from ρ are clearly w ∗ -continuous. Note that in that case, their ranges are w ∗ -closed, so that θ(A), π(B) are dual operator algebras and Φ(X) is a dual operator space. Moreover the inverse mappings θ−1 : θ(A) → A, π −1 : π(B) → B and Φ−1 : Φ(X) → X are all w∗ -continuous. These facts follow from a simple variant of A.2.5. 5.3 EXAMPLES OF OPERATOR ALGEBRA STRUCTURES The aim of this section (and of Section 5.5) is to ‘get our hands dirty’ with several very concrete examples; and thus to give the reader a feel for the kind of geometrical features of the matrix norms which determine whether the algebra is completely isomorphic to an operator algebra or not. 5.3.1 (Multiplication on p ) For any 1 ≤ p ≤ ∞, we equip the Banach space p with the pointwise product. Namely, if x = (xk )k≥1 and y = (yk )k≥1 are

210

Examples of operator algebra structures

arbitrary elements of p , we let xy = (xk yk )k≥1 . It is clear that this product is well-deﬁned and makes p a Banach algebra. It was proved by Davie [116] and Varopoulos [415] in the seventies that p is isomorphic to an operator algebra for every 1 ≤ p ≤ ∞. We consider here the question of determining some, or all, corresponding operator algebra structures on p . We will obtain the Davie– Varopoulos result en route (e.g. see 5.3.5 below). Let (en )n≥1 denote the canonical basis of p , and for any integer n ≥ 1 identify p n with the subspace of p spanned by e1 , . . . , en . Then pn is a subalgebra of p . Let mn : pn ⊗ pn → pn denote the corresponding product. We have " # mn aij ei ⊗ ej = aii ei , aij ∈ C. (5.10) 1≤i,j≤n

1≤i≤n

If p is equipped with an operator space structure then we regard pn as equipped with the induced structure. It clearly follows from Theorem 5.2.1 that the given structure on p is an operator algebra structure if and only if sup mn : pn ⊗h pn −→ pn cb < ∞. n

5.3.2 (The diagonal projection) We will often use the facts that Mn = Cn ⊗h Rn and that Sn1 = Mn∗ = Rn ⊗h Cn (see 1.5.14). If we think of Mn and Sn1 both as n × n matrices in the usual way, then the identiﬁcations in the last sentence are given by the purely algebraic isomorphism between Mn and Cn ⊗Cn taking Eij to ei ⊗ej . Via this last isomorphism, if Dn ⊂ Mn is the subspace of diagonal matrices (which we may also identify algebraically with Cn ), (5.10) simply says that mn corresponds to the canonical projection from Mn onto Dn . This projection is obviously completely contractive if Mn is given its usual structure Cn ⊗h Rn , and Dn has the corresponding subspace structure ∞ n . On the other hand, suppose that Mn is interpreted as the trace class matrices Sn1 = Rn ⊗h Cn . In this case, in the predual we have the canonical diagonal embedding ∞ n → Mn , which is a complete isometry, and the adjoint of this embedding is the canonical projection which we have identiﬁed with mn above. Thus in this case it follows that again ∗ 1 mn is completely contractive, but now as a map Rn ⊗h Cn → (∞ n ) = Max(n ). Proposition 5.1.7 shows that Min(∞ ) is, up to complete isomorphism, the only operator algebra structure on ∞ . It turns out that the situation for p = 1 is radically diﬀerent. Proposition 5.3.3 Any operator space structure on 1 is an operator algebra structure. Proof We let Id denote the canonical embedding of 1 into 2 . We shall ﬁrst prove that Id : Min(1 ) −→ R ∩ C = 1, (5.11) cb where R ∩ C is the Hilbert operator space deﬁned in 1.2.24. Indeed let a1 , . . . , an be in Mm , for some m ≥ 1. Then by 1.2.21 we have

Completely isomorphic theory of operator algebras ak ⊗ e k = sup θ a k k 1 Mm (Min( ))

k

whereas

1 2π

a∗k ak =

k

Mm

k

!

2π

0

ak eikt

211

∗

k

: θk ∈ C, |θk | ≤ 1 ,

ak eikt dt .

k

Therefore, a∗k ak k

Mm

1 ≤ 2π

!

2π

0

2 2 ak eikt dt ≤ ak ⊗ e k

k

Mm (Min(1 ))

k

.

Similarly we have ak a∗k k

Mm

2 ≤ ak ⊗ e k k

Mm (Min(1 ))

.

Hence (5.11) follows from (1.16). Let n ≥ 1 be an integer. By the discussion at the end of 5.3.2, we have mn : Rn ⊗h Cn −→ Max(1n ) ≤ 1. (5.12) cb Combining with (5.11) applied twice, we deduce that mn : Min(1n ) ⊗h Min(1n ) −→ Max(1n ) ≤ 1. cb This yields the result, by the discussion at the end of 5.3.1.

2

5.3.4 (The operator space Op ) The Banach space p has a natural operator space structure, denoted by Op , which was introduced by Pisier (see [331] and [328]). Its deﬁnition is based on complex interpolation and on the fact that p = [∞ , 1 ] p1 isometrically for any 1 ≤ p ≤ ∞. We let O∞ = Min(∞ ), O1 = Max(1 ) and according to 1.2.30, we deﬁne Op = [Min(∞ ), Max(1 )] p1

(5.13)

for 1 < p < ∞. The operator space O2 is usually denoted by OH. This operator Hilbert space has various remarkable properties regarding duality and interpolation, which are studied in great detail in [328]. In the sequel, the n-dimensional versions of the above spaces will be denoted by Opn (resp. OHn ). Corollary 5.3.5 For any 1 ≤ p ≤ ∞, Op is completely isomorphic to an operator algebra. Proof This clearly follows from Proposition 5.3.3 and 5.1.10 (2). An alternate proof is obtained by combining (5.14) below with Theorem 5.2.1. 2

212

Examples of operator algebra structures

5.3.6 (A counterexample) We show here that the BRS theorem 2.3.2 is not valid if we remove the hypothesis that A has an identity of norm 1 or a cai. Indeed it follows from the proof of 5.3.3 that mn is completely contractive from Max(1n ) ⊗h Max(1n ) to Max(1n ). Since it is also completely contractive from ∞ ∞ Min(∞ n ) ⊗h Min(n ) to Min(n ), it follows from 5.2.15 that mn : Opn ⊗h Opn −→ Opn ≤ 1. (5.14) cb

However if p is ﬁnite and n ≥ 2, then pn is not isometrically isomorphic to an operator algebra. Indeed take n = 2 and assume that ρ : p2 → B(H) is an isometric homomorphism, for some Hilbert space H. Then ρ(e1 ) and ρ(e2 ) are disjoint contractive (and hence orthogonal) projections. Thus for any complex numbers θ1 , θ2 , we have θ1 e1 + θ2 e2 = θ1 ρ(e1 ) + θ2 ρ(e2 ) = max{|θ1 |, |θ2 |}, a contradiction. Note that e1 + e2 is an identity for Op2 , with norm 21/p . Proposition 5.3.7 Let 1 ≤ p ≤ ∞, then: (1) Min(p ) is completely isomorphic to an operator algebra if and only if p = 1 or p = ∞. (2) Max(p ) is completely isomorphic to an operator algebra if and only if we have 1 ≤ p ≤ 2. p Proof Suppose that Min(p ) is an operator algebra structure on , where 1 < p < ∞. Let n ≥ 1 be an integer, and let un = 1≤k≤n ek ⊗ ek , regarded as an element of pn ⊗ pn . Then mn (un ) = 1≤k≤n ek , whose norm in pn is equal to n1/p . We deduce that there is a constant C > 0 (not depending on n) such that n ek ⊗ ek . n1/p ≤ C p p Min(n )⊗h Min(n )

k=1

By (1.44) and (A.5), we have n ek ⊗ ek p k=1

Min(n )⊗h Min(p n)

n 2 ≤ ek ⊗ ek 2 k=1

ˇ p n ⊗ n

2 = Id : 2n → pn .

If p ≥ 2, the right side of the above inequality is equal to 1. If p ≤ 2, it is equal 2 to n p −1 . This yields a contradiction and proves the ‘only if’ part of (1). The ‘if’ part follows from 5.3.3. We now turn to the proof of (2). Assume that 1 ≤ p ≤ 2 and let n ≥ 1 be an integer. It follows from (5.12) that mn : Max(2n ) ⊗h Max(2n ) −→ Max(1n ) ≤ 1. cb

Since the identity mappings 1n → pn and pn → 2n are contractive, we deduce using (1.12) that mn : Max(pn ) ⊗h Max(pn ) −→ Max(pn ) ≤ 1. cb Hence Max(p ) is completely isomorphic to an operator algebra by 5.2.1.

Completely isomorphic theory of operator algebras

213

Assume now that 2 < p ≤ ∞ and let q be the conjugate of p (p−1 + q −1 = 1). For any n ≥ 1, the norm of the identity mapping qn → 1n is equal to n1/p . Hence by (5.11) and (1.10) we have Id : Min(qn ) −→ Rn ∩ Cn ≤ n1/p . cb By duality, we obtain that Id : Cn −→ Max(p ) ≤ n1/p n cb

and Id : Rn −→ Max(pn )cb ≤ n1/p .

Here we used (1.30), (1.15), and (1.17). Thus Id ⊗ Id : Cn ⊗h Rn −→ Max(pn ) ⊗h Max(pn ) ≤ n2/p . cb Hence if Max(p ) is an operator algebra structure on p , then there is a constant C > 0 such that mn : Cn ⊗h Rn −→ Max(pn ) ≤ Cn2/p cb for any n ≥ 1. Restricting to the diagonal of Cn ⊗h Rn = Mn (see 5.3.2), we readily obtain that Id : Min(∞ ) −→ Max(p ) ≤ Cn2/p (5.15) n n cb for any n ≥ 1. To reach a contradiction, we will use a ‘spin system’ in M 2n , that is, a ﬁnite sequence w1 , . . . , wn of selfadjoint unitaries in M2n such that wi wj + wj wi = 0 for any i = j. Such a sequence satisﬁes n n √ 1/2 tk wk ≤ 2 |tk |2 , k=1

t1 , . . . , tn ∈ C;

(5.16)

= n.

(5.17)

k=1 n wk ⊗ wkt

M2n ⊗min M2n

k=1

E.g. see [337, p. 76] for the existence of such a sequence and a proof of (5.16) and (5.17). According to 1.2.21, the inequality (5.16) may be rephrased as n wk ⊗ ek

M2n (Min(2n ))

k=1

Hence

n wk ⊗ ek k=1

M2n (Min(qn ))

≤

≤

√ 2.

√ 1−1 2 n2 p .

Let vn : Min(qn ) → M2n be the linear mapping taking ek to wkt for any k. Then

214 n

wk ⊗ wkt = IM2n

k=1

Examples of operator algebra structures n ⊗ vn wk ⊗ ek , k=1

and so by (5.17) we have n n ≤ vn cb wk ⊗ ek

M2n (Min(qn ))

k=1

≤

√ 1−1 2 n 2 p vn cb .

On the other hand, each wk has norm 1, and so nk=1 wk ⊗ ek has norm 1 in M2n (∞ n ). By (5.15), this implies that n wk ⊗ ek ≤ Cn2/p . p M2n (Max(n ))

k=1

However the last norm is equal to vn cb by (1.30), hence we ﬁnally obtain that √ 1 1 n ≤ C 2 n 2 + p , which is impossible for p > 2. 2 Proposition 5.3.8 With pointwise product, R, C, and R ∩ C are completely isomorphic to operator algebras. Proof In the identiﬁcation discussed in 5.3.2, we have Rn ⊗h Rn = (Sn2 )r by 1.5.14 (8). Since the projection onto diagonal matrices is contractive on S n2 , it is automatically completely contractive from (Sn2 )r onto Rn , and so it follows that mn : Rn ⊗h Rn → Rn cb ≤ 1. This shows that R is completely isomorphic to an operator algebra. The argument for C is identical, and the result for R ∩ C follows at once. 2 5.3.9 (Schur product of matrices) Let H = 2 and let (ek )k≥1 denote its usual basis. We let V : H → H ⊗2 H be the linear isometry deﬁned by letting V (ek ) = ek ⊗ ek for any k ≥ 1. Given a, b ∈ B(H), we regard a ⊗ b as an element of B(H ⊗2 H) and we deﬁne the Schur product of a and b as a ∗ b = V ∗ (a ⊗ b)V.

(5.18)

If we represent a and b by their (inﬁnite) matrices [aij ]i,j≥1 and [bij ]i,j≥1 with respect to the basis (ek )k≥1 , then (5.18) can be written as [aij ]i,j≥1 ∗ [bij ]i,j≥1 = [aij bij ]i,j≥1 . Indeed for any i, j ≥ 1, we have (a ∗ b)ej , ei =(a ⊗ b)V ej , V ei = (a ⊗ b)(ej ⊗ ej ), (ei ⊗ ei ) =aej , ei bej , ei = aij bij . It is clear from (5.18) that for any a, b ∈ B(H), we have a ∗ b ≤ V 2 a ⊗ b = ab. Thus equipped with the Schur product ∗, B(H) is a commutative Banach algebra. If the matrix [aij ]i,j≥1 representing a has a ﬁnite number of nonzero elements, then the same holds for the matrix [aij bij ]i,j≥1 representing a ∗ b. Hence compact operators S ∞ (H) form an ideal of B(H) for the Schur product.

Completely isomorphic theory of operator algebras

215

5.3.10 (Abstract Schur product) It is worthwhile to observe that the above construction is a special case of joint multiplication. If A, B are two algebras, then their algebraic tensor product A ⊗ B is an algebra with the product deﬁned in (2.6). We showed in 2.2.2 that A ⊗min B with this product is an operator algebra if A and B are operator algebras. It follows from this fact, and from the functoriality of the minimal tensor product, that if A and B are operator spaces and are completely isomorphic to operator algebras, then A ⊗ min B with the joint multiplication is completely isomorphic to an operator algebra. It is easy to check that if we let A and B be equal to 2 equipped with its pointwise product (see 5.3.1), and if we identify 2 ⊗ 2 with ﬁnite rank operators on H = 2 in the usual way, then the joint multiplication on 2 ⊗ 2 coincides with the Schur product introduced in 5.3.9. Theorem 5.3.11 Equipped with their natural operator space structure and the Schur product, B(H) and S ∞ (H) are completely isomorphic to operator algebras. Proof The result for S ∞ (H) immediately follows from the discussion in 5.3.10. Indeed recall from 1.5.14 (5) that S ∞ (H) = C ⊗min R completely isometrically. Since C and R are both operator algebra structures on 2 by 5.3.8, S ∞ (H) is completely isomorphic to an operator algebra for the joint multiplication, which turns out to be the Schur product. We now deduce the result for B(H). It is clear from (5.18) that the Schur product ∗ on B(H) is separately w ∗ -continuous. Hence by 2.5.3, (B(H), ∗) is the second dual of (S ∞ (H), ∗) equipped with the Arens product. It is therefore isomorphic to an operator algebra by 2.7.3. Indeed (B(H), ∗) is w ∗ -homeomorphic and completely isomorphic to a dual operator algebra by 5.2.16. 2 5.4 Q-ALGEBRAS 5.4.1 (Introduction to Q-algebras) By deﬁnition, a (concrete) Q-algebra is a Banach algebra A of the form C/J, where C is a function algebra (that is, a subalgebra of a commutative C ∗ -algebra), and J ⊂ C is a closed, two-sided ideal. We will regard such an algebra as an operator space, with the quotient structure A = Min(C)/Min(J). Then A is an operator algebra by 2.3.4. Conversely, we will say that an operator algebra is an (abstract) Q-algebra if it is completely isometrically isomorphic to a concrete Q-algebra. Any Q-algebra is commutative but there exist commutative operator algebras which are not Q-algebras (see the Notes to Section 5.4). The deﬁnitions and properties from 5.1.4 and 5.1.6 may be considered for the special class of Q-algebras; and thus we may talk about algebras isomorphic or completely isomorphic to a Q-algebra. We observe that subalgebras and quotients of Q-algebras are again Q-algebras. Indeed let A = C/J be a Q-algebra and let B ⊂ A be a subalgebra. Then B = D/J, where D ⊂ C is the subalgebra of all c ∈ C such that c˙ ∈ B. If further B is an ideal of A, then D is an ideal of C and we see that A/B = (C/J) (D/J) C/D (using standard facts about

216

Q-algebras

‘quotients of quotients’). On the other hand, any direct sum ⊕∞ λ Aλ of a family {Aλ : λ ∈ I} of Q-algebras is a Q-algebra. Indeed if we write Aλ = Cλ /Jλ for ∞ each λ ∈ I, then ⊕∞ λ Aλ is a quotient of ⊕λ Cλ . It follows from these observations that if A is an algebra which is also a Banach space, and if A is isomorphic to a Q-algebra, then any subalgebra of A, any quotient of A, and any direct sum ∞ I (A) also is isomorphic to a Q-algebra. Similarly, if further A is an operator space which is completely isomorphic to a Q-algebra, then the same holds for quotients, subalgebras, and direct sums of A. 5.4.2 (Interpolation of Q-algebras) Let (A0 , A1 ) be a compatible couple of Banach algebras. If A0 and A1 are both Q-algebras, then we claim that for any θ ∈ (0, 1), the operator algebra Aθ = [A0 , A1 ]θ is a Q-algebra. Indeed it follows from 2.3.6 and the proof of 2.3.7 that Aθ is a quotient of a subalgebra of C0 (R; A0 ) ⊕∞ C0 (R; A1 ). The claim follows from this and the stability properties of Q-algebras discussed in the second paragraph of 5.4.1. As in 5.1.10, the functorial properties of interpolation ensure that if A0 , A1 are isomorphic to Qalgebras, then Aθ is isomorphic to a Q-algebra. Similarly, if A0 and A1 are also operator spaces and if A0 , A1 are completely isomorphic to Q-algebras, then Aθ is completely isomorphic to a Q-algebra. 5.4.3 (Polynomial inequalities) An operator algebra is a Q-algebra if and only if it is a quotient of a subalgebra of some direct sum ∞ I of the one-dimensional algebra C. Applying 5.2.10 and 5.2.11, we obtain the following polynomial characterizations: (1) If A is an operator space which is also an algebra, then A is completely isomorphic to a Q-algebra if and only if there exist M, δ > 0 such that F A,δ ≤ M F C ,

n, N ≥ 1, F ∈ MN (P n ).

(5.19)

In fact it suﬃces to prove (5.19) for homogeneous polynomials. Moreover A is (completely isometrically isomorphic to) a Q-algebra if and only if the latter holds with M = δ = 1. (2) If A is a Banach space which is also an algebra, then A is isomorphic to a Q-algebra if and only if there exist M, δ > 0 such that F A,δ ≤ M F C for any integer n ≥ 1 and any F ∈ P[Z1 , . . . , Zn ]. Moreover A is isometrically isomorphic to a Q-algebra if and only if the latter holds with M = δ = 1. 5.4.4 The next result is a handy criterion for when an algebra is completely isomorphic to a Q-algebra. We will need the following notation. Let n, N, r ≥ 1 be three integers and let u : Mn × · · · × Mn → MN be an r-linear map. For any algebra A, denote by uA : Mn (A) × · · · × Mn (A) −→ MN (A) the unique r-linear map deﬁned by letting uA (x1 ⊗ a1 , . . . , xr ⊗ ar ) = u(x1 , . . . , xr ) ⊗ a1 · · · ar ,

xk ∈ Mn , ak ∈ A.

Completely isomorphic theory of operator algebras

217

Theorem 5.4.5 Let A be a commutative algebra which is also an operator space. Then A is completely isomorphic to a Q-algebra if and only if there is a positive constant K > 0 such that for any integers n, N, r ≥ 1 and any r-linear map u : Mn × · · · × Mn → MN , we have uA ≤ K r u. Proof We assume that uA ≤ K r u for all u as above. Let F ∈ MN (P n ) be a homogeneous polynomial of degree r ≥ 1, the integers n, N, r being arbitrary. According to 5.4.3 (1), it will suﬃce to show that F satisﬁes (5.19). Using notation from 5.2.4 as in the proof of Theorem 5.2.8, we write λα ⊗ Xα1 · · · Xαr , F = α∈Λr

with λα ∈ MN . Then we let v : Mn ×· · ·×Mn → MN be the r-linear map deﬁned by v(Eα1 , . . . , Eαr ) = λα for any α = (α1 , . . . , αr ) ∈ Λr , so that F (a) = vA (a, . . . , a),

a ∈ Mn (A).

(5.20)

Denote by S r the permutation group on {1, . . . , r}. We introduce the symmetric r-linear map u : Mn × · · · × Mn → MN associated to v by letting 1 v(xσ(1) , . . . , xσ(r) ), xp ∈ Mn . u(x1 , . . . , xr ) = r! σ∈S r We shall now prove that vA (a, . . . , a) = uA (a, . . . , a),

a ∈ Mn (A);

(5.21)

and u ≤ er F C . (5.22) k Let a = p=1 xp ⊗ ap be an arbitrary element of Mn (A). Then vA (a, . . . , a) is equal to v(xp1 , . . . , xpr ) ⊗ ap1 · · · apr , where the sum is over all the r-tuples (p1 , . . . , pr ) valued in {1, . . . , k} (k r terms). Permuting the indices we see that for σ ∈ S r we have vA (a, . . . , a) = v xpσ(1) , . . . , xpσ(r) ⊗ apσ(1) · · · apσ(r) . (5.23) 1≤pj ≤k

Then we obtain uA (a, . . . , a) =

u(xp1 , . . . , xpr ⊗ ap1 · · · apr

1≤pj ≤k

# "1 ⊗ ap1 · · · apr = v xpσ(1) , . . . , xpσ(r) r! 1≤pj ≤k σ∈S r 1 = v xpσ(1) , . . . , xpσ(r) ⊗ apσ(1) · · · apσ(r) r! σ∈S r 1≤pj ≤k because A is commutative. Then (5.21) follows from (5.23).

218

Q-algebras

To prove (5.22), we choose any x1 , . . . , xr ∈ Ball(Mn ), and let (ε1 , . . . , εr ) be an r-tuple of independent ±1-valued random variables on a probability space (Ω, µ), with µ{εp = 1} = µ{εp = −1} = 1/2. Then v

r p=1

and

εp xp , . . . ,

r

εp xp = εp1 · · · εpr v(xp1 , . . . , xpr ); 1≤pj ≤r

p=1

! Ω

ε1 · · · εr εp1 · · · εpr dµ = 1

if {p1 , . . . , pr } = {1, . . . , r},

=0

if {p1 , . . . , pr } = {1, . . . , r}.

Consequently, the following integral representation formula holds: ! r r 1 u(x1 , . . . , xr ) = ε1 · · · εr v εp xp , . . . , εp xp dµ . r! Ω p=1 p=1 Since xp ≤ 1 for any p, we have u(x1 , . . . , xr ) ≤

r

p=1 εp xp

≤ r. Hence

rr sup v(x, . . . , x) : x ∈ Mn , x ≤ 1 . r! r

By (5.20) (taking A = C) the latter supremum equals F C . Also, rr! ≤ er . This yields (5.22). Combining (5.20) and (5.21), we obtain that F (a) = uA (a, . . . , a) for any a ∈ Mn (A). Hence F (a) ≤ uA ar ≤ K r uar ≤ (Ke)r F C ar by (5.22). This shows that F satisﬁes (5.19) with δ = (Ke)−1 and M = 1, which concludes the proof of the ‘if’ part. The converse direction is easier. Clearly we may assume that A is a Q-algebra. Thus A = C/J, where C ⊂ C(Ω) is a subalgebra of C(Ω) for some compact space Ω, and J ⊂ C is an ideal. Let u : Mn × · · · × Mn → MN be any r-linear map. We let q : C → A denote the canonical quotient map. Given any a1 , . . . , ar in Mn (A), with ai < 1, let c1 , . . . , cr in Mn (C) be such that qn (ci ) = ai and ci < 1 for any 1 ≤ i ≤ r. We have uA (a1 , . . . , ar ) = qN uC (c1 , . . . , cr ) . Using the isometric identiﬁcations Mm (C(Ω)) = C(Ω; Mm ) (e.g. from 1.2.3), we may regard c1 , . . . , cr and uC (c1 , . . . , cr ) as elements of C(Ω; Mn ) and C(Ω; MN ) respectively. Then uC (c1 , . . . , cr )(ω) = u(c1 (ω), . . . , cr (ω)),

ω ∈ Ω.

We deduce that uC (c1 , . . . , cr ) ≤ u, whence uA (a1 , . . . , ar ) ≤ u. This 2 shows that uA ≤ u.

Completely isomorphic theory of operator algebras

219

5.4.6 (Q-spaces) We say that an operator space X is a Q-space if there exist Banach spaces F ⊂ E such that X = Min(E)/Min(F ) completely isometrically. If X is merely completely isomorphic to a Q-space, we let dQ (X) = inf v : X −→ Y cb v −1 : Y −→ Xcb , where the inﬁmum runs over all possible v : X → Y such that Y is a Q-space and v is a complete isomorphism. The parameter dQ (X) measures the ‘completely bounded distance’ from X to a Q-space. If X is a Q-space, and if u : Mn → MN is a linear mapping, then we have u ⊗ IX : Mn (X) → MN (X) ≤ u. Indeed, assume that X = Min(E)/Min(F ), let q : E → X be the associated quotient map, and let x ∈ Mn (X) with norm < 1. Then there exists a z ∈ Mn (Min(E)) with z < 1 and qn (z) = x. Since ˇ and MN (Min(E)) = Mn ⊗E ˇ isometrically (see (1.11)), Mn (Min(E)) = Mn ⊗E we have (u ⊗ IE )z ≤ uz ≤ u by the ‘functoriality’ of the Banach space injective tensor product A.3.1. Since (u ⊗ IX )x = qN (u ⊗ IE )z, we see that (u ⊗ IX )x ≤ u, which proves the announced estimate. 5.4.7 (Comparing Q-spaces and Q-algebras) Obviously any Q-algebra is a Qspace. Conversely, let X be a Q-space and let A be X with the ‘zero product’ (see 2.2.9). Then A is a Q-algebra. Tosee this, let F be any element of M N (P n ) and consider its decomposition F = r≥1 Fr , with Fr ∈ MN (P n ) homogeneous of degree r (see 5.2.4). Since the product on A is zero, we have F A = F1 A . Moreover we may write F1 = 1≤i,j≤n λij ⊗ Xij with λij ∈ MN , and we have λij ⊗ xij F1 A = sup i,j

MN (X)

: xij ∈ X, [xij ]Mn (X) ≤ 1 .

Thus if we let u : Mn → MN be the linear mapping deﬁned by u(Eij ) = λij for any 1 ≤ i, j ≤ n, then F1 A coincides with u ⊗ IX : Mn (X) → MN (X) whereas F1 C coincides with u. Applying 5.4.6 and 5.2.4, we obtain that F A = F1 A ≤ F1 C ≤ F C . By 5.4.3 (1), this shows that A is a Q-algebra. Proposition 5.4.8 (Junge) Let X be an operator space. Then X is completely isomorphic to a Q-space if and only if there exists a constant K ≥ 1 such that for any n ≥ 1 and any linear map u : Mn → Mn , we have u ⊗ IX : Mn (X) −→ Mn (X) ≤ Ku. (5.24) Moreover dQ (X) coincides with the smallest K having that property.

220

Q-algebras

Proof The ‘only if’ direction follows from 5.4.6. Conversely if X satisﬁes (5.24), then arguing as in 5.4.7 with A = X with ‘zero product’, we obtain that F A ≤ KF C for any F ∈ MN (P n ). As in the last line of 5.4.7 we deduce that there is a Q-algebra B and an algebra isomorphism ρ : A → B such that ρcbρ−1 cb ≤ K. Since X = A completely isometrically, this is equivalent to saying that dQ (X) ≤ K. 2 5.4.9 (Interpolation of Q-spaces) Let (X0 , X1 ) be a compatible couple of operator spaces (see 1.2.30). If E0 and E1 are both completely isomorphic to Q-spaces, then the interpolation space Xθ = [X0 , X1 ]θ is completely isomorphic to a Qspace as well. This can be proved by the argument in 5.4.2. Alternatively, one may appeal to 5.4.8, which yields the estimate dQ (Xθ ) ≤ dQ (X0 )1−θ dQ (X1 )θ . Indeed this follows by interpolation from (1.24). Theorem 5.4.10 Let A be a commutative algebra which is also an operator space, and assume that A is completely isomorphic to a Q-space. If the multiplication on A extends to a completely bounded map m : A ⊗min A −→ A, then A is completely isomorphic to a Q-algebra. Proof We shall prove that A satisﬁes the suﬃcient condition of Theorem 5.4.5. By assumption, there exist a Banach space E and a completely bounded map q : Min(E) → A inducing a complete isomorphism from Min(E)/Min(Ker(q)) onto A. We let K > 0 be such that for any n ≥ 1 and any a ∈ Mn (A) with a < 1, there exists z ∈ Mn (Min(E)) such that qn (z) = a and z < K. We ﬁx an r-linear map u : Mn × · · · × Mn → MN and shall estimate the norm of uA . For that purpose we shall ﬁrst estimate the norm of the auxiliary r-linear mapping ˇ ⊗ ˇ · · · ⊗E ˇ u : Mn (Min(E)) × · · · × Mn (Min(E)) −→ MN ⊗E deﬁned by letting u(x1 ⊗ z1 , . . . , xr ⊗ zr ) = u(x1 , . . . , xr ) ⊗ z1 ⊗ · · · ⊗ zr for any 1 r ], . . . , z r = [zij ] ∈ Ball(Mn (Min(E))). xp in Mn and any zp in E. Let z 1 = [zij Using notation from 5.2.4, we set λα = u(Eα1 , . . . , Eαr ) for any α ∈ Λr . Then λα ⊗ zα1 1 ⊗ · · · ⊗ zαr r . u(z 1 , . . . , z r ) = α∈Λr

Given any functionals ϕ1 , . . . , ϕr ∈ Ball(E ∗ ), we have ) 1 * λα ϕ1 , zα1 1 · · · ϕr , zαr r u(z , . . . , z r ), ϕ1 ⊗ · · · ⊗ ϕr = α∈Λr

=u

ϕ1 , zα1 1 , . . . , ϕr , zαr r .

Completely isomorphic theory of operator algebras

221

Hence ) 1 * u(z , . . . , z r ), ϕ1 ⊗ · · · ⊗ ϕr ≤ u ϕ1 , zα1 1 · · · ϕr , zαr r ≤ u. MN By A.3.1, this shows that u ≤ u. On the other hand, ˇ · · · ⊗E) ˇ Min(E) ⊗min · · · ⊗min Min(E) = Min(E ⊗ completely isometrically by 1.5.3 (2). From this and the ‘functoriality’ of ⊗ min , for any r ≥ 1, the r-fold tensor product of q satisﬁes q ⊗ · · · ⊗ q : Min(E ⊗ ˇ · · · ⊗E) ˇ −→ A ⊗min · · · ⊗min Acb ≤ qrcb. (5.25) Next, we denote as usual by mr : A ⊗min · · · ⊗min A → A the r-fold multiplication on A. Let a1 , . . . , ar in Mn (A), with ai < 1, and choose z1 , . . . , zr in Mn (Min(E)) such that qn (zp ) = ap and zp < K for any 1 ≤ p ≤ r. Then it follows from the respective deﬁnitions of uA and u that u(z1 , . . . , zr ) . uA (a1 , . . . , ar ) = IMN ⊗ mr ◦ (q ⊗ · · · ⊗ q) Hence combining (5.25) with the inequality u ≤ u, we ﬁnd that r uA (a1 , . . . , ar ) ≤ qrcb mr cb K r u ≤ qrcb mr−1 cb K u.

By taking the supremum over all possible such a1 , . . . , ar , we may deduce that r uA ≤ qrcb mr−1 2 cb K u. This yields the result by Theorem 5.4.5. Corollary 5.4.11 Let A be a commutative algebra which is also a Banach space and assume that the multiplication on A extends to a bounded map ˇ A −→ A. m : A⊗ Then Min(A) is completely isomorphic to a Q-algebra. Proof A minimal operator space is a Q-space, and by (1.10) and 1.5.3 (2), the ˇ → A is bounded if and only if m : Min(A) ⊗min Min(A) → Min(A) map m : A⊗A is completely bounded. Hence Min(A) satisﬁes 5.4.10, whence the result. 2 5.4.12 We now come back to the example of p with pointwise product. This commutative Banach algebra is isomorphic to an operator algebra. In Section 5.3 we gave various examples of corresponding operator algebra structures on p . In fact Davie [116] and Varopoulos [415] showed that p is isomorphic to a Q-algebra. This will be established in 5.4.13 below using operator spaces. Also we wish to consider the question of whether the operator algebra structures on p exhibited in the previous section make p completely isomorphic to a Qalgebra. It follows from a deep result of Junge–Pisier (see [209, Theorem 3.2]) that Max(E) cannot be a Q-space if E is an inﬁnite-dimensional Banach space. The other cases will be settled in 5.4.15 and 5.4.16.

222

Q-algebras

Corollary 5.4.13 For any 1 ≤ p ≤ ∞, p with pointwise product is isomorphic to a Q-algebra. Indeed the operator space [Min(∞ ), Min(1 )] p1 (whose underlying Banach space is p ) is completely isomorphic to a Q-algebra. Proof The result is obvious for p = ∞. Hence by interpolation, (see 5.4.2) it suﬃces to prove that Min(1 ) is completely isomorphic to a Q-algebra. According to Theorem 5.3.3 and 5.2.14, the multiplication on 1 extends to a bounded map m : 1 ⊗γ2 1 → 1 . However the injective tensor norm and γ2 are equivalent on 1 ⊗ 1 . Indeed, this is one of the equivalent forms of Grothendieck’s inequality ˇ 1 → 1 is bounded, and the result follows (see [324, Chapters 4–6]). Thus m : 1 ⊗ from 5.4.11. 2 5.4.14 (A noncommutative Grothendieck inequality) In the next proof, we will use a remarkable inequality due to Pisier, which is a consequence of the so-called ‘noncommutative Grothendieck inequality’ (see [324, Corollary 9.5]): If u : A → B is a bounded operator between C ∗ -algebras, then for any x1 , . . . , xn in A, we have max u(xk )∗ u(xk ) , u(xk )u(xk )∗ xk x∗k , x∗k xk , ≤ K 2 u2 max where K is an absolute constant. Assume that A = B = MN for some N ≥ 1. Then according to (1.16), this is equivalent to u ⊗ IR∩C : MN (R ∩ C) −→ MN (R ∩ C) ≤ Ku. Corollary 5.4.15 R ∩ C is completely isomorphic to a Q-algebra. Proof First, the operator space R ∩ C is completely isomorphic to a Q-space. This follows by applying 5.4.14 above, and Junge’s characterization 5.4.8. Let m denote the multiplication mapping on 2 . We know from 5.3.8 and its proof that m : R ⊗h R → R is completely contractive. Since R ⊗h R = R ⊗min R completely isometrically (see 1.5.14 (2)), we deduce that m : (R ∩ C) ⊗min (R ∩ C) → R is completely contractive. Likewise, the product m : (R ∩ C) ⊗min (R ∩ C) → C is completely contractive, and so m : (R ∩ C) ⊗min (R ∩ C) −→ R ∩ C ≤ 1. cb The result therefore follows from Theorem 5.4.10.

2

Proposition 5.4.16 (1) Up to complete isomorphism, the minimal operator space structure is the only one for which 1 is completely isomorphic to a Q-space. (2) R and C are not completely isomorphic to Q-spaces. Indeed for any n ≥ 1, √ dQ (Rn ) = dQ (Cn ) = n.

Completely isomorphic theory of operator algebras

223

(3) For any 1 ≤ p < ∞, Op is not completely isomorphic to a Q-space. Indeed for any n ≥ 1, we have 1

1 n 2p √ ≤ dQ (Opn ) ≤ n 2p . 2

Proof (1) Assume that 1 is equipped with an operator space structure such that there exist a Banach space E, an operator space X, a complete quotient map q : Min(E) → X, and an isomorphism u : 1 → X such that u−1 is completely bounded. By the lifting property of 1 , there exists a bounded map σ : 1 → E such that qσ = u. This map is automatically completely bounded from Min(1 ) into Min(E), and so I1 = u−1 qσ : Min(1 ) → 1 is completely bounded. (2) Let n, m be two positive integers and let u : Mm → Mm be a linear map. 1/2 n If a1 , . . . , an in Mm , then k=1 u(ai )∗ u(ai ) is dominated by n k=1

u(ai )2

1/2

≤ u

n

ai 2

1/2

≤

n 1/2 √ n u a∗i ai .

k=1

k=1

This √ shows that u ⊗ ICn : Mm (Cn ) → Mm (C√n ) has norm less than or equal to n u. Thus by 5.4.8 we have dQ (Cn ) ≤ n. Conversely, let u : Mn → Mn be equal to the transposition map. Under the identiﬁcations Rn (Cn ) = Mn and Rn (Rn ) = Sn2 (see 1.5.14), the restriction of u ⊗ ICn to Rn (Cn ) √ coincides with 2 → S . The latter has norm equal to n, giving the the identity mapping M n n √ equality dQ (Cn ) = n. The same argument works as well for Rn . (3) Let n, N ≥ 1 be integers, let a1 , . . . , an in MN , and let u be the linear ) to M taking e to a for any k. Then u(θ) = map from Min(∞ N k k n k ak θk for any θ = (θ1 , . . . , θn ) ∈ ∞ n , hence n 21 √ ucb ≤ ak a∗k n k=1

by the (easy) converse of the representation theorem of completely bounded maps 1 1.2.8. Since CB(Min(∞ n ), MN ) = MN (Max(n )) isometrically, this shows that 2 1 the identity mapping Id : n → n satisﬁes √ Id : Rn −→ Max(1n )cb ≤ n. (5.26) Combining with (5.11), this implies that √ I1 : Min(1n ) −→ Max(1n ) ≤ n. n cb

√ Since Min(1n ) is a Q-space, this yields dQ (Max(1n )) ≤ n. By interpolation 1 (see 5.3.4 and 5.4.9), we deduce that dQ (Opn ) ≤ n 2p . We now turn to the lower estimates (which show that Op is not completely isomorphic to a Q-space). Let

224

Applications to the isomorphic theory

w1 , . . . , wn ∈ M2n be unitaries satisfying (5.16) and (5.17), and let u : Cn → M2n be the linear map taking the basis vector ek to wk , for any 1 ≤ k ≤ n. We clearly 1 have u : Min(∞ n ) → M2n cb ≤ n, and u : Max(n ) → M2n cb ≤ 1 (see (1.12)). By interpolation, this yields u : Opn −→ M2n ≤ n1− p1 . cb By (5.17) we have n n ≤ ek ⊗ wkt k=1

M2n (Op n)

u : Opn −→ M2n , cb

hence we obtain that n 1 ek ⊗ wkt np ≤

M2n (Op n)

k=1

.

(5.27)

Let q be the conjugate of p (so p−1 +√q −1 = 1). Passing to the adjoint in (5.26), n. Since Id : Max(1n ) → Cn cb = 1, we we have Id : Min(∞ n ) → Cn cb = 1 obtain by interpolation that Id : Oqn → Cn cb ≤ n 2p . Equivalently, n ek ⊗ ek k=1

Cn (Op n)

1

≤ n 2p .

(5.28)

n ek , if τ denotes the transposition Clearly k=1 ek ⊗ wkt = I ⊗ τ ◦ u k ek ⊗ √ map on M2n . Moreover u : Cn → M2n ≤ 2 by (5.16). Thus we deduce from the easy direction of 5.4.8 that n n √ p ek ⊗ wkt ≤ 2 d (O ) e ⊗ e . Q k k n p p k=1

M2n (On )

k=1 1

Combining with (5.27) and (5.28), we see that n 2p ≤

Cn (On )

√ 2 dQ (Opn ).

2

5.5 APPLICATIONS TO THE ISOMORPHIC THEORY 5.5.1 (Strategy) In this section we will see that operator space theory can be sometimes used as a simple but eﬃcient tool in the ‘isomorphic theory’ of operator algebras. Assume that we are given a Banach algebra A and suppose that we wish to show that A is isomorphic to an operator algebra. According to Theorem 5.2.1, it suﬃces to ﬁnd one operator space structure on A (not necessarily the ‘natural one’), such that m : A ⊗h A → A is completely bounded. This reduces the task to creating the appropriate operator space structure on A. To illustrate this principle, consider A = S 1 (2 ) equipped with the usual product. We will show in the proof of 5.5.7, that Max(2 )⊗h Max(2 ) is an operator algebra structure on S 1 (2 ). Hence indeed S 1 (2 ) is isomorphic to an operator algebra. However we show in 5.5.8 that S 1 (2 ) is not completely isomorphic to an operator algebra when equipped with its usual operator space structure.

Completely isomorphic theory of operator algebras

225

5.5.2 (Schatten spaces) For any 1 ≤ p < ∞, we let S p = S p (2 ) denote the Schatten p-class on 2 (see A.1.3), considered as consisting of inﬁnite matrices with scalar entries. We equip this space with the Schur product. If a, b ∈ S p (2 ), then a ⊗ b ∈ S p (2 ⊗2 2 ), with a ⊗ bp = ap bp . Indeed |a ⊗ b|p = |a|p ⊗ |b|p . Hence (5.18) shows that a ∗ b belongs to S p , with a ∗ bp ≤ V 2 a ⊗ bp = ap bp Thus (Sp , ∗) is a Banach algebra. On the other hand we observe that S p is also a Banach algebra for the usual product of operators. Indeed this multiplication satisﬁes abp ≤ ap b∞ if a ∈ S p and b ∈ S ∞ , and hence abp ≤ ap bp if a, b ∈ S p . We shall show below that S p is isomorphic to an operator algebra, either for the Schur or the usual product. We will need the well-known fact that the Schatten spaces form an interpolation scale. Namely for any 1 < p < ∞ we have S p = [S ∞ , S 1 ] p1 .

(5.29)

Theorem 5.5.3 Let A be an algebra and a Banach space, and assume that A is ˆ isomorphic to an operator algebra. Equip S 1 with the Schur product. Then S 1 ⊗A is isomorphic to an operator algebra for the joint multiplication (see 5.3.10). Proof We let m1 , m2 , and m denote the multiplications on S 1 , A, and S 1 ⊗ A respectively. It is clear that if A and B are isomorphic as normed algebras, then ˆ and S 1 ⊗B ˆ are isomorphic as well, and so we may assume that A is an S 1 ⊗A operator algebra. Thus m2 : A ⊗h A → A is completely contractive (see 2.3.1). ˆ By 1.5.12 (1), Max(S 1 ) ⊗ A is an operator space structure on S 1 ⊗A. Thus by ˆ is isomorphic to an operator algebra provided Theorem 5.2.1, S 1 ⊗A m : (Max(S 1 ) ⊗ A) ⊗h (Max(S 1 ) ⊗ A) −→ Max(S 1 ) ⊗ A ≤ 1. (5.30) cb As Banach algebras, S 2 with Schur product equals 2N2 with pointwise multiplication. Hence (5.12) yields a complete contraction from (S 2 )r ⊗h (S 2 )c → Max(1N2 ). Moreover the ‘identity mapping’ from 1N2 into S 1 is contractive, and hence completely contractive with respect to the maximal operator space structures. Thus we obtain a complete contraction s : (S 2 )r ⊗h (S 2 )c −→ Max(S 1 ), whose restriction to S 1 ⊗ S 1 coincides with the Schur product m1 . By 1.5.14, 2 r (S ) ⊗h (S 2 )c ⊗ A = (S 2 )r ⊗h A ⊗h (S 2 )c completely isometrically. Thus s ⊗ IA extends to a complete contraction from (S 2 )r ⊗h A ⊗h (S 2 )c into Max(S 1 ) ⊗ A. Since the ‘identity mapping’ from S 1 into S 2 is contractive, we ﬁnally obtain that

226

Applications to the isomorphic theory m1 ⊗ IA : Max(S 1 ) ⊗h A ⊗h Max(S 1 ) −→ Max(S 1 ) ⊗ A ≤ 1. cb

Since ⊗ is commutative and dominates ⊗h (see 1.5.13), and since ⊗h is associative, the ‘identity mapping’ induces a complete contraction

(Max(S 1 ) ⊗ A) ⊗h (Max(S 1 ) ⊗ A) −→ Max(S 1 ) ⊗h (A ⊗h A) ⊗h Max(S 1 ). We deduce that the following are complete contractions:

(Max(S 1 ) ⊗ A) ⊗h (Max(S 1 ) ⊗ A)

IS 1 ⊗m2 ⊗IS 1

−→

m1 ⊗IA

−→

Max(S 1 ) ⊗h A ⊗h Max(S 1 )

Max(S 1 ) ⊗ A.

Thus (5.30) holds, which completes our proof.

2

We shall now derive a few consequences of Theorem 5.5.3. Corollary 5.5.4 For every 1 ≤ p ≤ ∞, the Schatten space S p equipped with the Schur product is isomorphic to an operator algebra. Proof The result holds for p = 1 by applying 5.5.3 with A = C. It holds true as well for p = ∞ by Theorem 5.3.11. Thus the result holds for any p by interpolation, using (5.29) and 5.1.10 (1). 2 Corollary 5.5.5 Let A be isomorphic to an operator algebra and let 1 ≤ p ≤ ∞ be a number. Equip p (A) with the ‘pointwise A-multiplication’ (that is, deﬁne αβ = (ak bk )k≥1 , for α = (ak )k≥1 and β = (bk )k≥1 in p (A)). Then p (A) is isomorphic to an operator algebra as well. Proof The pointwise A-multiplication on p ⊗ A is just the joint multiplication, if p is equipped with its usual pointwise multiplication. Regard 1 ⊂ S 1 as the subspace consisting of all diagonal operators. This subspace is the range of a contractive idempotent S 1 → S 1 (namely, the ‘diagonal projection’), and it easily ˆ ⊂ S 1 ⊗A ˆ isometrically. Since 1 (A) = 1 ⊗A ˆ isometrically (e.g. follows that 1 ⊗A ˆ and see [121, VIII, Example 10]), we obtain that 1 (A) is a subalgebra of S 1 ⊗A, so Theorem 5.5.3 ensures that 1 (A) is isomorphic to an operator algebra if A is. The result thus holds for p = 1. It is plain that it holds as well for p = ∞, and the general case follows again by interpolation. 2 ˆ · · · ⊗ ˆ 2 ) Regard 2 ⊂ S 1 as the subspace of all matrices 5.5.6 (The algebra 2 ⊗ with non-zero entries only on the ﬁrst row. Then this subspace is the range of a ˆ is isocontractive idempotent S 1 → S 1 . Arguing as in 5.5.5, we ﬁnd that 2 ⊗A morphic to an operator algebra provided that A is. By induction we deduce that ˆ · · · ⊗ ˆ 2 of for any integer n ≥ 1, the Banach space projective tensor product 2 ⊗ 2 n copies of is isomorphic to an operator algebra. Theorem 5.5.7 For every 1 ≤ p ≤ ∞, the Schatten space S p equipped with the usual product is isomorphic to an operator algebra.

Completely isomorphic theory of operator algebras

227

Proof The result is obvious for p = ∞, and hence by interpolation again it suﬃces to prove the result for p = 1. From (A.8) and (A.7), Max( 2 )⊗h Max(2 ) is an operator space structure on S 1 . It therefore suﬃces to show that this operator space satisﬁes Theorem 5.2.1. Indeed we will show that the usual product on S 1 induces a complete contraction m : Max(2 ) ⊗h Max(2 ) ⊗h Max(2 ) ⊗h Max(2 ) → Max(2 ) ⊗h Max(2 ). Note that the domain of m reordered by associativity is Max(2 ) ⊗h Max(2 ) ⊗h Max(2 ) ⊗h Max(2 ). For any x, y, z, t in 2 , we have m (x ⊗ y) ⊗ (z ⊗ t) = y, z2 x ⊗ t. Letting tr : 2 ⊗ 2 → C denote the usual trace, we have m = I2 ⊗ tr ⊗ I2

on 2 ⊗ 2 ⊗ 2 ⊗ 2 .

(5.31)

Since Max(2 ) ⊗h Max(2 ) = S 1 , the trace tr extends to a contractive (hence 2 completely contractive) functional on Max(2 )⊗h Max( ). Hence m is completely 2 2 2 2 contractive on Max( ) ⊗h Max( ) ⊗h Max( ) ⊗h Max(2 ). Proposition 5.5.8 With either the usual or the Schur product, S 1 ∼ = R ⊗h C is not completely isomorphic to an operator algebra. Proof Assume, by way of contradiction, that R ⊗h C, with the usual product, is completely isomorphic to an operator algebra. Writing (R ⊗h C) ⊗h (R ⊗h C) = R ⊗h (C ⊗h R) ⊗h C,

(5.32)

we see from (5.31) that the trace functional has to be bounded on C ⊗R equipped with the Haagerup tensor norm. Since C ⊗h R = S ∞ , this is a contradiction. Now let m denote the Schur product and assume that m : (R ⊗h C) ⊗h (R ⊗h C) −→ (R ⊗h C) is completely bounded. Consider the following isometric identiﬁcations, which follow successively from (5.32), (1.51), and 1.5.14 (4): CB((R ⊗h C) ⊗h (R ⊗h C),(R ⊗h C)) = CB(R ⊗h S ∞ ⊗h C, S 1 ) = CB(S ∞ , (R ⊗h S ∞ ⊗h C)∗ ) = CB(S ∞ , CB(S ∞ , B(2 ))). Let u : S ∞ → CB(S ∞ , B(2 )) be the completely bounded map corresponding to m in this identiﬁcation. Then it is easy to check (we leave this to the reader) that for any a = [aij ]i,j≥1 and b = [bij ]i,j≥1 in S ∞ , we have u(a) (b) = a ∗ bt = [aij bji ]i,j≥1 .

228

Notes and historical remarks

The desired contradiction follows since u is not bounded. To see this, given an integer n ≥ 1, let 1 a = √ [θjk ]1≤j,k≤n ∈ Mn ⊂ S ∞ , n iπ j−k n .) where θjk ∈ C with |θjk | = 1 and a = 1. (Take, for example, θ = e jk For any 1 ≤ j, k ≤ n, we have u(a) (Ekj ) = √1n θjk Ejk . Thus 1 IMn ⊗ u(a) θjk Ekj = √ [Ejk ] ∈ Mn (Mn ). n √ However θjk Ekj = 1, whereas [Ejk ]M (M ) = n. Hence u(a)cb ≥ na, n n which proves the result. 2 5.6 NOTES AND HISTORICAL REMARKS The study of Banach algebras which are/are not isomorphic to operator algebras or Q-algebras was initiated in the seventies. Among the many contributions from that time, we mention the works of Carne [81, 82], Craw and Davie [116], Dixon [123, 124], Tonge [411] and Varopoulos [414–417], which played a role in the development of the completely isomorphic theory presented in this chapter. If A is a Banach space which is also an algebra, and if α is a tensor norm on A ⊗ A, we say that A is an α-algebra if the multiplication on A extends to a bounded map m : A ⊗α A → A. With this terminology, Tonge’s result 5.2.13 says that any g2 -algebra is isomorphic to an operator algebra. Indeed several authors tried to ﬁnd a speciﬁc tensor norm α such that the class of α-algebras coincides with the class of operator algebras up to isomorphism. However Carne [81] showed that no such α exists by constructing a γ2∗ -algebra (see (A.7)) which is not isomorphic to an operator algebra. Switching from Banach spaces to operator spaces, the completely isomorphic characterization of operator algebras provided by Theorem 5.2.1 allows to ﬁnd one operator space tensor norm, namely the Haagerup norm, characterizing operator algebras. 5.1: Paulsen’s similarity result 5.1.2 is from [305, 306], and was originally motivated by the case when A = A(D) is the disc algebra. Theorem 5.1.2 for A(D) says that an operator T on a Hilbert space is similar to a contraction if (and only if) it is completely polynomially bounded, i.e. there exists a constant K ≥ 1 such that [fij (T ) ≤ K sup{[fij (z)] : z ∈ C, |z| ≤ 1} for any matrix [fij ] of polynomial. We recall here that T is said to be polynomially bounded if we merely have f (T ) ≤ K sup{|f (z)| : z ∈ C, |z| ≤ 1} for any polynomial f . Theorem 5.1.2 in the case when A is a C ∗ -algebra had been previously established by Haagerup [178]; thus a bounded homomorphism π : A → B(H) is similar to a ∗-representation if (and only if) it is completely bounded. Two famous similarity problems are attached to the situations considered above. The ﬁrst one, known as the Halmos problem, asks whether any polynomially bounded operator is similar to a contraction. This was solved in the negative by Pisier in 1997 [330]. The

Completely isomorphic theory of operator algebras

229

second one, known as the Kadison similarity problem, asks whether any bounded representation on a C ∗ -algebra A is similar to a ∗-representation. This is still open in general but has been settled in several important cases. For example, the answer is positive if A = B(H) [178], or if A is nuclear [78, 91] (as noted in 5.1.3). The main references for similarity problems and their relationships with operator spaces are [335] (see also [332] and Pisier’s series of papers on the similarity degree of an operator algebra [333,334,336]), and [314] (and references therein). Other developments may be found in [250, 251]. The result 5.1.9 is from [44]. Its proof also shows that CB(X) is completely isometrically isomorphic to an operator algebra if and only if X is a (completely isometric to) a column Hilbert space. Similar proofs show that given a Banach space E, the Banach algebra B(E) is isomorphic (resp. isometrically isomorphic) to an operator algebra if and only if E is isomorphic (resp. isometrically isomorphic) to a Hilbert space. Alternatively, note that if B(E) is isometric to an operator algebra then it satisﬁes von Neumann’s inequality. But this was shown by Foia¸s to imply that X is Hilbertian [29, p. 230]. 5.2: Theorem 5.2.1 is due to Blecher [44]. Some of its variants discussed here (e.g. 5.2.5) are taken from [62]. The proof presented here emphasizes the analogy with techniques developed in the seventies for the study of operator algebras up to isomorphism. A breakthrough result from that period is Craw’s Lemma (from [116]) characterizing Q-algebras up to isomorphism by polynomial inequalities. That lemma formally corresponds to 5.4.3 (2). Its extension to operator algebras presented in 5.2.11 was obtained by Dixon [124] and Varopoulos [417] independently. In this respect, Proposition 5.2.5 should be regarded as an operator space version of Craw’s Lemma. The idea of reducing computations to homogeneous polynomials goes back to Davie [116]. Lemma 5.2.7 was inspired by [417], and 5.2.12 is taken from the same paper. It is remarkable that the latter result was proved before operator spaces existed. Another key result used throughout this section is the fact that quotients of operator algebras are again operator algebras—see the Notes to Section 2.3 for historical comments on that result. Pisier presents in [337] a variant on the proof of 5.2.1 that ﬁts within the context of his ‘universal operator algebra of an operator space’. The dual version 5.2.16 is due to Le Merdy [248], whereas the completely isomorphic characterization of operator bimodules in 5.2.17 is due to Blecher [46]. Any ﬁnite-dimensional algebra (resp. ﬁnite-dimensional algebra with an operator space structure) is isomorphic (resp. completely isomorphic) to an operator algebra, and indeed to a subalgebra of Mn . This follows easily from 5.2.1, but in fact has a simple direct proof (e.g. see the proof of 2.11 in [155]). 5.3: The examples and results that we present in this section (except 5.3.11) are mostly from [62]. Other examples of operator algebra structures can be found in that paper. That p equipped with the pointwise multiplication is isomorphic to a Q-algebra (and hence to an operator algebra) was obtained independently by Davie [116] in the case when p ≤ 2, and Varopoulos [415] in the general case. Both proofs use Craw’s Lemma; however the idea of using interpolation as in

230

Notes and historical remarks

the proof of 5.3.5 is from [415]. As a consequence of his isomorphic characterization of operator algebras (see 5.2.12), Varopoulos proved in [417] that B(H) with the Schur product is isomorphic to an operator algebra. In [69] it was checked that the Schur multiplication ∗ on B(H) is completely contractive, and it was shown that (B(H), ∗) cannot be isometrically isomorphic to an operator algebra. The proof is similar to the one in 5.3.6. 5.4: There exist commutative operator algebras which are not isomorphic to a Q-algebra. This follows from some remarkable work of Varopoulos [416] on von Neumann’s inequality in more than three variables. Namely Varopoulos showed that for any constant K ≥ 1, there exist an integer n = n(K), a homogeneous polynomial F of degree 3 in n variables, and an n-tuple (a1 , . . . , an ) of commuting contractions on some Hilbert space, such that F (a1 , . . . , an ) ≥ K sup |F (z1 , . . . , zn )| : zk ∈ C, |zk | ≤ 1 = K F C . Apply the above result for each integer K ≥ 1, and let AK be the operator algebra generated by a1 , . . . , an . Then their direct sum A = ⊕K AK is a commutative operator algebra, which cannot be isomorphic to a Q-algebra. Indeed if A were isomorphic to a Q-algebra, there would exist by 5.4.3 (2) two constants δ, M > 0 such that F A,δ ≤ M F C for any n ≥ 1 and any F ∈ P[Z1 , . . . , Zn ]. If F is homogeneous of degree 3, then F A,δ = δ 3 F A,1 . Hence we ﬁnd that for any K ≥ 1, for any n ≥ 1, and for any homogeneous polynomial F of degree 3, we have F A,1 ≤ M δ −3 F C . This contradicts the above estimate. Our main characterization of Q-algebras (Theorem 5.4.5) is taken from [62]. Again the proof borrows some ideas from the isomorphic theory developed in the 1970s. In particular, the idea of symmetrization is from [116]. Recent interest in Q-spaces apparently started from the noncommutative Khintchine inequalities of Lust-Piquard and Pisier [260], on the Schatten space S 1 . Let (gk )k≥1 be a sequence of complex valued independent standard Gaussian random variables on a probability space (Ω, µ), and let G ⊂ L1 (Ω) be the closed linear span of the gk ’s. As explained in [331, Section 8.3], the main result from [260] has the following consequence: if we regard G as an operator subspace of Max(L1 (Ω)), then the linear mapping taking ek to gk for each k ≥ 1 extends to a complete isomorphism G ≈ R + C, where R + C denotes the dual operator space of R ∩ C. By duality, one obtains that R ∩ C ≈ L∞ (Ω)/G⊥ is a Q-space. Shortly after this nontrivial example was discovered, Pisier conjectured the characterization of Q-spaces stated as 5.4.8. This question was eventually settled by Junge, but published in [327]. The latter paper also characterizes quotients of subspaces of C ∗ -algebras of the form MN (C(Ω)) for a ﬁxed integer N ≥ 1. Further developments may be found in [206]. The short proof of 5.4.8, based on the relationship between Q-spaces and Q-algebras, is from [62]. The interplay between the noncommutative Grothendieck inequality and the representation of R ∩ C as a Q-space, appears in [260]. New results on Q-spaces related to 5.4.5 appear in Ricard’s work [357, 358].

Completely isomorphic theory of operator algebras

231

Theorem 5.4.10 and its applications to the spaces [Min(∞ ), Min(1 )] p1 and to R ∩ C were obtained in [62]. In 5.4.16, part (2) is a well-known consequence of 5.4.8, whereas parts (1) and (3) are from [62]. The proof of the latter part is related to some ideas from [308]. In particular, the fact that the completely bounded norm of the identity mapping from Min(1n ) into Max(1n ) is dominated √ by n is due to Paulsen. 5.5: The fact that S p is isomorphic to an operator algebra with either the usual or the Schur product was proved in [62], where the reader will ﬁnd more discussion on operator algebra structures on S p . This includes the negative result 5.5.8. The matrix appearing in the proof of that result, showing that u is not bounded, was provided by Smith. That S 1 with its ‘natural’ operator space structure, and either the usual or the Schur product, is not completely isomorphic to an operator algebra may explain why 5.5.4 and 5.5.7 remained unnoticed in the seventies. As an application of Theorem 5.2.16, we show now that S 1 with either the usual or the Schur product is actually w ∗ -homeomorphic and isomorphic to a dual operator algebra. Indeed the Schur product and the usual product are both separately w∗ -continuous on S 1 . Hence by 5.2.16 and the proofs of 5.5.3 and 5.5.7, it suﬃces to check that Max(S 1 ) and Max(2 ) ⊗h Max(2 ) are both dual operator spaces. This is clear for the ﬁrst of these spaces, which is the dual of Min(S ∞ ) by 1.4.12. The second case is a bit more delicate. Arguing as in the proof of (1.60), one can show that Max(2 ) ⊗w∗ h X = Max(2 ) ⊗h X for any operator space X. In particular Max(2 ) ⊗w∗ h Max(2 ) = Max(2 ) ⊗h Max(2 ). Hence the latter space is the dual of Min(2 ) ⊗h Min(2 ), by (1.58). Theorem 5.5.3 and its consequences 5.5.5 and 5.5.6 are new (but easy) extensions of results from [62]. Since S p with the Schur product is isomorphic to an operator algebra and is commutative, it is natural to ask if it is actually isomorphic to a Q-algebra. This problem was ﬁrst raised by Varopoulos in [417] in the case p = ∞. It is proved that S p is indeed isomorphic to a Q-algebra for 2 ≤ p ≤ 4 in [247] and for 1 ≤ p ≤ 2 in [322]. The question for p > 4 is apparently still open at the time of this writing.

6 Tensor products of operator algebras

6.1 THE MAXIMAL AND NORMAL TENSOR PRODUCTS Tensor products and C ∗ -norms play a prominent role in the theory of C ∗ algebras, in particular in the study of nuclear C ∗ -algebras and semidiscrete (or injective) von Neumann algebras. Our major goal in this chapter is to extend part of that theory to nonselfadjoint operator algebras, and to give some applications. Recall that if A and B are operator algebras, then their algebraic tensor product A ⊗ B is an algebra with the product deﬁned by ai ⊗ b i cj ⊗ dj = ai cj ⊗ bi dj , i

j

i,j

for ﬁnite families (ai )i , (cj )j in A, and (bi )i , (dj )j in B. It follows from 2.2.2 that A ⊗min B with this product is an operator algebra. In this ﬁrst section, we shall consider some other operator algebra tensor norms on A⊗ B. For simplicity in this chapter, we will usually assume that our operator algebras are at least approximately unital. 6.1.1 (Maximal tensor product) Suppose that H is a Hilbert space, that X and Y are operator spaces, and that Φ : X → B(H) and Ψ : Y → B(H) are completely bounded maps. Following 5.2.6, we write Φ • Ψ : X ⊗ Y −→ B(H) for the linear mapping taking x ⊗ y to Φ(x)Ψ(y), for any x ∈ X and y ∈ Y . In this chapter we are often concerned with the case that Φ and Ψ have commuting ranges, that is, Φ(x)Ψ(y) = Ψ(y)Φ(x) for all x ∈ X, y ∈ Y . If A and B are approximately unital operator algebras, deﬁne I = I(A, B) to be the collection of all pairs (π, ρ) of completely contractive representations π : A → B(H) and ρ : B → B(H), for a Hilbert space H, such that π and ρ have commuting ranges. For any integer n ≥ 1, and any y ∈ Mn (A ⊗ B), we deﬁne

(6.1) |||y|||n = sup IMn ⊗ (π • ρ) (y)Mn (B(H)) : (π, ρ) ∈ I . It is clear that each |||· |||n is a seminorm. To show that this is a norm, assume that A and B are represented as operator algebras on H1 and H2 respectively. Thus

Tensor products of operator algebras

233

A ⊗min B ⊂ B(H1 ⊗2 H2 ) completely isometrically, by 1.5.2. The two mappings π : A → B(H1 ⊗2 H2 ) and ρ : B → B(H1 ⊗2 H2 ) deﬁned by π(a) = a ⊗ IH2 and ρ(b) = IH1 ⊗b, are completely isometric homomorphisms with commuting ranges. By the above, (IMn ⊗ (π • ρ))(y) = ymin, for y ∈ Mn (A ⊗ B). Hence |||· |||n dominates the minimal norm, and is therefore also a norm on Mn (A ⊗ B). We let A⊗max B be the completion of A⊗B in the norm |||· |||1 , and we call A⊗max B the maximal tensor product of A and B. By 2.3.3, A ⊗max B is an operator algebra. The matrix norms on A ⊗max B will be often denoted by · max . It is easily seen that A ⊗max B = B ⊗max A as operator algebras. Proposition 6.1.2 Let A and B be approximately unital operator algebras. Then A ⊗max B is an approximately unital operator algebra, and the identity mapping on A ⊗ B extends to complete contractions A ⊗h B −→ A ⊗max B −→ A ⊗min B. If further A and B are unital, then A⊗max B is unital as well (with unit 1A ⊗1B ). Proof Most of this was shown in 6.1.1. Arguing as in 2.2.2 shows that A⊗ max B is approximately unital. By the easy direction of the CSPS theorem (see 1.5.7), together with (6.1), the canonical map from A × B to A ⊗max B is completely contractive. Thus by 1.5.4, we see that the identity mapping on A ⊗ B extends 2 to a completely contractive linear map from A ⊗h B to A ⊗max B. 6.1.3 (Normal tensor product) Let A be an approximately unital operator algebra and let M be a unital dual operator algebra. We let I nor = I nor (A, M ) be the collection of all pairs (π, ρ) ∈ I(A, M ) such that ρ is w ∗ -continuous. For any n ≥ 1, and any y ∈ Mn (A ⊗ M ), we deﬁne

ynor = sup IMn ⊗ (π • ρ) (y)Mn (B(H)) : (π, ρ) ∈ I nor . As in 6.1.1 we see that these are matrix norms on A ⊗ B. We let A ⊗ nor M be the resulting completion. Arguing as in 6.1.1, we see that A ⊗nor M is an approximately unital operator algebra, which is unital if A is unital. Moreover we have a canonical sequence of completely contractive homomorphisms A ⊗max M −→ A ⊗nor M −→ A ⊗min M.

(6.2)

For an approximately unital operator algebra B, it easily follows from the extension principle 2.5.5 that A ⊗max B ⊂ A ⊗nor B ∗∗ completely isometrically. Consequently we also have A ⊗max B ⊂ A ⊗max B ∗∗ completely isometrically. 6.1.4 (The selfadjoint case) Assume that A and B are C ∗ -algebras. Then according to 1.2.4, I(A, B) is the collection of all pairs of commuting ∗-representations π : A → B(H) and ρ : B → B(H). Hence in this case, A ⊗max B is the classical maximal tensor product from the C ∗ -theory (e.g. see [407, IV.4]). Indeed in this case, · max is the maximal C ∗ -norm on A ⊗ B, and A ⊗max B is a C ∗ algebra. Likewise, if A is a C ∗ -algebra and M is a W ∗ -algebra, then A ⊗nor M is the normal tensor product considered in [140].

234

The maximal and normal tensor products

Lemma 6.1.5 Suppose that A and B are closed approximately unital subalgebras of an operator algebra D, such that ab = ba for all a ∈ A, b ∈ B. Let C = AB, also a closed subalgebra of D. If θ : C → B(H) is a completely contractive homomorphism, then there exist two commuting completely contractive homomorphisms π : A → B(H) and ρ : B → B(H), such that θ(ab) = π(a)ρ(b) for a ∈ A and b ∈ B, and [θ(C)] = [π(A)] ∩ [ρ(B)] . If, further, θ is nondegenerate and completely isometric (see 2.1.5), then π and ρ map into the canonical copy (see Section 2.6) of the multiplier algebra M (C) inside B(H). Proof If A and B are unital with identities e and f respectively, then ef is an identity for C. In this case this result is then obvious, by simple algebra. In the general case we will use second duals and their units, viewing A∗∗ and B ∗∗ as subalgebras of D ∗∗ . The canonical separately w ∗ -continuous product on D∗∗ restricts to a separately w ∗ -continuous map on A∗∗ × B ∗∗ . This map has range inside C ∗∗ by a routine density argument using (2.21). For θ as above, let θ˜: C ∗∗ → B(H) be the canonical w∗ -continuous extension of θ. We deﬁne w : A∗∗ ×B ∗∗ → B(H) to be the composition of the last two maps. Let e and f be the units of A∗∗ and B ∗∗ respectively. Deﬁne π : A → B(H) and ρ : B → B(H) by π(a) = w(a, f ) and ρ(b) = w(e, b), for a ∈ A and b ∈ B. Let (et )t and (fs )s be cai’s for A and B respectively. By Proposition 2.5.8, et → e in the w∗ -topology of A∗∗ . If a ∈ A and b ∈ B, then ρ(b) is the w ∗ limit of w(et , b) = θ(et b), and π(a) is the w∗ -limit of w(a, fs ) = θ(afs ). Since θ(aet fs b) = θ(afs )θ(et b), we deduce by taking iterated w ∗ -limits over s and t, that θ(ab) = π(a)ρ(b). A similar argument, beginning with θ(bfs et a), shows that θ(ab) = ρ(b)π(a). Thus π and ρ have commuting ranges. Suppose that θ is a nondegenerate embedding. For a ∈ A we have π(a )θ(ab) = w∗ − lim θ(a fs )θ(ab) = w∗ − lim θ(a afs b) = θ(a ab). Thus π(a )θ(C) ⊂ θ(C), and a similar argument shows that θ(C)π(a ) ⊂ θ(C). Thus π maps into M (C). Similarly, ρ maps into M (C). The arguments for the remaining assertions of the lemma are similar, using iterated limits involving the cai’s, and are left as an exercise. 2 6.1.6 (Reduction to the unital case) Tensor products of operator algebras A and B are a little easier to deal with if A and B are unital, mostly because in this case there are canonical embeddings of A and B into the tensor product. In fact there are useful techniques to reduce many situations to the unital case, which are similar to what happens in the well-known selfadjoint case. We will not take the time to develop this topic fully, in fact we will content ourselves with proving here just a little more than is needed for this chapter. Our ﬁrst observation is that if C 1 is the unitization from Section 2.1 of an algebra C, then it is easy to see from Meyer’s theorem from that section, together with (6.1), that A⊗max B ⊂ (A⊗max B)1 ⊂ A1 ⊗max B 1 , completely isometrically as subalgebras, if A and B are nonunital. A similar result holds for the minimal tensor product.

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Next we notice that if A and B are not unital, but are approximately unital, then there are useful embeddings of A and B into the multiplier algebra of the tensor product. To see this in the case of the maximal tensor product, apply the previous lemma with C = A ⊗max B, and D = A1 ⊗max B 1 . Here we are identifying A with A ⊗ 1 and B with 1 ⊗ B. We obtain canonical commuting completely contractive homomorphisms π : A → M (C) and ρ : B → M (C). To see that π is isometric, for example, note that if b ∈ B with b = 1, and if (e t ) is a cai for A, then π(a) ≥ π(a)(et ⊗ b) = (aet ) ⊗ b = aet . Taking the limit shows that π is isometric. A similar argument shows that π is completely isometric. Again, similar arguments establish the analoguous facts for A ⊗min B. The following follows immediately from the above: Corollary 6.1.7 Let A and B be approximately unital operator algebras, let H be a Hilbert space and let θ : A ⊗max B → B(H) be a completely contractive homomorphism. Then there exist two commuting completely contractive homomorphisms π : A → B(H) and ρ : B → B(H) such that θ = π • ρ and [θ(A ⊗ B)] = [π(A)] ∩ [ρ(B)] . If further A, B and θ are unital, this may be achieved with π and ρ being unital. 6.1.8 (Universal properties) Suppose that A and B are approximately unital operator algebras, set C = A ⊗max B, and let i be the canonical map from A ⊗ B to C. Let ν : A → M (C) and κ : B → M (C) be the canonical embeddings (see 6.1.6), then i = ν • κ. In fact, C = A ⊗max B is uniquely determined by the following universal property. The algebra C is a matrix normed algebra, there exist completely contractive homomorphisms ν : A → M (C) and κ : B → M (C), with commuting ranges, the mapping i = ν • κ : A ⊗ B → M (C) has range in C, and indeed which is dense in C, and the following property holds: Given any operator algebra D, and any completely contractive homomorphisms π : A → D and ρ : B → D whose ranges commute, there exists a (necessarily unique) completely contractive homomorphism θ : A ⊗max B → D such that θ ◦ i = π • ρ. That this property indeed characterizes the maximal tensor product up to completely isometric isomorphism, follows easily using the usual type of algebraic argument for proving such results. Of course if A and B are unital, then C = M (C). Let B1 , B2 be two unital matrix normed algebras, and consider the unital algebra B1 ⊗ B2 . By an argument similar to that of 3.1.5 (3), B1 ⊗ B2 is a matrix normed algebra. Combining 2.4.7 and the above paragraph, it is easy to deduce that O(B1 ⊗ B2 ) = O(B1 ) ⊗max O(B2 ) as operator algebras. Now let G1 , G2 be discrete semigroups, and apply that result with B1 = Max(1G1 ) and B2 = Max(1G2 ). Using 2.4.9, we deduce that O(G1 ) ⊗max O(G2 ) is equal to the

enveloping operator algebra of Max(1G1 ) ⊗ Max(1G2 ). The latter space is equal

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ˆ 1G ) by 1.5.12 (2), and hence equals Max(1G ×G ) by [121, VIII, to Max(1G1 ⊗ 2 1 2 Example 10]. Thus we obtain that as operator algebras, we have O(G1 × G2 ) = O(G1 ) ⊗max O(G2 ). 6.1.9 (Functoriality) Let (A, B) and (C, D) be two pairs of approximately unital operator algebras and let π : A → C and ρ : B → D be completely contractive homomorphisms. Then the mapping π ⊗ ρ : A ⊗ B → C ⊗ D extends to a completely contractive homomorphism from A ⊗max B into C ⊗max D. We leave this as an exercise using either the deﬁnition of the tensor product, or its universal property above. In particular if A and B are subalgebras of C and D respectively, then the embedding A ⊗ B ⊂ C ⊗ D extends to a complete contraction A ⊗max B → C ⊗max D. In general, this mapping is not an isometry. That is, the maximal tensor product is not ‘injective’. This point will be discussed in more detail in Section 6.2. We note however that if B is a C ∗ -algebra, then we have ∗ A ⊗max B ⊂ Cmax (A) ⊗max B

completely isometrically.

(6.3)

Indeed let y ∈ Mn (A ⊗ B) for some n ≥ 1, and consider two commuting completely contractive representations π : A → B(H) and ρ : B → B(H). We let D = [ρ(B)] ⊂ B(H) be the commutant of the range of ρ. Since B is a C ∗ algebra, the mapping ρ is a ∗-representation (by 1.2.4), hence D is a C ∗ -algebra. ∗ (A). Since π is valued in D, it follows from Proposition 2.4.2 that it Let C = Cmax extends to a ∗-representation π ˜ : C → D ⊂ B(H). Thus π ˜ and ρ are commuting representations, and hence IMn ⊗ (π • ρ) (y) = IMn ⊗ (˜ π • ρ) (y) ≤ yMn (C⊗max B) . We deduce that yMn (A⊗max B) ≤ yMn(C⊗max B) by taking the supremum over all possible pairs (π, ρ). This proves (6.3). The above results also have ‘normal’ versions. Thus if π : A → C is a completely contractive homomorphism between approximately unital operator algebras, and if ρ : M → N is a w∗ -continuous completely contractive homomorphism between unital dual operator algebras, then we obtain a canonical completely contractive homomorphism π ⊗ ρ : A ⊗nor B → C ⊗nor N . Moreover arguing as above, we see that if M is a W ∗ -algebra, we have ∗ A ⊗nor M ⊂ Cmax (A) ⊗nor M

completely isometrically.

(6.4)

If u : B → C is a completely bounded linear map between operator algebras, then in general IA ⊗ u need not be bounded as a map from A ⊗max B into A ⊗max C. (See the Notes to Section 6.5 for more on this). However, we have: Lemma 6.1.10 Let B and C be C ∗ -algebras, and let u : B → C be a completely positive map. Then for any approximately unital operator algebra A, we have IA ⊗ u : A ⊗max B −→ A ⊗max C ≤ ucb = u. (6.5) cb

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Proof This result is well-known if A is a C ∗ -algebra (e.g. see [407, IV.4.23]). In that case, IA ⊗ u : A ⊗max B → A ⊗max C is actually completely positive. Suppose now that A is an approximately unital operator algebra. By (6.3), ∗ we may regard IA ⊗ u as the restriction of ICmax (A) ⊗ u to A ⊗max B. Then the 2 result follows from the C ∗ -algebra case. 6.1.11 (Reduction to the nondegenerate case) Let A, B be approximately unital operator algebras, and let I 0 be the collection of all pairs (π, ρ) in I(A, B) such that π and ρ are nondegenerate (see 2.1.5). Then the deﬁnition (6.1) of the matrix norms on A ⊗max B is not changed if the supremum is taken over all (π, ρ) in I 0 (instead of over all (π, ρ) in I). To prove this, it suﬃces to check that if π : A → B(H) and ρ : B → B(H) are two commuting completely contractive homomorphisms, then there exists a subspace K of H, and commuting nondegenerate completely contractive homomorphisms π : A → B(K) and ρ : B → B(K), such that (π • ρ)(y) = (π • ρ )(y)PK , for y ∈ A ⊗ B. To see this, let Kπ = [π(A)H], Kρ = [ρ(B)H], and K = Kπ ∩ Kρ . We let p and q be the projections onto Kπ and Kρ respectively, and let (et )t and (fs )s be cai’s for A and B respectively. We have π(et )ρ(fs ) = ρ(fs )π(et ) for any t, s. By Lemma 2.1.9, π(et ) → p and ρ(fs ) → q in the w∗ -topology of B(H). We deduce that p and q commute. Then e = pq = qp is the projection onto K, and e = w∗ − lim lim ρ(es )π(et ) = w∗ − lim lim π(et )ρ(es ). s

t

t

s

(6.6)

We deﬁne completely contractive mappings π : A → B(K) and ρ : B → B(K) by setting π (a) = eπ(a)e and ρ (b) = eρ(b)e, for a ∈ A and b ∈ B. Since π and ρ commute, we deduce from (6.6) that eπ(a) = π(a)e. It is clear from this commutation property that π is a homomorphism. Likewise, ρ is a homomorphism, and ρ(b)e = eρ(b) for b ∈ B. Moreover, π and ρ have commuting ranges. We now check that π and ρ are nondegenerate. For a ∈ A and b ∈ B, we have lims limt ρ(fs )π(et )π(a)ρ(b) = π(a)ρ(b). Thus by (6.6), eπ(a)ρ(b) = π(a)ρ(b). On the other hand, π(et )ζ → ζ and ρ(es )ζ → ζ, for ζ ∈ K. Hence lims limt ρ(fs )π(et )ζ = ζ. Thus we have K = [π(A)ρ(B)H]. Since π (A)π(A)ρ(B) = π(A)ρ(B), we deduce the desired equality [π (A)K] = K. Likewise, [ρ (B)K] = K. Finally note that if a ∈ A, b ∈ B, then π(a)ρ(b) = eπ(a)ρ(b) = eπ(a)eρ(b)e = π (a)ρ (b)e, which is what was needed. 6.1.12 (The iterated maximal tensor product) Let A1 , . . . , AN be approximately unital operator algebras. Given pairwise commuting completely contractive homomorphisms πk : Ak → B(H), let π1 •· · ·•πN : A1 ⊗· · ·⊗AN → B(H) be the map taking a1 ⊗· · ·⊗aN to π1 (a1 ) · · · πN (aN ), for any a1 ∈ A1 , . . . , aN ∈ AN . Given n ∈ N, and y ∈ Mn A1 ⊗ · · · ⊗ AN ), deﬁne

ymax = sup IMn ⊗ (π1 • · · · • πN )(y)M (B(H)) , n

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The maximal and normal tensor products

where the supremum runs over all such families (π1 , . . . , πN ). Arguing as in 6.1.2, we see that max is a norm on each Mn A1 ⊗ · · · ⊗ AN ), and that the resulting completion is an approximately unital operator algebra. The latter is denoted by A1 ⊗max · · · ⊗max AN . The next result, states that ⊗max is associative. We leave the proof as an exercise (using Corollary 6.1.7). Lemma 6.1.13 For approximately unital operator algebras A, B, C, we have (A ⊗max B) ⊗max C = A ⊗max B ⊗max C = A ⊗max (B ⊗max C). completely isometrically. By deﬁnition, a C ∗ -algebra B is nuclear if A ⊗min B = A ⊗max B for any C ∗ -algebra A. Likewise, a W ∗ -algebra M is semidiscrete if A⊗min M = A⊗nor M for any C ∗ -algebra A (see 6.6.1 for more on this ‘deﬁnition’). We shall see next that these properties are unchanged if we allow A to be nonselfadjoint. Proposition 6.1.14 (1) A C ∗ -algebra B is nuclear if and only if A ⊗max B = A ⊗min B completely isometrically for any approximately unital operator algebra A. (2) A W ∗ -algebra M is semidiscrete if and only if A ⊗nor M = A ⊗min M completely isometrically for any approximately unital operator algebra A. Proof Assume that B is nuclear. By the injectivity of the minimal tensor ∗ product, we have A ⊗min B ⊂ Cmax (A) ⊗min B completely isometrically. Since ∗ ∗ Cmax (A) ⊗min B = Cmax (A) ⊗max B, the equality A ⊗min B = A ⊗max B follows from (6.3). The proof of (2) is similar using (6.4). 2 Corollary 6.1.15 Let A, B be approximately unital operator algebras, let M be a unital dual operator algebra, and let n ≥ 1 be an integer. The following identities hold completely isometrically: Mn (A ⊗max B) = Mn (A) ⊗max B;

(6.7)

Mn (A ⊗nor M ) = Mn (A) ⊗nor M = A ⊗nor Mn (M ).

(6.8)

Proof Since Mn is nuclear, (6.7) follows from 6.1.13 and 6.1.14. Let y ∈ Mn (A ⊗ M ) for some n ≥ 1, and let (π, ρ) ∈ I nor (A, M ) be a pair of representations on some Hilbert space H. We deﬁne ρ˜ : M → Mn (B(H)) by letting ρ˜(b) = 1 ⊗ ρ(b), for b ∈ M . Here 1 denotes the unit of Mn . Then we have IMn ⊗ (π • ρ) = (IMn ⊗ π) • ρ˜ on Mn (M ). Since ρ˜ is w∗ -continuous, we deduce that (IMn ⊗ (π • ρ))y ≤ yMn(A)⊗nor M . Taking the supremum over all (π, ρ) ∈ I nor (A, M ), we deduce that yMn (A⊗nor M) ≤ yMn (A)⊗nor M . Conversely, let (θ, ρ) ∈ I nor (Mn (A), M ) be a pair of representations on H, and recall the identiﬁcation Mn (A) = Mn ⊗min A from 2.2.3. Since Mn is a nuclear C ∗ -algebra, we may equally write Mn (A) = Mn ⊗max A by Proposition 6.1.14. By Lemma 6.1.7 we may therefore write θ = σ • π, where σ : M n → B(H) and π : A → B(H) are completely contractive representation such that σ, π and

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ρ have commuting ranges. By deﬁnition, π • ρ : A ⊗ M → B(H) extends to a completely contractive homomorphism on A ⊗nor M . Its range commutes with the range of σ. Hence in turn, θ•ρ = σ•(π•ρ) extends to a completely contractive homomorphism on Mn ⊗max (A ⊗nor M ). Again since Mn is nuclear, the latter space coincides with Mn (A ⊗nor M ). Thus θ • ρ(y) ≤ yMn (A⊗nor M) , which leads to yMn (A)⊗nor M ≤ yMn (A⊗nor M) . The above shows that Mn (A ⊗nor M ) = Mn (A) ⊗nor M isometrically. The complete isometry follows from the isometry by replacing n with mn, and re2 placing A by Mm (A). The proof for the second identity in (6.8) is similar. 6.2 JOINT DILATIONS AND THE DISC ALGEBRA In this section we will emphasize the interplay between dilations (or more precisely, joint dilations) and the maximal tensor product. A basic illustration of this phenomenon is provided by Ando’s theorem reviewed in 2.4.13. As observed in 2.4.14, Ando’s result is equivalent to the completely isometric equality O(N20 ) = A(D2 ). We also noticed that O(N) = A(D) by Nagy’s dilation theorem. Hence by the centered formulae in 6.1.8 and 2.4.13, we have A(D) ⊗max A(D) = A(D) ⊗min A(D)

(6.9)

completely isometrically. To investigate other such relationships, we will need the following general dilation principle due to Arveson. 6.2.1 (Arveson’s dilation theorem) Let A be a unital-subalgebra of a unital operator algebra B. Let π : A → B(H) be a unital completely contractive representation on some Hilbert space H. If in turn B is a unital-subalgebra of a unital C ∗ -algebra C, then we can regard A + A as an operator subsystem of C. By Lemma 1.3.6, π (uniquely) extends to a complete contraction φ : A+A → B(H). This extension is unital, hence completely positive (see 1.3.2). By Arveson’s extension theorem 1.3.5, φ extends to a (necessarily unital) completely positive map φˆ : C → B(H). Applying Stinespring’s factorization theorem (1.3.4) to this map, there is a Hilbert space K containing H as a subspace, and a ∗-representation ˆ ) = PH π ˆ (· )|H . Restricting to B ⊂ C, we have π ˆ : C → B(K) such that φ(· obtained a unital completely contractive representation π ˆ : B → B(K) and an ˆ (a)J, for a ∈ A. In this isometric embedding J : H → K such that π(a) = J ∗ π situation, we say that π ˆ : B → B(K) is a B-dilation of π. 6.2.2 (Completely contractive representations of A(D)) In 2.4.12 we showed that Nagy’s dilation theorem implies the ‘matricial von Neumann inequality’ described there, that is, the polynomial functional calculus f ∈ P → f (T ) extends to a unital completely contractive homomorphism uT : A(D) → B(H) if T ∈ B(H) is a contraction. Using 6.2.1 we now note that these two results are in fact equivalent. Indeed, if u : A(D) → B(H) is a completely contractive homomorphism, then T = u(z) is a contraction, Also, u = uT , and [T ] = [uT (A(D))] . Applying 6.2.1 with A = A(D) and B = C(T), and using the fact (obvious from

240

Joint dilations and the disc algebra

spectral theory) that unitaries in B(K) correspond to unital ∗-homomorphisms from C(T) to B(K), we recover Nagy’s dilation theorem. Proposition 6.2.3 Let A and B be unital-subalgebras of unital operator algebras C and D respectively. The following are equivalent. (i) A ⊗max B ⊂ C ⊗max D completely isometrically. (ii) For any Hilbert space H and for any pair of commuting unital completely contractive homomorphisms π : A → B(H) and ρ : B → B(H), there exist a Hilbert space K, an isometry J : H → K, and commuting unital completely contractive homomorphisms π ˆ : C → B(K) and ρˆ: D → B(K) such that ˆ (a)ˆ ρ(b)J, π(a)ρ(b) = J ∗ π

a ∈ A, b ∈ B.

(6.10)

Proof Assume (i), and let π : A → B(H) and ρ : B → B(H) be as in (ii). Then π • ρ : A ⊗ B → B(H) extends to a unital completely contractive homomorphism θ : A ⊗max B → B(H). Since A ⊗max B is a unital subalgebra of C ⊗max D, Arveson’s theorem 6.2.1 ensures that θ admits an (C ⊗max D)-dilation θˆ from C ⊗max D to B(K). By 6.1.7 we may write θˆ = π ˆ • ρˆ for a pair of commuting unital completely contractive representations π ˆ : C → B(K) and ρˆ : D → B(K). This pair clearly satisﬁes (6.10). Conversely, assume (ii). Let π : A → B(H) and ρ : B → B(H) be two commuting unital completely contractive homomorphisms, for some Hilbert space H. Let π ˆ : C → B(K) and ρˆ : D → B(K) be dilations of π and ρ provided by π • ρˆ(y))J, for y ∈ A ⊗ B, and hence (ii). Then π • ρ(y) = J ∗ (ˆ π • ρ(y) ≤ ˆ π • ρˆ(y) ≤ yC⊗maxD . Taking the supremum over all pairs (π, ρ) as above and applying 6.1.11, we deduce that yA⊗maxB ≤ yC⊗maxD . Thus the completely contractive embedding A ⊗max B → C ⊗max D provided by 6.1.9 is an isometry. A similar proof shows that it is a complete isometry. 2 The following provides a class of embeddings satisfying the equivalent conditions of the last result. We use Dirichlet operator algebras (see 4.3.9 and 4.3.10). Proposition 6.2.4 Let B be a Dirichlet operator algebra. Then for any C ∗ algebra A we have A ⊗max B ⊂ A ⊗max Ce∗ (B) completely isometrically. Proof Set D = Ce∗ (B). By assumption, the operator system B + B is dense in D. Let A be a C ∗ -algebra, and let π : A → B(H) and ρ : B → B(H) be commuting completely contractive representations. By 6.1.11 we may assume that ρ is unital. By Lemma 1.3.6, ρ admits a (unique) completely positive extension to B + B . We may extend ρ further by density to a completely positive contraction ρ˜ : D → B(H). On the other hand, let C = [π(A)] ⊂ B(H) be the commutant

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241

of the range of π. Since π is a ∗-representation (see 1.2.4), C is selfadjoint. Thus Ran(˜ ρ) ⊂ C (since ρ maps into C, and ρ˜(b∗ ) = ρ˜(b)∗ for b ∈ B). We deduce that IA ⊗ ρ˜ : A ⊗max D −→ A ⊗max C = 1, cb by Lemma 6.1.10. Let Φ : A ⊗max C → B(H) be the contraction taking a ⊗ c to π(a)c, for any a ∈ A and c ∈ C. Then π • ρ is the restriction of Φ ◦ (IA ⊗ ρ˜), and so (π • ρ)n (y) ≤ Φ((IA ⊗ ρ˜)(y)). Taking the supremum over all pairs (π, ρ) and applying 6.1.11, we obtain that yA⊗maxB ≤ yA⊗max D . Thus the complete contraction from A ⊗max B to A ⊗max D is an isometry. Replacing A by Mn (A), and applying Corollary 6.1.15, we conclude that it is actually a complete isometry. 2 6.2.5 (Dirichlet uniform algebras) Assume that B ⊂ C(Ω) is a uniform algebra on some compact space Ω. Let A be a unital-subalgebra of a unital operator algebra C. By 6.1.14 (1), and since commutative C ∗ -algebras are nuclear, we have C ⊗max C(Ω) = C ⊗min C(Ω). Hence by the injectivity of the minimal tensor product, we have A ⊗min B ⊂ C ⊗max C(Ω) completely isometrically. Thus the canonical complete contraction from A ⊗max B to C ⊗max C(Ω) is a complete isometry if and only if A ⊗max B = A ⊗min B

completely isometrically.

(6.11)

Hence (6.11) holds true if and only if the dilation property 6.2.3 (ii) is satisﬁed for D = C(Ω). Moreover 6.2.4 shows that (6.11) is valid if B is a Dirichlet algebra and A is selfadjoint. It is interesting to determine which unital operator algebras A have the property that A ⊗max A(D) = A ⊗min A(D) completely isometrically. Certainly this holds if A = A(D), by (6.9), or if A is a C ∗ -algebra, by the previous paragraph. Indeed, A(D) is a Dirichlet uniform algebra. Thus by the observations in the last paragraph, together with 6.2.3 and the discussion in 6.2.2, we deduce the following strengthening of Nagy’s dilation theorem. Corollary 6.2.6 Let A be a unital-subalgebra of a unital C ∗ -algebra C. Let H be a Hilbert space, let T ∈ B(H) be a contraction, and let π : A → B(H) be a unital completely contractive homomorphism such that π(a)T = T π(a) for a ∈ A. Then there exist a Hilbert space K, an isometry J : H → K, a ∗-representation π ˆ : C → B(K), and a unitary operator U ∈ B(K), which satisfy π ˆ (c)U = U π ˆ (c) for c ∈ C, and ˆ (a)U n J, a ∈ A, n ≥ 0. π(a)T n = J ∗ π 6.3 TENSOR PRODUCTS WITH TRIANGULAR ALGEBRAS The main goals of this section are to provide some concrete examples of tensor products of operator algebras; and also to indicate some remarkable connections to dilation theory and the well-known Sz–Nagy–Foia¸s commutant lifting theorem. Facts in this section will not be used later in the book.

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Tensor products with triangular algebras

6.3.1 We let T n denote the algebra of upper triangular n × n matrices. Arguing as in 6.2.5 and using the fact that Mn is nuclear, we see that whenever A is a unital-subalgebra of a unital operator algebra C, then A ⊗max T n ⊂ C ⊗max Mn completely isometrically if and only if A ⊗max T n = A ⊗min T n completely isometrically. The latter is valid if A is a C ∗ -algebra, by Proposition 6.2.4, since T n is a Dirichlet algebra with Ce∗ (T n ) = Mn (see 4.3.7 (1)). Other such examples will be discussed in 6.3.7 and in the Notes to this section. Let M denote the space of complex valued inﬁnite matrices (tij )i,j≥1 with uniformly bounded truncations (see 1.2.26). Recall that M ∼ = B(2 ). More pre2 cisely, any T ∈ B( ) is identiﬁed with (tij )i,j≥1 , where tij = T (ej ), ei , for i, j ≥ 1. Here (ej )j≥1 is the canonical orthonormal basis. Consider the ‘inﬁnite’ triangular algebra T ∞ = (tij ) ∈ M : tij = 0 for i > j ≥ 1 . Clearly T ∞ is a unital dual algebra; indeed if Eij are the usual ‘matrix units’, then T ∞ is the w∗ -closure of the linear span of {Eij : j ≥ i ≥ 1} in M. Since M∼ = B(2 ) is known to be a semidiscrete von Neumann algebra (see remarks in 6.6.1), an argument similar to the one in the previous paragraph, but using 6.1.14 (2), shows that for any unital-subalgebra A of C, the embedding of A ⊗nor T ∞ into C ⊗nor M is a complete isometry if and only if A ⊗nor T ∞ = A ⊗min T ∞ completely isometrically. For a ﬁnite integer n ≥ 1, we may regard T n as a subalgebra of T ∞ , namely T n = Span{Eij : 1 ≤ i ≤ j ≤ n}. 6.3.2 (Completely contractive representations of T n ) Let H be a Hilbert space and let w : T ∞ → B(H) be a unital and w∗ -continuous completely contractive homomorphism. For any i ≥ 1, we let pi = w(Eii ). Since Eii2 = Eii and Eii = 1, we see that p2i = pi and pi ≤ 1. Hence each pi is a projection. We let Hi ⊂ H be the range of pi . These spaces are pairwise orthogonal since pi pj = w(Eii Ejj ) = 0 when i = j. Moreover we have IH = w(1) = w∗ − lim w(E11 + · · · + Enn ) = w∗ − lim (p1 + · · · + pn ). n

n

Thus H is the Hilbert space direct sum of the Hi . Clearly Eii Ei(i+1) E(i+1)(i+1) equals Ei(i+1) , for i ≥ 1. Applying w, we see that pi w(Ei(i+1) )pi+1 = w(Ei(i+1) ). ⊥ ⊂ Ker(w(Ei(i+1) )) and that Ran(w(Ei(i+1) )) ⊂ Hi . We This means that Hi+1 may therefore identify w(Ei(i+1) ) with an operator θi from Hi+1 into Hi . Since w is a contraction we have θi : Hi+1 −→ Hi ≤ 1,

i ≥ 1.

(6.12)

For any j > i ≥ 1, we may write Eij = Ei(i+1) E(i+1)(i+2) · · · E(j−1)j . Applying w, we ﬁnd that j > i ≥ 1. (6.13) w(Eij ) = θi θi+1 · · · θj−1 ,

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243

It turns out that (6.12) and (6.13) characterize such homomorphisms w. Namely, let (Hi )i≥1 be a sequence of Hilbert spaces, and let H be their Hilbert space direct sum. For any i ≥ 1, let θi : Hi+1 → Hi be a contraction. Then there exists a (necessarily unique) unital and w∗ -continuous completely contractive homomorphism w : T ∞ → B(H) satisfying (6.13), such that w(Eii ) is the projection onto Hi , for i ≥ 1. This remarkable result is due to McAsey and Muhly (see [274]). In the ﬁnite-dimensional case of T n there is a similar description. Given an integer n ≥ 2 and a Hilbert space H, a linear mapping w : T n → B(H) is a unital completely contractive homomorphism if and only if there exist an orthogonal decomposition H = H1 ⊕ · · · ⊕ Hn and contractions θi : Hi+1 → Hi , for 1 ≤ i ≤ n − 1, such that w(Eij ) = θi θi+1 · · · θj−1 for 1 ≤ i < j ≤ n, and such that w(Eii ) is the projection onto Hi . Lemma 6.3.3 Let S ∈ B(2 ) be the ‘backwards shift’ operator, whose matrix (sij )i,j≥1 is given by si(i+1) = 1 if i ≥ 1, and sij = 0 if j = i + 1. Let uS be the associated representation of the disc algebra A(D) on 2 (see 6.2.2). Then uS is a complete isometry, and Ran(uS ) ⊂ T ∞ . Proof Since S ∈ T ∞ , it is clear that uS maps into T ∞ . For any f ∈ A(D), uS (f ) is the so-called Toeplitz operator with matrix [f (j − i)]i,j≥1 . Here f (k) denotes the kth Fourier coeﬃcient of f (see (2.9)). Since uS is a complete contraction and A(D) is a minimal operator space, it suﬃces to show that uS is an isometry. By density it therefore suﬃces to show that for any polynomial f ∈ P, we have f ∞ = f(j − i) i,j≥1 . Let g be an arbitrary trigonometric polynomial in Ball(L2 (T)). Then " " # 2 #1/2 2 1/2 + f g2 = f g(j) = f(j − i)g(i) j

j

i

# " 1/2 g(i)2 ≤ f(j − i) i,j≥1 = f(j − i) i,j≥1 . i

The result follows by taking the supremum over all such g.

2

In view of the last result, we may regard A(D) as a subalgebra of T ∞ . Proposition 6.3.4 For any approximately unital operator algebra A, A ⊗max A(D) ⊂ A ⊗nor T ∞

completely isometrically.

Proof By 6.1.15 it suﬃces to prove that A ⊗max A(D) ⊂ A ⊗nor T ∞ isometrically. This amounts to proving that for any ﬁnite sequence (ak )k≥0 in A, ak ⊗ z k ≤ ak ⊗ S k . (6.14) A⊗max A(D) A⊗nor T ∞ k≥0

k≥0

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To see this, note that IA ⊗ uS : A ⊗max A(D) → A ⊗nor T ∞ is a contraction, by 6.3.3, (6.2), A ⊗ P as a ﬁnite and 6.1.9. If we write an A-valued polynomial F ∈ sum F = k≥0 ak ⊗ z k , with ak ∈ A, then (IA ⊗ uS )(F ) = k≥0 ak ⊗ S k . If (6.14) is valid, then IA ⊗ uS acts as an isometry on polynomials, and hence by density IA ⊗ uS is an isometry. To prove (6.14), we consider two commuting completely contractive homomorphisms π : A → B(H) and ρ : A(D) → B(H). We may assume (using 6.1.11) that ρ is unital. Thus ρ = uT for a contraction T commuting with the range of π (see 6.2.2). According to the result of McAsey–Muhly cited in 6.3.2, there is a unital w∗ -continuous completely contractive homomorphism wT : T ∞ −→ B(2 (H)) = B(2 )⊗B(H) such that wT (Eij ) = Eij ⊗ T j−i , for j ≥ i ≥ 1. Next we deﬁne a mapping ˆ (a) = I2 ⊗ π(a), for a ∈ A. Then π ˆ is completely π ˆ : A → B(2 (H)) by letting π contractive and commutes with wT . We have S k ⊗ π(ak )T k = sup π(ak )T k z k : z ∈ C, |z| = 1 k≥0

B(2 (H))

B(H)

k≥0

by Lemma 6.3.3 and the displayed formula in 2.2.7. However, wT (S k ) = (S ⊗T )k , ˆ (ak )wT (S k ), and so for k ≥ 0. Thus S k ⊗ π(ak )T k = π S k ⊗ π(ak )T k 2 = π ˆ (ak )wT (S k ) ≤ ak ⊗ S k . B( (H)) A⊗nor T ∞ k≥0

k≥0

k≥0

The last two centered formulae imply that π(ak )ρ(z k ) = π(ak )T k k≥0

B(H)

k≥0

B(H)

≤ ak ⊗ S k k≥0

A⊗nor T

Taking the supremum over all π and ρ = uT as above, we obtain (6.14).

. ∞

2

Theorem 6.3.5 (Paulsen and Power) Let A be an approximately unital operator algebra. Then the following assertions are equivalent: (i) A ⊗max A(D) = A ⊗min A(D) completely isometrically. (ii) For every integer n ≥ 1, A ⊗max T n = A ⊗min T n completely isometrically. (iii) A ⊗nor T ∞ = A ⊗min T ∞ completely isometrically. Proof Since A ⊗min A(D) ⊂ A ⊗min T ∞ completely isometrically, the implication ‘(iii) ⇒ (i)’ follows from Proposition 6.3.4. Note moreover that when proving ‘(ii) ⇒ (iii)’ and ‘(i) ⇒ (ii)’, we will only need to prove that the desired identities hold isometrically. Indeed, as in the proof of 6.3.4, the completely isometric statements follow from the isometric ones and Corollary 6.1.15. We assume (ii), and shall prove that A ⊗nor T ∞ = A ⊗min T ∞ . We regard T n as a subalgebra of T ∞ , and we let αn : T ∞ → T n denote the canonical

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‘truncation’ operator. We clearly have b = w ∗ -limn αn (b), for b ∈ T ∞ . Hence for any (π, w) ∈ I nor (A, T ∞ ), and any y = k ak ⊗ bk ∈ A ⊗ T ∞ , we have π(ak )w(bk ) = w∗ − lim π(ak )w αn (bk ) . π • w(y) = n

k

k

For n ≥ 1, let wn be the restriction of w to T n . Our hypothesis (ii) ensures that π • wn is completely contractive on A ⊗min T n . Hence π(ak )w αn (bk ) ≤ ak ⊗ αn (bk ) . A⊗min T n k k Moreover αn is a complete contraction, and so the right side of this inequality is less than or equal to ymin. We deduce that π • w(y) ≤ ymin. Hence ymax = ymin as expected. Assume (i), and let n ≥ 2. To show the isometric version of (ii), we consider two commuting completely contractive homomorphisms π : A → B(H) and w : T n → B(H), with w unital, and we will show that π • w : A ⊗min T n −→ B(H) ≤ 1.

(6.15)

Then taking the supremum over all possible (π, w) and applying 6.1.11 yields A ⊗min T n = A ⊗max T n isometrically. Recall from 6.3.2 that there exist a orthogonal decomposition H = H1 ⊕ · · · ⊕ Hn , such that w(Eii ) = pi is the projection onto Hi , and such that the contraction w(Eij ) maps Hj into Hi and vanishes on Hj⊥ , for 1 ≤ i ≤ j ≤ n. Deﬁne T = w E12 + E23 + · · · + E(n−1)n . Then T k = w E1(k+1) + E2(k+2) + · · · + E(n−k)n , for k ≥ 0. Applying this with k = j − i, we deduce that for any ζ = (ζ1 , . . . , ζn ) and any η = (η1 , . . . , ηn ) in H = H1 ⊕ · · · ⊕ Hn , we have T j−i (ζj ), ηi = w(Eij )ζ, η,

1 ≤ i ≤ j ≤ n.

(6.16)

Here of course we are identifying ζj with pj (ζ). Deﬁne V : H → 2n (H) by ei ⊗ ζi , ζ = (ζ1 , . . . , ζn ) ∈ H. V (ζ) = 1≤i≤n

Clearly V is an isometry. We claim that for any a ∈ A, we have 1 ≤ i ≤ j ≤ n. π(a)w(Eij ) = V ∗ Eij ⊗ T j−i π(a) V, To see this, let ζ, η in H. Then ) ∗ * V Eij ⊗ T j−i π(a) V (ζ), η = T j−i π(a)(ζj ), ηi .

(6.17)

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Since π(a) commutes with each projection pj , we have π(a)(ζj ) = pj π(a)(ζ). Therefore using (6.16) we have T j−i π(a)(ζj ), ηi = T j−i pj π(a)(ζ), ηi = w(Eij )π(a)ζ, η, which shows (6.17). We now consider y ∈ A ⊗ T n . We may write aij ⊗ Eij , y= 1≤i≤j≤n

for some aij ’s in A. Then by (6.17), we have π • w(y) = π(aij )w(Eij ) ≤ 1≤i≤j≤n

Eij ⊗ T j−i π(aij )

.

Mn (B(H))

1≤i≤j≤n

It is clear that T is a contraction. Consider uT : A(D) → B(H) (see 6.2.2). Note that T , and hence also uT , commutes with π. Then # " j−i j−i Eij ⊗ T π(aij ) = IMn ⊗ (π • uT ) Eij ⊗ aij ⊗ z . 1≤i≤j≤n

1≤i≤j≤n

We therefore deduce from our hypothesis (i) that j−i π • w(y) ≤ E ⊗ a ⊗ z ij ij B(H)

Mn (A⊗min A(D))

1≤i≤j≤n

.

Now recall that Mn (A⊗min A(D)) = A(D; Mn (A)) isometrically (see 2.2.7). Hence the right side of the last inequality is equal to sup Eij ⊗ z j−i aij : z ∈ C, |z| = 1 . (6.18) 1≤i≤j≤n

Mn (A)

For z ∈ T, consider the diagonal matrix Uz = Diag(z, z 2 , . . . , z n ) ∈ Mn . Then Uz is a unitary and " # Eij ⊗ z j−i aij = Uz∗ Eij ⊗ aij Uz = Uz∗ y Uz . 1≤i≤j≤n

1≤i≤j≤n

Since Uz∗ y Uz min = ymin for z ∈ T, we see that the supremum in (6.18) is equal to ymin. This shows that π • w(y)B(H) is less than or equal to ymin, which completes the proof of (6.15). 2 The following ‘normal’ analogue of Theorem 6.3.5 has a similar proof. Theorem 6.3.6 For a unital dual operator algebra M , the following assertions are equivalent:

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(i) A(D) ⊗nor M = A(D) ⊗min M completely isometrically. (ii) For every integer n ≥ 1, T n ⊗nor M = T n ⊗min M completely isometrically. Corollary 6.3.7 The following identities hold completely isometrically: (1) If A is a C ∗ -algebra, and n ≥ 1 is any positive integer, A ⊗max T n = A ⊗min T n

and

A ⊗nor T ∞ = A ⊗min T ∞ .

and

A(D) ⊗nor T ∞ = A(D) ⊗min T ∞ .

and

T m ⊗nor T ∞ = T m ⊗min T ∞ .

(2) For any n ≥ 1, A(D) ⊗max T n = A(D) ⊗min T n (3) For any n, m ≥ 1, T m ⊗max T n = T m ⊗min T n

Proof By Theorem 6.3.5, we obtain the ﬁrst two assertions from the corresponding results in Section 6.2, that is, (6.9) and 6.2.5. In turn, (3) follows from (2) by applying 6.3.5 and 6.3.6. 2 Tensor products with triangular algebras are closely related to dilation theory, and in particular to the well-known commutant lifting theorem of Sz–Nagy–Foia¸s (see [405]). This is well illustrated by the following consequence of 6.2.3: Proposition 6.3.8 (Paulsen and Power) Let A be a unital-subalgebra of a unital C ∗ -algebra C. The following are equivalent: (i) A ⊗max T 2 = A ⊗min T 2 completely isometrically. (ii) For any pair of unital completely contractive homomorphisms π : A → B(H) and θ : A → B(H ), and any contraction T : H → H with the property that π(a)T = T θ(a) for a ∈ A, there exist Hilbert spaces K, K , isometries J : H → K and J : H → K , unital ∗-representations π ˆ : C → B(K) and ˆ ˆ (c)U = U θ(c) θˆ: C → B(K ), as well as a unitary U : K → K, such that π ∗ ∗ ˆ (a)U J , π(a) = J π ˆ (a)J and for all c ∈ C, and such that π(a)T = J π ∗ˆ θ(a) = J θ(a)J , for a ∈ A. Proof Assume (i). Then A ⊗max T 2 ⊂ C ⊗max M2 completely isometrically (see 6.3.1). Let H, H , π, θ, and T be as in (ii), and deﬁne σ : A → B(H ⊕ H ) and τ : T 2 → B(H ⊕ H ) by letting " # π(a) 0 λ1 µ λ1 IH µT σ(a) = and τ = . (6.19) 0 θ(a) 0 λ2 0 λ2 IH Then σ and τ are unital completely contractive homomorphisms (see 2.2.11). Moreover our assumption that T intertwines π(a) and θ(a) for a ∈ A, ensures that σ and τ have commuting ranges. By the implication ‘(i) ⇒ (ii)’ of Proposition 6.2.3, there exist a Hilbert space F , an isometry j : H ⊕ H → F , and commuting unital completely contractive homomorphisms σ ˆ : C → B(F ) and

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Pisier’s delta norm

τˆ : M2 → B(F ), such that σ(a)τ (b) = j ∗ σ ˆ (a)ˆ τ (b)j, for a ∈ A, b ∈ T 2 . Since τˆ is a ∗-representation of M2 , there is a decomposition F = K ⊕ K and a unitary operator U : K → K such that # " λ1 IK µU λ1 µ = , τˆ ν λ2 νU ∗ λ2 IK ˆ and τˆ commute, we see that K and K are both for λ1 , λ2 , µ, ν in C. Since σ σ ˆ (C)-invariant. Thus we may write σ ˆ=π ˆ ⊕ θˆ for some unital ∗-representations ˆ ˆ (c) and θ(c), for π ˆ : C → B(K) and θˆ: C → B(K ). Moreover U intertwines π c ∈ C. Next we check that j maps H into K and H into K . Since j ∗ τˆ(b)j = τ (b), we have j ∗ PK j = PH . For ζ ∈ H, this implies that ζ ≤ PK j(ζ) ≤ ζ, and j(ζ) = PK j(ζ). Hence j(H) ⊂ K. Similarly, j(H ) ⊂ K . Let J : H → K and J : H → K be the corresponding restrictions. Then the last three assertions of (ii) follow from the relation σ(a)τ (b) = j ∗ σ ˆ (a)ˆ τ (b)j above, for b = E11 , E12 , E22 respectively. To prove conversely that (ii) implies (i), it suﬃces to reverse the arguments and to appeal to the implication ‘(ii) ⇒ (i)’ of 6.2.3. Given a Hilbert space E, and unital completely contractive homomorphisms σ : A → B(E) and τ : T 2 → B(E), whose ranges commute, there is a decomposition E = H ⊕ H , a contraction T : H → H, as well as π and θ as in (ii), such that τ and σ have the form (6.19), and such that T intertwines π(a) and θ(a) for a ∈ A. Appealing to (ii), and using simple algebra, yields the desired maps in 6.2.3 (ii). 2 6.4 PISIER’S DELTA NORM 6.4.1 (Deﬁnitions) Let X be an operator space and let B be an approximately unital operator algebra. In this section we introduce a tensor norm on X ⊗ B, denoted by δ, and called the delta norm. Let Q0 : B ⊗ X ⊗ B → X ⊗ B be the linear mapping deﬁned by letting Q0 (a ⊗ x ⊗ b) = x ⊗ ab, for x ∈ X and a, b ∈ B. Recall that any c ∈ B can be written as c = ab, for some a and b in B (by A.6.2). Thus Q0 is onto. Hence for any z ∈ X ⊗ B, we may deﬁne δ(z) = inf{yh}, where the inﬁmum runs over all y ∈ B ⊗ X ⊗ B such that Q0 (y) = z. Here yh denotes the Haagerup tensor norm of y (see 1.5.4 and 1.5.5). A more explicit deﬁnition of δ is provided by the ‘3-variable’ version of (1.40). Namely,

δ(z) = inf [xij ]Mn (X) [a1 · · · an ]Rn (B) [b1 · · · bn ]t Cn (B) , where the inﬁmum runs over all decompositions of z of the form xij ⊗ ai bj . z=

(6.20)

1≤i,j≤n

It is clear that δ is a seminorm, and it is not hard to check that this is actually a norm on X ⊗ B. Indeed we shall show in 6.4.2 below that zmin ≤ δ(z), for z in

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249

X ⊗ B. We denote the resulting completion by X ⊗δ B. By construction, and by an elementary Banach space argument, Q0 uniquely extends to a quotient map Q : B ⊗h X ⊗h B −→ X ⊗δ B.

(6.21)

6.4.2 (Comparisons with other tensor norms) Let H be a Hilbert space, let θ : X → B(H) be a complete contraction, and let π : B → B(H) be a completely contractive homomorphism. If θ and π have commuting ranges, then we claim: θ • π(z) ≤ δ(z), z ∈ X ⊗ B. (6.22) B(H) Indeed we have θ(x)π(ab) = π(a)θ(x)π(b), for x ∈ X and a, b ∈ B. Hence if z is represented as in (6.20), we have θ • π(z) = θ(xij )π(ai bj ) = π(ai )θ(xij )π(bj ). The latter quantity is dominated by ai ⊗ xij ⊗ bj h (see the ﬁrst paragraph of 1.5.8). Taking the inﬁmum over all such representations of z, we obtain (6.22). By a similar argument to that in 6.1.1, equation (6.22) implies that ymin ≤ δ(y),

y ∈ X ⊗ B.

On the other hand, δ is dominated by the Haagerup norm on X ⊗ B. To see this, ﬁrst observe that δ(x ⊗ b) ≤ xb, for x ∈ X and b ∈ B. Indeed if x < 1 and b < 1, then by A.6.2 we can write b = b b , with b < 1 and b < 1. Hence x ⊗ b = Q0 (b ⊗ x ⊗ b ), and we conclude that δ(x ⊗ b) ≤ b xb < 1. Now let (et )t be a cai for B, and consider z = xk ⊗ bk ∈ X ⊗ B, where (xk )k and xk ⊗ et bk for (bk )k are ﬁnite sequences in X and B respectively. We let zt = any t. Then the above point implies that δ(z − z) → 0, and hence δ(z t t ) → δ(z). Moreover zt = Q0 ( k et ⊗ xk ⊗ bk ), and hence et ⊗ xk ⊗ bk ≤ xk ⊗ bk = zh. δ(zt ) ≤ k

h

k

h

We deduce that δ(z) ≤ zh. It is also clear from (6.22) and 6.1.1 that if X = A is an approximately unital operator algebra, then the maximal tensor norm · max on A ⊗ B is dominated by δ. The ﬁrst main result of this section is Corollary 6.4.4 below, which is a ‘converse’ to (6.22). According to this result, δ can be regarded as an genuine analogue of the maximal tensor norm. We will deduce 6.4.4 from the following description of the dual space of X⊗δ B: Theorem 6.4.3 (Pisier) Let X be an operator space, and let B be an approximately unital operator algebra. Given a functional ϕ : X ⊗ B → C, the following are equivalent: (i) ϕ extends to an element of (X⊗δ B)∗ , with ϕ ≤ 1.

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Pisier’s delta norm

(ii) There exist a Hilbert space H, vectors ζ1 , ζ2 ∈ Ball(H), a completely contractive map θ : X → B(H), and a completely contractive homomorphism π : B → B(H), such that θ and π have commuting ranges, and such that ϕ(x ⊗ b) = θ(x)π(b)ζ2 , ζ1 ,

x ∈ X, b ∈ B.

Proof If (ii) holds, then ϕ(z) = (θ • π)(z)ζ2 , ζ1 , for z ∈ X ⊗ B. This clearly implies (i) by (6.22). Assume conversely that (i) holds. Composing ϕ with the contractive mapping Q from (6.21), we obtain ∗ with ϕ ◦ Q ≤ 1. ϕ ◦ Q ∈ B ⊗h X ⊗h B , By the CSPS theorem for trilinear maps (see the second paragraph of 1.5.8), there exist Hilbert spaces K1 , K2 , vectors e1 ∈ Ball(K1 ) and e2 ∈ Ball(K2 ), a completely contractive map v : X → B(K2 , K1 ), and completely contractive representations π1 : B → B(K1 ) and π2 : B → B(K2 ), such that ϕ(x ⊗ ab) = π1 (a)v(x)π2 (b)e2 , e1 ,

x ∈ X, a, b ∈ B.

(6.23)

By 2.1.10, we may assume that π1 and π2 are nondegenerate. Consider a, b, c in B. Then writing acb = (ac)b or acb = a(cb), and computing ϕ(x ⊗ acb) by means of (6.23), we ﬁnd that π1 (a)π1 (c)v(x)π2 (b)e2 , e1 = π1 (a)v(x)π2 (c)π2 (b)e2 , e1 .

(6.24)

Since π1 and π2 are nondegenerate, the vectors e1 and e2 belong to [π1 (B)∗ e1 ] and [π2 (B)e2 ] respectively (see the last line of 2.2.8). Replacing K1 and K2 by [π1 (B)∗ e1 ] and [π2 (B)e2 ] if necessary, we may assume that the spaces π1 (B)∗ e1 and π2 (B)e2 are dense in K1 and K2 respectively. Under these conditions, we deduce from (6.24) that π1 (c)v(x) = v(x)π2 (c),

x ∈ X, c ∈ B.

(6.25)

Let H = K1 ⊕ K2 , and write any element of B(H) as a 2 × 2 matrix with respect to that decomposition. We deﬁne π : B → B(H) and θ : X → B(H) by letting " # " # π1 (b) 0 0 v(x) π(b) = and θ(x) = , (6.26) 0 0 0 π2 (b) for x ∈ X and b ∈ B. Then (6.25) ensures that θ and π have commuting ranges. It is clear that θ and π are both completely contractive, and that π is a homomorphism. Let ζ1 = (e1 , 0) and ζ2 = (0, e2 ), elements in H. Then ζ1 ≤ 1 and ζ2 ≤ 1. Moreover, θ(x)π(b)ζ2 , ζ1 = v(x)π2 (b)e2 , e1 , for x ∈ X and b ∈ B. If (et )t is a cai for B, then π1 (et ) → IK1 strongly. Hence θ(x)π(b)ζ2 , ζ1 = limπ1 (et )v(x)π2 (b)e2 , e1 = lim ϕ(x ⊗ et b) = ϕ(x ⊗ b) t

by (6.23). This proves (ii).

t

2

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251

Corollary 6.4.4 (Pisier) Let X be an operator space and let B be an approximately unital operator algebra. Then for any z ∈ X ⊗ B, we have δ(z) = sup θ • π(z) , where the supremum runs over all triples (H, θ, π), in which H is a Hilbert space, θ : X → B(H) is a complete contraction, π : B → B(H) is a completely contractive homomorphism, and θ and π have commuting ranges. Proof This follows from (6.22), 6.4.3, and the Hahn–Banach theorem.

2

6.4.5 (Functoriality) Let u : X → Y be a completely contractive map between two operator spaces, and let π : B → D be a completely contractive homomorphism between approximately unital operator algebras. Then the mapping u ⊗ π : X ⊗ B → Y ⊗ D extends to a contraction from X⊗δ B into Y ⊗δ D. This follows either directly from the deﬁnition of the delta norm, or from the above corollary, arguing as in 6.1.9. In particular if X is a subspace of Y , and if B is a subalgebra of D, then we have a canonical complete contraction X⊗δ B → Y ⊗δ D. 6.4.6 (The normal delta norm) For the purpose of later applications of the delta norm, we will need the normal delta norm σδ. This is deﬁned analoguously to the normal Haagerup tensor product of 1.6.8. Let X be a dualoperator space, ∗ ∗ and let M be a unital dual operator algebra. We deﬁne X⊗δ M σ ⊂ X⊗δ M to be the subspace of all functionals whose associated bilinear form X × M → C is separately w∗ -continuous. Then we deﬁne ∗ the normal delta tensor product X⊗σδ M to be the dual space of (X⊗δ M σ . The norm of any z in X⊗σδ M will be denoted by σδ(z). We can regard X ⊗M as a subspace of X⊗σδ M in a obvious way. Clearly σδ(z) ≤ δ(z), for z ∈ X ⊗ M . In the notes for Section 6.6 we will show that this inequality may be strict. Lemma 6.4.7 If z ∈ X⊗σδ M then σδ(z) ≤ 1 if and only if there exists a net (zt )t in X ⊗ M such that δ(zt ) < 1 for any t, and ϕ, z = limt ϕ, zt , for ∗ ϕ ∈ X⊗δ M σ . ∗ Proof If σδ(z) ≤ 1, then the functional z : X⊗δ M σ → C admits a contrac∗∗ by the Hahn–Banach theorem. By Goldstine’s tive extension z ∈ X⊗δ M Lemma (A.2.1), we can ﬁnd a net (zt )t ⊂ X ⊗ M converging to z in the w ∗ ∗∗ topology of X⊗δ M , and such that δ(zt ) < 1 for any t. This yields the ‘only if’ part. The converse is obvious. 2 Theorem 6.4.8 Let X be a dual operator space and let M be a unital dual operator algebra. Given a functional ϕ : X⊗B → C, the following are equivalent: ∗ (i) ϕ extends to an element of X⊗δ B σ , with ϕ ≤ 1. (ii) There exist a Hilbert space H, vectors ζ1 , ζ2 ∈ Ball(H), a w∗ -continuous completely contractive map θ : X → B(H), and a w ∗ -continuous completely

252

Pisier’s delta norm contractive homomorphism π : M → B(H), such that θ and π have commuting ranges, and such that ϕ(x ⊗ b) = θ(x)π(b)ζ2 , ζ1 ,

x ∈ X, b ∈ M.

Proof Clearly (ii) implies (i) by Theorem 6.4.3. For the other direction, we assume (i), and we will mimic the proof of the analoguous implication in Theorem 6.4.3, with B = M . We will show that (6.23) can be achieved with the additional property that π1 : M → B(K1 ), π2 : M → B(K2 ), and v : X → B(K2 , K1 ), are w∗ -continuous. Clearly, ϕ ◦ Q is a contractive functional on M ⊗h X ⊗h M . ∗ Since the product on ∗ M is separately w -continuous (see 2.7.4 (1)), and since ϕ belongs to X⊗δ B σ , we see that the trilinear mapping associated to ϕ ◦ Q is separately w∗ -continuous as well. According to 1.6.10, there exist Hilbert spaces E1 , E2 , and w∗ -continuous completely contractive maps w : X → B(E2 , E1 ), u2 : M → B(C, E2 ) = E2c , and u1 : M → B(E1 , C) = E1∗r such that ϕ(x ⊗ ab) = (ϕ ◦ Q)(a ⊗ x ⊗ b) = u1 (a)w(x)u2 (b), for a, b ∈ M and x ∈ X. Applying a slight modiﬁcation of Theorem 2.7.10, we ﬁnd a Hilbert space K2 , a vector e2 ∈ K2 with e2 ≤ 1, a contraction W2 : K2 → E2 , and a unital w∗ -continuous completely contractive representation π2 : M → B(K2 ) such that u2 (b) = W2 π2 (b)e2 , for b ∈ M . Likewise, u1 admits a factorization u1 (a) = ξ ◦π1 (a)W1 , where K1 is a Hilbert space, ξ is the functional on K1 given by taking the inner product with a vector e1 ∈ Ball(K1 ), and π1 : M → B(K1 ) is a unital w∗ -continuous completely contractive representation, and W1 : E1 → K1 is a contraction. The linear map v = W1 w(· )W2 is a w∗ continuous complete contraction, and it is clear from above that π1 , π2 , v, e1 and e2 satisfy (6.23) as expected. If we now follow the proof of 6.4.3, we see that π 2 and θ deﬁned by (6.26) are w∗ -continuous. Hence that proof yields (ii). Corollary 6.4.9 Let X be a dual operator space and let M be a unital dual operator algebra. Then for any z ∈ X ⊗ M , we have σδ(z) = sup θ • π(z) , where the supremum runs over all Hilbert spaces H, all w ∗ -continuous completely contractive maps θ : X → B(H), and all w ∗ -continuous completely contractive homomorphisms π : M → B(H), such that θ and π have commuting ranges. Proof Pick z ∈ X ⊗ M . By deﬁnition, σδ(z) = sup ϕ, z , where the supre ∗ mum is over all ϕ ∈ X⊗δ B σ with ϕ ≤ 1. By Theorem 6.4.8 we deduce that σδ(z) is dominated by the supremum in the centered equation above. To check the converse inequality, we assume that σδ(z) ≤ 1. It suﬃces to show that θ • π(z) ≤ 1 for all θ and π as above. If ζ1 , ζ2 ∈ Ball(H), deﬁne ϕ in ∗ X⊗δ M σ by ϕ(x ⊗ b) = θ(x)π(b)ζ2 , ζ1 , for x ∈ X and b ∈ M . Clearly ϕ is contractive. Let (zt )t be the net in X ⊗ M given by 6.4.7. Then ϕ, zt < 1 for any t, and so ϕ, z ≤ 1. Taking the supremum over ζ1 , ζ2 as above, we see that θ • π(z) ≤ 1. 2

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Proposition 6.4.10 Let X be an operator space and let B be an approximately unital operator algebra. ∗ (1) Given any ϕ ∈ X⊗δ B , the restriction of ϕ to X ⊗ B admits a unique ∗ extension to some Φ ∈ (X ∗∗ ⊗δ B ∗∗ σ . The resulting mapping ϕ → Φ is a linear isometric isomorphism, that is, ∗ ∗ X⊗δ B = (X ∗∗ ⊗δ B ∗∗ σ isometrically. (2) We have (X⊗δ B)∗∗ = X ∗∗ ⊗σδ B ∗∗ isometrically. (3) The following canonical embeddings are isometries: X⊗δ B ⊂ X ∗∗ ⊗δ B ∗∗ ,

X⊗δ B ⊂ X ∗∗ ⊗δ B,

X⊗δ B ⊂ X⊗δ B ∗∗ .

Proof There is a canonical contraction j : X⊗δ B → X ∗∗ ⊗δ B ∗∗ given by 6.4.5. The map Φ → Φ ◦ j from (X ∗∗ ⊗δ B ∗∗ )∗σ to (X⊗δ B)∗ is a one-to-one ∗ linear contraction. To prove that it is a surjective isometry, let ϕ ∈ X⊗δ B . Recall from 6.4.2 that δ(z) ≤ zh, for z ∈ X ⊗ B. It therefore follows from 1.6.7 that ϕ : X ⊗ B → C extends to a (necessarily unique) separately w ∗ -continuous Φ : X ∗∗ ⊗ B ∗∗ → C. Our aim is to show that ∗ Φ ∈ (X ∗∗ ⊗δ B ∗∗ , with Φ ≤ ϕ. (6.27) We may assume that ϕ = 1. Fix z ∈ X ∗∗ ⊗ B ∗∗ , with δ(z) < 1. By 6.4.1, ∗∗ ∗∗ ∗∗ ∗∗ there exists matrices a∗∗ = [a∗∗ = [x∗∗ 1 · · · an ] ∈ Rn (B ), x ij ] ∈ Mn (X ), ∗∗ ∗∗ ∗∗ t ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ and b = [b1 · · · bn ] ∈ Cn (B ), such that z = xij ⊗ ai bj with b < 1, x∗∗ < 1 and a∗∗ < 1. Since Mn (X ∗∗ ) = Mn (X)∗∗ isometrically (see 1.4.11), Goldstine’s Lemma ensures that there is a net (xr )r in the unit ball of Mn (X) converging to x∗∗ in the w∗ -topology of Mn (X)∗∗ . Similarly, there exist nets (as )s in Ball(Rn (B)) and (bs )s in Ball(Cn (B)) converging to a∗∗ and b∗∗ in the w∗ -topologies of Rn (B)∗∗ and Cn (B)∗∗ respectively. For any r, s, s , we have " # r s s xij ⊗ ai bj ≤ 1, δ 1≤i,j≤n

and so

, - r s s ϕ, xij ⊗ ai bj ≤ 1 . 1≤i,j≤n

Taking the limit over r, we deduce from the w ∗ -continuity of Φ that , - ∗∗ s s Φ, x ⊗ a b ij i j ≤ 1 1≤i,j≤n

for any s, s . Since the product on B ∗∗ is separately w∗ -continuous, we have

∗∗ ∗ a∗∗ asi bsj , i bj = w − lim lim s

s

1 ≤ i, j ≤ n.

Then we ﬁnally deduce from the w∗ -continuity of Φ in the second variable that |Φ, z| ≤ 1. This shows (6.27).

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Item (2) follows from (1) at once. The ﬁrst assertion in (3) follows from the ﬁrst line of the proof of (1), together with (2) and the fact that σδ(z) ≤ δ(z). The proofs of the rest of (3) are similar. 2 6.5 FACTORIZATION THROUGH MATRIX SPACES 6.5.1 (Deﬁnition of ∆(u)) In this section we will describe the delta norm in terms of factorizations through matrix spaces, by ﬁnite rank operators. Let Y be an operator space and let B be an approximately unital operator algebra. We will use the canonical identiﬁcation between Y ∗ ⊗ B and the space of ﬁnite rank operators from Y into B. If u : Y → B is such a map, and if z ∈ Y ∗ ⊗ B is associated to u, we deﬁne ∆(u) = δ(z). If z = xij ⊗ ai bj , for [xij ] ∈ Mn (Y ∗ ), and a1 , . . . , an , b1 , . . . , bn in B, then y ∈ Y. (6.28) u(y) = xij , yai bj , Let α : Y → Mn be associated to [xij ] via the relation Mn (Y ∗ ) = CB(Y, Mn ). Then (6.28) asserts that u = βα, where β : Mn → B is the mapping deﬁned by β(Eij ) = ai bj ,

1 ≤ i, j ≤ n.

(6.29)

Here as usual (Eij ) are the matrix units of M n . Thus ∆(u) is given by the t , where the inﬁmum runs over expression inf αcb [a1 · · · an ] [b1 · · · bn ] all n ≥ 1, and all possible factorizations u = βα, with β deﬁned by (6.29). The following gives an alternative description of ∆(u) if Y = Mn : Proposition 6.5.2 Let n ≥ 1 be an integer, let B be an approximately unital operator algebra, and let u : Mn → B be a linear mapping. Then ∆(u) < 1 if and only if there exist an integer m ≥ 1, and two ﬁnite families (aik )1≤i≤n,1≤k≤m , (bkj )1≤j≤n,1≤k≤m in B such that ∗ ∗ aik aik < 1, bkj bkj (6.30) < 1, i,k

j,k

and for s = [sij ] in Mn , u(s) =

aik sij bkj .

(6.31)

i,j,k

Proof We shall use the dual operator space Sn1 = Mn∗ = Rn ⊗h Cn . By the associativity of the Haagerup tensor product and 1.5.14 (3), we can write B ⊗h Sn1 ⊗h B ∼ = Rn (B) ⊗h Cn (B). Let u : Mn → B with ∆(u) < 1, and let z ∈ Sn1 ⊗ B be associated to u. Then there exist w ∈ B ⊗ Sn1 ⊗ B such that wh < 1 and Q(w) = z, where Q is the

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quotient map considered in (6.21) for X = Sn1 . Suppose that w ˜ ∈ Rn (B)⊗Cn (B) corresponds to w via the last centered equation. Thus w ˜ h < 1, and so there exist an integer m ≥ 1, and m-tuples c1 , . . . , cm in Rn (B) and d1 , . . . , dm in Cn (B), such that ck ⊗ dk , (ck )k Rm (Rn (B)) < 1, and (dk )k Cm (Cn (B)) < 1. w ˜= 1≤k≤m

However Rm (Rn (B)) = Rnm (B) and Cm (Cn (B)) = Cnm (B) isometrically. Writing each ck ∈ Rn (B) as ck = i aik ⊗ ei , with a1k , . . . , ank ∈ B, we have 12 ∗ (ck )k = aik aik . Rm (Rn (B)) i,k

Likewise, writing dk =

j

ej ⊗ bkj , with bk1 , . . . , bkn ∈ B, we have

12 ∗ (dk )k = bkj bkj . Cm (Cn (B)) j,k

It is now easy to check that u satisﬁes (6.31), which completes the proof of the ‘only if’ part. The ‘if’ part is obtained by simply reversing the arguments. 2 6.5.3 (A factorization property) We may regard the above result as a special factorization property for maps u : Mn → B. If a = [aik ] ∈ Mn,m (B), and if b = [bkj ] ∈ Mm,n (B), then we will say that the mapping u : Mn → B deﬁned by (6.31) is ‘implemented’ by the pair (a, b). Let W and V be a and b respectively, considered as ‘columns’ in Cnm (B). If C is any unital C ∗ -algebra containing B, consider the ∗-representation π : Mn → Mnm (C) taking s ∈ Mn to s ⊗ Im ⊗ IC . It is easy to check that u = W ∗ π(· )V . In particular, if B is a C ∗ -algebra and if b = a∗ , then V = W and u is completely positive. In this case, 1.3.4 gives aik a∗ik . u = ucb ≤ V 2 = i,k

The following result reduces the computation of ∆(u) for general ﬁnite rank operators, to the case considered in 6.5.2. Lemma 6.5.4 For any ﬁnite rank operator u : Y → B, we have ∆(u) = inf αcb ∆(β) , where the inﬁmum runs over all integers n ≥ 1, and all possible factorizations β α u = βα, with Y −→ Mn −→ B. Proof That ∆(u) dominates the inﬁmum is clear from the last line in 6.5.1, together with 6.5.2. For the reverse inequality, we assume that u = βα is a

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factorization of u through Mn , and we ﬁx > 0. There exist an m ∈ N, a linear map γ : Mn → Mm , and a1 , . . . , am , b1 , . . . , bm in B, with γcb [a1 · · · an ] [b1 · · · bn ]t < ∆(β) + . Also, β = β γ, where the map β : Mm → B is deﬁned by β (Epq ) = ap bq , for 1 ≤ p, q ≤ m. Write u = β (γα). Since γαcb ≤ γcbαcb , we have ∆(u) ≤ αcb γcb [a1 · · · an ] [b1 · · · bn ]t . Hence ∆(u) ≤ αcb(∆(β) + ). Letting ε → 0 yields the result.

2

6.5.5 (Factorization of w∗ -to-norm continuous maps) Let u : Y → B be a ﬁnite rank operator from an operator space Y into an approximately unital operator algebra B. Assume that Y = Z ∗ is a dual operator space and that u is w ∗ -tonorm continuous. It is easy to argue that this is equivalent to u being canonically associated to some z ∈ Z ⊗ B. It follows from 6.4.10 (3) that ∆(u) = z Z⊗δ B . This implies that ∆(u) = inf αcb ∆(β) , where the inﬁmum runs over possible factorizations u = βα, where α : Y → Mn is w∗ -continuous, and β : Mn → B. 6.5.6 (Decomposable maps) In the rest of this section, we focus on the case when B is a C ∗ -algebra. It turns out that in this case, the delta norm and its companion ∆, are closely related to Haagerup’s decomposable norm. We shall only give a brief introduction to this topic and refer the reader to [180] for details and complements (see also [149, Section 5.4] and the Notes to this section). Let B and C be two C ∗ -algebras. A linear mapping u : C → B is said to be decomposable provided that u is a linear combination of completely positive maps from C into B. It is plain that in that case, u may be written as u = (u1 − u2 ) + i(u3 − u4 ),

for completely positive uj : C → B.

(6.32)

We deﬁne DEC(C, B) to be the vector space of all such maps. Note, for example, that any ﬁnite rank operator between C ∗ -algebras is decomposable. Recall for any u : C → B, the mapping u : C → B deﬁned by u (c) = u(c∗ )∗ (see 1.2.25). A key observation is that u : C → B is decomposable if and only if it is the ‘1-2-corner’ of a completely positive corner-preserving map Φ1 u (6.33) : M2 (C) −→ M2 (B). w= u Φ2 Here Φ1 , Φ2 are (necessarily completely positive) maps from C into B. To check this observation, observe that if (6.33) deﬁnes a completely positive map, then v(λ) = Φ1 + λu + λu + Φ2 : C → B is completely positive, for any complex number λ such that |λ| = 1. Hence (6.32) holds, with 4u1 = v(1), 4u2 = v(−1), 4u3 = v(i), and 4u4 = v(−i). To see the other direction, let λ be as above, and let U (λ) be the diagonal unitary 2 × 2 matrix with entries 1 and λ. If Φ : C → B

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257

is a map which is completely positive, then so is U (λ)∗ Φ2 (·)U (λ). Hence if u satisﬁes (6.32), then the four operators u1 u1 u2 −u2 u3 iu3 u4 −iu4 , , , u1 u1 −u2 u2 −iu3 u3 iu4 u4 are all completely positive. Their sum w : M2 (C) → M2 (B) is a completely positive map of the required form (6.33), with Φ1 = Φ2 = u1 + u2 + u3 + u4 . For any u ∈ DEC(C, B), we deﬁne udec = inf{w}, where the inﬁmum runs over all completely positive corner-preserving maps w : M2 (C) → M2 (B) with 1-2-corner equal to u. It is clear that · dec is a norm on DEC(C, B). For example, to see that λudec = |λ|udec , ﬁrst note that this is easy if λ ≥ 0. On the other hand, if |λ| = 1, and if w is as in (6.33) and if U (λ) is as above, then U (λ)∗ w(·)U (λ) is completely positive too, with 1-2-corner λu. Thus λudec ≤ udec; and the other direction follows by symmetry. Let u : C → B be decomposable, and let Φ1 , Φ2 : C → B be such that the mapping w deﬁned by (6.33) is completely positive. Let A be a C ∗ -algebra. Then IA ⊗ w extends to a completely positive map from A ⊗max M2 (C) into A ⊗max M2 (B), whose norm is dominated by w (see 6.1.10 and its proof). Since A ⊗max M2 (C) = M2 (A ⊗max C) and A ⊗max M2 (B) = M2 (A ⊗max B) (see (6.7)), we deduce that IA ⊗ u is bounded from A ⊗max C into A ⊗max B, with norm dominated by w. Taking the inﬁmum over all such w’s, we obtain: IA ⊗ u : A ⊗max C −→ A ⊗max B ≤ udec. (6.34) cb Arguing as in the proof of Lemma 6.1.10, it is easy to deduce that (6.34) holds as well for a nonselfadjoint approximately unital operator algebra A. Proposition 6.5.7 Let n ≥ 1 be a positive integer, let B be a C ∗ -algebra, and let u : Mn → B be a linear mapping. Then ∆(u) = udec . Proof We will prove that ∆(u) ≤ udec as a consequence of Corollary 6.4.4. Let z ∈ Sn1 ⊗ B be associated to u. Let H be a Hilbert space, let θ : Sn1 → B(H) be a complete contraction, let π : B → B(H) be a ∗-representation, and assume that θ and π have commuting ranges. We let M = [π(B)] ⊂ B(H) be the commutant of the range of π, so that θ is valued in M . Thus θ : Sn1 → M is associated with some y ∈ Mn ⊗ M . We write ϕk ⊗ bk and y= σp ⊗ cp , z= k

p

where (ϕk )k ⊂ Sn1 , (bk )k ⊂ B, (σp )p ⊂ Mn , and (cp )p ⊂ M . Then

258

Factorization through matrix spaces θ • π(z) =

θ(ϕk )π(bk ) =

k

σp , ϕk π(bk )cp .

k,p

On the other hand, u(σp ) ⊗ cp = σp , ϕk bk ⊗ cp . u ⊗ IM (y) = p

k,p

taking Let Ψ : B ⊗max M → B(H) be the completely contractive mapping any b ⊗ c to π(b)c. The above calculation shows that θ • π(z) = Ψ (u ⊗ IM )(y) , hence θ • π(z) ≤ Ψ udec ymax, by property (6.34). Since Mn ⊗max M = Mn ⊗min M = CB(Sn1 , M ), we have ymax = ymin = θcb ≤ 1. Since Ψ ≤ 1, we have proved that θ • π(z) ≤ udec. We deduce by 6.4.4 that δ(z) ≤ udec . That is, ∆(u) ≤ udec. Conversely, if ∆(u) < 1, then by 6.5.2 there are matrices [aik ] ∈ Mn,m ⊗ B and [bkj ] ∈ Mm,n ⊗ B satisfying (6.30) and (6.31). For any 1 ≤ i ≤ n and 1 ≤ k ≤ m, consider the elements of M2 (B) deﬁned by cik = E11 ⊗ aik

and

cik = E21 ⊗ b∗ki .

Here E11 and E22 denote the diagonal matrix units of M2 . Deﬁne (cik ) ∈ M2n,m ⊗ M2 (B). c= (cik ) Let w : M2n → M2 (B) be the linear mapping implemented by the pair (c, c∗ ), in the notation of 6.5.3. Then the discussion in 6.5.3 shows that w is completely positive. Furthermore, by the last line of 6.5.3, b∗kj bkj < 1. w ≤ max aik a∗ik , Under the identiﬁcation M2n = M2 (Mn ), the mapping w is corner-preserving and its 1-2-corner is equal to u. For example, to see the last statement, note that for ∗ i, j ∈ {1, . . . , n}, we have by careful inspection that w(E12 ⊗ Eij ) = k cik cjk . However this last quantity equals

(E11 ⊗ aik )(E12 ⊗ bkj ) = E12 ⊗

k

as expected. We deduce that udec < 1.

aik bkj = E12 ⊗ u(Eij ),

k

2

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259

Corollary 6.5.8 Let Y be an operator space, let B be a C ∗ -algebra, and let u : Y → B be any ﬁnite rank operator. Then ∆(u) = inf{αcb βdec }, where the inﬁmum runs over all possible factorizations of u of the form u = βα, β α with Y −→ Mn −→ B. 2

Proof Combine Lemma 6.5.4 and Proposition 6.5.7. 6.6 NUCLEARITY AND SEMIDISCRETENESS FOR LINEAR OPERATORS

6.6.1 (Nuclear C ∗ -algebras and semidiscrete W ∗ -algebras) We brieﬂy review some fundamental results from C ∗ -theory, which will serve as a motivation and a guide for the results presented in this section. Let B be a C ∗ -algebra. Then B is nuclear (i.e. A ⊗min B = A ⊗max B for every C ∗ -algebra A) if and only α

βt

t if there exist nets of completely positive contractive maps B −→ Mnt −→ B such that b = limt βt αt (b) for every b ∈ B. This result is due to Choi and Eﬀros [89], and to Kirchberg [227]. The above property is usually referred to as the completely positive approximation property. Smith showed in [390] that replacing the words ‘completely positive contractive’ in the above, by ‘completely contractive’, gives an equivalent characterization of nuclear C ∗ -algebras. A proof of Smith’s result may be found in 7.1.12 below. Likewise, if M is a W ∗ -algebra, then A ⊗min M = A ⊗nor M for every C ∗ -algebra A if and only if there exist nets βt αt Mnt −→ M such that for b ∈ M we of completely positive contractions M −→ have b = w∗ -limt βt αt (b). This was proved by Eﬀros and Lance [140], who called such W ∗ -algebras semidiscrete. Furthermore, it is known that a W ∗ -algebra is semidiscrete if and only if it is injective [419]; and that a C ∗ -algebra B is nuclear if and only if its second dual B ∗∗ is semidiscrete [89]. These two results rely on Connes’ famous work on injectivity [99]. Delta norms were introduced by Pisier (see [337]) in order to give a new proof of the Choi–Eﬀros–Kirchberg characterization of nuclearity. We will see that they can also be used to obtain approximation properties for certain classes of operator algebra valued linear maps.

6.6.2 (K-Nuclear operators) Let Y be an operator space, let B be an approximately unital operator algebra, and let u : Y → B be a bounded operator. Given any constant K ≥ 0, we say that u is K-nuclear if whenever A is an approximately unital operator algebra, then IA ⊗ u extends to a bounded operator from A ⊗min Y into A ⊗max B, with IA ⊗ u : A ⊗min Y −→ A ⊗max B ≤ K. (6.35) (We warn the reader that this notion of ‘nuclearity’ for maps is quite diﬀerent from the class of maps of the same name introduced by Grothendieck, which

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Nuclearity and semidiscreteness for linear operators

there are also operator space variants of (e.g. see [149, Section 12.1]). See also 7.1.1 below, for a diﬀerent notion yet). Some comments on our deﬁnition: (1) Replacing A by Mn (A) and applying Corollary 6.1.15 and (1.36), we see that if u : Y → B is a K-nuclear map, then for any A as above, the tensor map IA ⊗u is completely bounded from A⊗min Y into A⊗max B, with IA ⊗ucb ≤ K. In particular, any K-nuclear map u is completely bounded, with ucb ≤ K. (2) We say that an operator algebra B is K-nuclear if the identity mapping IB is K-nuclear. If B is a C ∗ -algebra, then 6.1.14 says that B is nuclear in the classical sense if and only if it is 1-nuclear. Moreover if a C ∗ -algebra B is K-nuclear for some K ≥ 1, then it is 1-nuclear. Indeed if A is a C ∗ -algebra, and if the identity mapping I : A ⊗min B → A ⊗max B is bounded, then it is a ∗-representation, and hence is a contraction. Thus all of these notions of nuclearity coincide for C ∗ -algebras. The situation is rather diﬀerent if B is nonselfadjoint. We will see in 7.1.8 that if an approximately unital operator algebra B is 1-nuclear, then it has to be a C ∗ -algebra. We have met (for example, in 6.2.4) nonselfadjoint operator algebras B which satisfy A ⊗max B = A ⊗min B isometrically for any C ∗ -algebra A; however such an algebra B need not be 1-nuclear. On the other hand, any nonselfadjoint ﬁnite-dimensional operator algebra is clearly K-nuclear for some K > 1, but not 1-nuclear (by 7.1.8). (3) Let u : Y → B be a bounded operator, let K ≥ 0, and assume that B is a C ∗ -algebra. One can show by a variant of the proofs of 6.1.10 or 6.1.14, that u is K-nuclear provided that (6.35) holds for any C ∗ -algebra A. 6.6.3 (K-Semidiscrete operators) Let Y be an operator space, let M be a unital dual operator algebra, and let u : Y → M be a bounded operator. Given any constant K ≥ 0, we say that u is K-semidiscrete if whenever A is an approximately unital operator algebra, then IA ⊗ u extends to a bounded operator from A ⊗min Y into A ⊗nor M , with IA ⊗ u : A ⊗min Y −→ A ⊗nor M ≤ K. Similar comments to those in 6.6.2 apply to this deﬁnition. In particular, if M is a W ∗ -algebra, then M is semidiscrete in the classical sense if and only if it is 1-semidiscrete, and if and only if it is K-semidiscrete for some K ≥ 1. Theorem 6.6.4 (Pisier) Let Y be an operator space and let B be an approximately unital operator algebra. We consider a bounded operator u : Y → B and a constant K ≥ 0. Then the following assertions are equivalent: (i) u is K-nuclear (in the sense of 6.6.2). (ii) For any operator space X, the mapping IX ⊗u extends to a bounded operator from X ⊗min Y into X⊗δ B, with IX ⊗ u : X ⊗min Y −→ X ⊗δ B ≤ K. (ii)’ Same as (ii) for ﬁnite-dimensional operator spaces X.

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261

(iii) There exists a net ut : Y → B of ﬁnite rank operators converging to u in the point-norm topology, such that ∆(ut ) < K for each t. α

βt

t (iv) There exist nets of maps Y −→ Mnt −→ B, such that βt αt converges to u in the point-norm topology, and such that αt cb ∆(βt ) < K for each t.

If further B is a C ∗ -algebra, these conditions are equivalent to: α

βt

t (iv)’ There exist nets of maps Y −→ Mnt −→ B, such that βt αt converges to u in the point-norm topology, and such that αt cb βt dec < K for each t.

Proof The equivalence between (iii) and (iv) (and (iv)’ in the C ∗ -case) is an obvious consequence of Lemma 6.5.4 (and Corollary 6.5.8). In order to prove ‘(iii) ⇒ (i)’, we shall ﬁrst show that if u : Y → B is ﬁnite rank, then it is ∆(u)-nuclear. Going back to the deﬁnition of ∆(u), we assume that u = βα, where β : Mn → B is deﬁned by (6.29) for some [a1 · · · an ] in Rn (B) and [b1 · · · bn ]t in Cn (B), and α : Y → Mn is a linear mapping. If A is an approximately unital operator algebra and [cij ] ∈ Mn (A), we have (IA ⊗ β)([cij ]) =

cij ⊗ ai bj .

i,j

Hence δ (IA ⊗β)([cij ]) ≤ [cij ][a1 · · · an ] [b1 · · · bn ]t . Since the delta norm dominates the maximal tensor norm on A ⊗ B (see 6.4.2), this shows that IA ⊗ β : Mn (A) −→ A ⊗max B ≤ [a1 · · · an ] [b1 · · · bn ]t . Since Mn (A) = Mn ⊗min A, we have IA ⊗ α : A ⊗min Y −→ Mn (A) ≤ αcb . Hence we obtain by composition that IA ⊗ u : A ⊗min Y −→ A ⊗max B ≤ αcb [a1 · · · an ] [b1 · · · bn ]t . According to 6.5.1, this shows that u is ∆(u)-nuclear. Assume (iii), let A be an approximately unital operator algebra, and let z ∈ A ⊗ Y with ymin ≤ 1. Applying the above to each ut , we obtain that (IA ⊗ ut )(z) < K. Since (IA ⊗ ut )(z) converges to (IA ⊗ u)(z) in A ⊗max B, max this implies that (IA ⊗ u)(z) ≤ K. max This shows that u is K-nuclear. Assume (i), and let X be an operator space. To show (ii), it will suﬃce by 6.4.4 to show that for any pair of commuting completely contractive maps θ : X → B(H) and π : B → B(H) such that π is a homomorphism, we have (θ • π) ◦ (IX ⊗ u) : X ⊗min Y −→ B(H) ≤ K. (6.36)

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For such θ and π, consider the commutant A = [π(B)] ⊂ B(H). Then θ ⊗ IY : X ⊗min Y −→ A ⊗min Y ≤ 1.

(6.37)

We let Φ : A ⊗max B → B(H) be the map taking a ⊗ b to aπ(b), which is clearly contractive. Also, (θ • π) ◦ (IX ⊗ u) = Φ ◦ (IA ⊗ u) ◦ (θ ⊗ IY ) on X ⊗ Y . Hence applying (6.37) and appealing to (6.35), we deduce the desired estimate (6.36). This yields (ii). Finally, we assume (ii)’ and will show (iii). Let F be the set of all ﬁnitedimensional subspaces E of Y , ordered by inclusion. Our net (ut )t will be indexed by F. We let (εt )t be a net of positive real numbers with limt εt = 0. Given any t = E in F, we apply (ii)’ with X = E ∗ . If j ∈ X⊗Y is associated to the canonical embedding of E into Y , then (IX ⊗ u)(j) ∈ X ⊗ B is associated to the restriction u|E : E → B of u to E. Since jmin = 1 (by 1.35), we have ∆(u|E ) ≤ K. By 6.5.4, there exist n ≥ 1, and linear maps α : E → Mn and β : Mn → B, such that u|E = βα, αcb ≤ 1, and ∆(β) < K(1 + εt ). By the injectivity of Mn (see 1.2.10), α admits a completely contractive extension α : Y → Mn . Deﬁne ut = (1 + 2εt )−1 β α : Y −→ B. By 6.5.4 we have ∆(ut ) ≤ (1+2εt )−1 αcb ∆(β) < K. For y ∈ Y , the net (ut (y))t is eventually equal to (1 + 2εt )−1 u(y), and hence converges to u(y). 2 Theorem 6.6.5 Let Y be an operator space and let M be a unital dual operator algebra. We consider a bounded operator u : Y → M and a constant K ≥ 0. Then the following assertions are equivalent: (i) u is K-semidiscrete (in the sense of 6.6.3). (ii) For any ﬁnite-dimensional operator space X, the mapping IX ⊗ u extends to a bounded operator from X ⊗min Y into X⊗σδ B, with IX ⊗ u : X ⊗min Y −→ X⊗σδ B ≤ K. (iii) There exists a net ut : Y → M of ﬁnite rank operators converging to u in the point-w∗ topology, such that ∆(ut ) < K for each t. α

βt

t (iv) There exist nets of maps Y −→ Mnt −→ M such that βt αt converges to u ∗ in the point-w topology, such that αt cb ∆(βt ) < K for each t.

If further M is a W ∗ -algebra, these conditions are equivalent to: α

βt

t (iv)’ There exist nets of maps Y −→ Mnt −→ M so that βt αt converges to u in ∗ the point-w topology, and such that αt cb βt dec < K for each t.

Proof As in 6.6.4, we only need to show the equivalence between (i), (ii), and (iii). We omit the proof that (i) implies (ii), since this is similar to the corresponding implication in Theorem 6.6.4 (but uses 6.4.9 in place of 6.4.4).

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To see that (iii) implies (i), let A be an approximately unital operator algebra. We consider an arbitrary element k ak ⊗ yk in A ⊗ Y , (ak )k and (yk )k being ﬁnite families of A and Y respectively. It follows from 6.6.4 that ak ⊗ ut (yk ) ≤K ak ⊗ y k , max

k

min

k

for any t. Let π : A → B(H) and ρ : M → B(H) be two commuting completely contractive homomorphisms, such that ρ is w ∗ -continuous. Then π(ak )ρ ut (yk ) ≤ K ak ⊗ y k , (6.38) k

min

k

for any t. The w∗ -continuity of ρ ensures that ρ u(yk ) = w∗ -limt ρ ut (yk ) , for any k. We deduce that π•ρ ak ⊗ u(yk ) = π(ak )ρ u(yk ) = w∗ − lim π(ak )ρ ut (yk ) . k

t

k

k

Taking (6.38) into account, this yields ak ⊗ u(yk ) ≤ K ak ⊗ y k π • ρ k

k

. min

Taking the supremum over all such pairs (π, ρ), we deduce that ak ⊗ u(yk ) ≤ K ak ⊗ y k . k

nor

k

min

This proves that u is K-semidiscrete. We now assume (ii), and will show (iii). Let F 1 be the set of all ﬁnite subsets I1 of Y , let F 2 be the set of all ﬁnite subsets I2 of M∗ , and set F = F 1 ×F 2 ×(0, 1). We endow this set with its canonical order, given by (I1 , I2 , ε) ≤ (I1 , I2 , ε ) ⇐⇒ I1 ⊂ I1 , I2 ⊂ I2 , ε ≥ ε . We shall deﬁne a net (ut )t indexed by F which satisﬁes (iii). Fix t = (I1 , I2 , ε) in F, and let E ⊂ Y be the ﬁnite-dimensional subspace spanned by I1 . Applying (ii) with X = E ∗ , and arguing as in the proof of Theorem 6.6.4, we ﬁnd that the tensor z ∈ X ⊗ M associated to the restriction u|E of u to E satisﬁes σδ(z) ≤ K. By Lemma 6.4.7, there exists a net (zs )s of X ⊗ M with δ(zs ) < K converging to z in the w∗ -topology of X⊗σδ M . Let ws : E → M be associated to zs ; then we have ∆(ws ) < K. The convergence of zs to z is easily seen to imply that (ws ) converges to u|E in the point w∗ -topology. Thus we may ﬁnd an operator w : E → M such that ∆(w) < K, and u(y), ϕ − w(y), ϕ < ε, (6.39) y ∈ I1 , ϕ ∈ I2 . As in the proof of 6.6.4, we may extend w to some ut : Y → M with ∆(ut ) < K. 2 It clearly follows from (6.39) that the net (ut ) satisﬁes (iii).

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6.6.6 (A characterization of nuclear and semidiscrete operator algebras) Let B be an approximately unital operator algebra and let K ≥ 1. Applying Theorem 6.6.4 to IB , we see that B is K-nuclear (in the sense of 6.6.2 (2)) if and only α

βt

t if there exist nets of maps B −→ Mnt −→ B such that αt cb ∆(βt ) < K, and b = limt βt αt (b), for b ∈ B. If B is a 1-nuclear C ∗ -algebra then with some more work (e.g. see [337, Section 12]) one can also ensure that αt , βt are completely positive, recovering one direction of the Choi–Eﬀros–Kirchberg result mentioned in 6.6.1. In any case, since βt cb ≤ ∆(βt ) (as is easily argued from 6.5.1 say, or from the comparison · min ≤ δ(·) proved in 6.4.2), we can deduce as a special case of what we did above, that if B is 1-nuclear then there exists an ‘approximate factorization’ of IB as above, where αt , βt are completely contractive. Likewise, if M is a unital dual operator algebra, Theorem 6.6.5 implies that βt αt Mnt −→ M M is K-semidiscrete if and only if there exist nets of maps M −→ such that αt cb ∆(βt ) < K, and b = w∗ -limt βt αt (b), for b ∈ M .

6.6.7 (Local reﬂexivity) Let Y be an operator space. For a ﬁnite-dimensional operator space X, we may canonically identify (X ⊗min Y )∗∗ and X ⊗min Y ∗∗ as topological vector spaces. The corresponding ‘identity mapping’ JX : (X ⊗min Y )∗∗ −→ X ⊗min Y ∗∗

(6.40)

is a contraction. To see this, consider z ∈ X ⊗ Y ∗∗ and let u : X ∗ → Y ∗∗ be the associated linear mapping. If z(X⊗minY )∗∗ ≤ 1, then there exists by Goldstine’s Lemma (see A.2.1), a net (zt )t in X ⊗ Y which converges to z in the w ∗ -topology, such that zt X⊗min Y ≤ 1 for any t. If we let ut : X ∗ → Y be the linear mapping associated to zt , this is equivalent to saying that each ut is a complete contraction (by (1.32)), and that u(ϕ) is the w ∗ -limit of ut (ϕ), for ϕ ∈ X ∗ . These immediately imply that ucb ≤ 1, that is, zX⊗minY ∗∗ ≤ 1. −1 is not a contraction in general, that is, the two The inverse mapping JX ∗∗ norms on X ⊗ Y given by (X ⊗min Y )∗∗ and X ⊗min Y ∗∗ may be diﬀerent (see the Notes to this section for more). Given a constant K ≥ 1, we say that Y is −1 ≤ K for any ﬁnite-dimensional operator space X. K-locally reﬂexive if JX Corollary 6.6.8 Let Y be an operator space, let B be an approximately unital operator algebra, and let u : Y → B be a bounded operator. (1) Let K ≥ 0. If the second adjoint u∗∗ : Y ∗∗ → B ∗∗ is K-semidiscrete, then u is K-nuclear. (2) Let K ≥ 1 and K ≥ 1, and assume that Y is K -locally reﬂexive. If u is K-nuclear, then u∗∗ is KK -semidiscrete. Proof Assume that u∗∗ is K-semidiscrete, and let X be a ﬁnite-dimensional operator space. Since the mapping JX in (6.40) is a contraction, it follows from Theorem 6.6.5 that IX ⊗ u∗∗ : (X ⊗min Y )∗∗ −→ X⊗σδ B ∗∗ ≤ K. (6.41)

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265

Hence by 6.4.10 (2), we deduce that IX ⊗ u : X ⊗min Y −→ X⊗δ B ≤ K. This shows that u is K-nuclear by Theorem 6.6.4. Assume conversely that u is K-nuclear. Reversing the arguments above, we see that (6.41) holds true for any ﬁnite-dimensional operator space X. Composing −1 , we deduce that with JX IX ⊗ u∗∗ : X ⊗min Y ∗∗ −→ X⊗σδ B ∗∗ ≤ K J −1 . X Now if Y is K -locally reﬂexive, this is less than or equal to KK . Hence u∗∗ is 2 KK -semidiscrete by Theorem 6.6.5. We will continue our discussion of nuclearity and semidiscreteness in the ﬁrst section of the next chapter. 6.7 NOTES AND HISTORICAL REMARKS 6.1: If A and B are C ∗ -algebras, then their algebraic tensor product A ⊗ B is a ∗-algebra: we set (a⊗b)∗ = a∗ ⊗b∗ , for a ∈ A and b ∈ B. By deﬁnition a C ∗ -norm is a norm · α on A ⊗ B such that xyα ≤ xα yα , y ∗ yα = y2α , and yα = y ∗ α for any x, y ∈ A⊗B. Thus the resulting completion A⊗α B is a C ∗ algebra. The study of C ∗ -norms goes back at least to the work of Turumaru [413], Takesaki [406], who proved that · min is the smallest C ∗ -norm on A ⊗ B, and Guichardet [174], who proved that · max is the greatest one. The term ‘nuclear’ was introduced by Lance in the study [238]. The reader is referred, for example, to [407, Chapter IV], or to [420], for the basics on C ∗ -norms, and their history. We will not attempt below to reference carefully the huge number of papers devoted to the selfadjoint versions of topics in this chapter. The maximal tensor product for nonselfadjoint operator algebras was introduced by Paulsen and Power [316]. This is a generalization of the maximal C ∗ -norm for C ∗ -algebras (see 6.1.4). However there seems to be no good analogue of the theorem of Takesaki which asserts that · min ≤ · α if · α is a C ∗ -tensor norm. Indeed in [316], an example is given of two unital operator algebras A and B and an algebra homomorphism Φ : A ⊗ B → B(H) such that each of the maps a ∈ A → Φ(a ⊗ 1B ) and b ∈ B → Φ(1A ⊗ b) is a complete isometry, but for which Φ(a ⊗ b) < ab for some a ∈ A, b ∈ B. The normal tensor product for C ∗ -algebras was introduced by Eﬀros and Lance in [140]. Its extension to nonselfadjoint operator algebras appears in [316] under another name, see also [249]. There is also a binormal tensor product on pairs of W ∗ -algebras which was introduced in [140]. It can be extended to the nonselfadjoint setting as follows. Let N and M be two unital dual operator algebras. For any n ≥ 1 and any y ∈ Mn (N ⊗ M ), deﬁne ybin = sup{(IMn ⊗ (π • ρ))(y)}, where the inﬁmum runs over all pairs of commuting w ∗ -continuous completely contractive homomorphisms π : N → B(H) and ρ : M → B(H). These are matrix norms on

266

Notes and historical remarks

N ⊗ M , and the resulting completion N ⊗bin M is a unital operator algebra. It follows from 6.1.7 that if A, B are approximately unital operator algebras, and if θ : A ⊗max B → M is a completely contractive homomorphism into a dual operator algebra M , then θ extends to a separately w ∗ -continuous completely contractive homomorphism θ˜: A∗∗ ⊗bin B ∗∗ → M . Applying that property with M = (A⊗max B)∗∗ , one may deduce that A∗∗ ⊗bin B ∗∗ ⊂ (A⊗max B)∗∗ completely isometrically. The two statements 6.1.2 and 6.1.14 are due to Paulsen and Power [316]. This paper only considers unital operator algebras, in which case Lemma 6.1.7, and hence Lemma 6.1.13 are quite transparent. We could not ﬁnd any reference for these statements in the nonunital case; or for the result in 6.1.11. Lemma 6.1.10 is from [252]. Variants of the maximal tensor product, and nuclearity and semidiscreteness, for other classes of operator spaces such as TROs (see 4.4.1), have been introduced and studied by Kirchberg, and Kaur and Ruan, and others. See [226], for example. 6.2: Besides Ando’s theorem, the main sources for joint dilations and their relationships with the maximal tensor product are the two papers [316] and [315] by Paulsen and Power. Related work also appears in [114] and [109]. Proposition 6.2.3 is from [316], which also contains the identity (6.9). As noticed in the latter paper, the examples by Crabb and Davie [103] and Varopoulos [416] showing that von Neumann’s inequality fails for triples of commuting contractions, imply that A(D2 ) ⊗min A(D) and A(D2 ) ⊗max A(D) are not isometric. Arveson’s dilation theorem discussed in 6.2.1 is from [21]. Proposition 6.2.4 seems to be new, but the fact that A ⊗max A(D) = A ⊗min A(D) completely isometrically for any C ∗ -algebra A is well-known. Indeed, Corollary 6.2.6 is an old result due to Arveson (unpublished). A remarkable generalization was obtained by Parrott [299], as follows. Let A be a C ∗ -algebra, and assume for simplicity that A is unital. Let π : A → B(H) be a unital ∗-representation, let ρ : A(D 2 ) → B(H) be a unital completely contractive homomorphism, and assume that π and ρ have commuting ranges. If (z, w) denote the variables in D2 , let T = ρ(z) and S = ρ(w). These are commuting contractions on H, and since A is selfadjoint, the range of π commutes with S, S ∗ , T, T ∗. Hence by [299, Theorem 4], there exist a Hilbert space K, an isometry J : H → K, commuting unitaries U, V ∈ B(K), and a unital ∗-representation π ˆ : A → B(K) with U and whose range commutes ˆ (anm )U n V m J for any ﬁnite V , and such that n,m π(anm )T n S m = n,m J ∗ π family (an,m )n,m≥0 in A. If we let ρˆ : C(T2 ) → B(K) denote the ∗-representation ˆ (a)ˆ ρ(f )J, for deﬁned by letting π ˆ (f ) = f (U, V ), we obtain that π(a)ρ(f ) = J ∗ π a ∈ A and f ∈ A(D2 ). Thus 6.2.3 (ii) is satisﬁed for C = A, B = A(D2 ), and D = C(T2 ). According to 6.2.5, this implies that A ⊗max A(D2 ) = A ⊗min A(D2 ) completely isometrically. It is not hard to deduce that A(D 2 ; A1 ⊗max A2 ) and A(D; A1 ) ⊗max A(D; A2 ) coincide completely isometrically, for C ∗ -algebras A1 and A2 . We leave this to the reader. 6.3: The main result of this section, Theorem 6.3.5, is from [316]. Some of

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267

its consequences, such as assertions (2) and (3) in 6.3.7 appeared earlier in [315]. The McAsey–Muhly theorem mentioned in 6.3.2 admits a generalization to nest algebras due to Paulsen, Power, and Ward [317]. See also [109] for related work. In fact, many results in Section 6.3 extend to the case when triangular algebras are replaced by nest algebras. For example, if A is an approximately unital operator algebra and if A ⊗min A(D) = A ⊗max A(D) completely isometrically, then we have A ⊗min A = A ⊗max B for every ﬁnite-dimensional nest algebra B (see [316, Theorem 3.4]), and A ⊗min B = A ⊗nor B for every nest algebra A (by [315, Theorem 3.1]). It is also possible to prove a variant of 6.3.6 involving the binormal tensor product considered in the Notes to Section 6.1. Namely if M is a unital dual operator algebra, then the conditions (i) and (ii) in 6.3.6 are also equivalent to T ∞ ⊗min M being completely isometric to T ∞ ⊗bin M . As a consequence, one obtains that T ∞ ⊗min T ∞ = T ∞ ⊗bin T ∞ . More generally, if A and B are any two nest algebras, then A ⊗min B = A ⊗bin B completely isometrically. This result is implicit in [315]. Proposition 6.3.8 is from [316]. 6.4: Delta norms ﬁrst appeared in 1996 in an early and preliminary version of Pisier’s book [337]. The material from 6.4.1 to 6.4.5 dates back to that time. In fact, the δ norm was introduced for the purpose of giving an operator space proof of the fact that a C ∗ -algebra is nuclear if and only if it has the completely positive approximation property (see 6.6.1). We refer the reader to [337, Chapter 12] for that proof. There are other interesting proofs of 6.4.4, e.g. [265, p. 133]. The normal delta norm was introduced by Le Merdy in [249], from which the second part of this section is taken. Oikhberg and Pisier considered a variant of the δ norm for pairs of operator spaces [294]. If X and Y are operator spaces, deﬁne µ(z) = sup{θ • σ(z)} for z ∈ X ⊗ Y , where the supremum runs over all pairs of complete contractions θ : X → B(H) and σ : Y → B(H) with commuting ranges. In analogy with 6.4.4, it is showed in [294] that µ(z) < 1 if and only if there are z1 , z2 in X ⊗ Y such that z = z1 + z2 , and z1 X⊗h Y + z2f Y ⊗h X < 1. Here z2f denotes the image of z2 under the canonical ‘ﬂip’ from X ⊗ Y to Y ⊗ X. 6.5: The decomposable norm was introduced by Haagerup in [180]. This norm has several remarkable properties that we brieﬂy review. First, we have ucb ≤ udec for any decomposable map u : C → B between C ∗ -algebras. Next, we have udec = ucb = u if u is positive, and we also have udec = ucb if B is injective. Indeed, this follows from Paulsen’s proof of 1.2.8. Furthermore if A is a third C ∗ -algebra and if u : C → B and v : B → A are decomposable, then v ◦ u is decomposable and v ◦ udec ≤ vdec udec. The main result of [180] asserts that if M is any W ∗ -algebra, and if C is any inﬁnite-dimensional C ∗ -algebra, then M is injective if (and only if) every completely bounded map u : C → M is decomposable. We have seen in (6.34) that decomposable maps ‘tensorize’ for the maximal tensor product. This result turns out to be optimal. Indeed, Kirchberg showed that if M is a W ∗ -algebra, C is a C ∗ -algebra, K ≥ 0 is a constant, and u : C → M is a bounded map such that for any C ∗ -algebra A, IA ⊗ u : A ⊗max C → A ⊗nor M ≤ K, then u is decomposable and udec ≤ K.

268

Notes and historical remarks

We refer to [337, Chapter 14] for a proof. Thus if M is not injective and C is inﬁnite-dimensional, then there exist completely bounded maps u : C → M and C ∗ -algebras A such that IA ⊗ u is unbounded with respect to the maximal tensor norms. The ﬁrst part of Section 6.5 is composed of several simple observations. Results 6.5.7 and 6.5.8 are due to Pisier (see [337]). In [208], Junge and Le Merdy proved a generalization of 6.5.7, namely, if u : C → B is a ﬁnite rank operator between C ∗ -algebras, then ∆(u) = udec. Indeed the proof of 6.5.7 presented here is a simpliﬁcation of the proof of this more general result. There is no good analogue of the above mentioned result with the completely bounded norm replacing the decomposable one. To see this, if B, C are C ∗ -algebras and u : C → B is ﬁnite rank, deﬁne γ(u) = inf{αcb βcb}, where the inﬁmum runs over all inα

β

tegers n ≥ 1, and all possible factorizations u = βα, with C −→ Mn −→ B. If B is a W ∗ -algebra, it is shown in [208] that B is injective if and only if γ(u) = u cb for any C, and any ﬁnite rank operator u : C → B. 6.6: Theorem 6.6.4 is from the preliminary version of [337] alluded to in the Notes to 6.4. However the terms ‘K-nuclear’ and ‘K-semidiscrete’, and their use in a setting involving nonselfadjoint algebras, are from [249]. See 6.6.1, 6.6.2, and 6.6.3 for the connection with the usual notions of nuclear C ∗ -algebras and semidiscrete W ∗ -algebras. Theorem 6.6.5 and its corollary 6.6.8 are taken from [249]. In general, δ and σδ are not comparable norms. To see this, let M be a semidiscrete W ∗ -algebra which is not a nuclear C ∗ -algebra (for instance take M = B(2 )), and let K ≥ 1. Then M cannot be K-nuclear (see 6.6.2 (2)), hence Theorem 6.6.4 ensures that there exist a ﬁnite-dimensional operator space X, and z ∈ X ⊗ M such that zmin ≤ 1 and δ(z) ≥ K. However we have σδ(z) = zmin by 6.6.5. Thus δ(z) ≥ Kσδ(z). Local reﬂexivity (see 6.6.7) is a fundamental notion for operator spaces, although we will barely touch this topic in this book. See [149, Part IV] for a comprehensive treatment. We merely recall that local reﬂexivity was deﬁned for C ∗ -algebras by Eﬀros and Haagerup in [136]. It is quite remarkable that there are operator spaces which are not K-locally reﬂexive for any K. The full group C ∗ -algebra over the free group F2 , and B(H) (if H is inﬁnite-dimensional) are such examples. On the other hand, it follows from a pioneering paper of Archbold and Batty [15] that any nuclear C ∗ -algebra is 1-locally reﬂexive. Other fundamental papers on locally reﬂexive operator spaces include [137,141]. See also the cited papers of Kirchberg, and [296], and references therein.

7 Selfadjointness criteria

There is a well-known book by Burckel with the title Characterizations of C(X) among its subalgebras [80]. This chapter may be viewed as having a similar aim, namely to discuss several interesting criteria which force an operator algebra to be selfadjoint. Such results are in some sense ‘negative’ in nature, showing that certain results or themes which are important for C ∗ -algebras, may not be transferred to general operator algebras, or at least not in a literal way. Nonetheless, they are sometimes quite useful, for example when in the middle of some proof one needs to show that a certain algebra is a C ∗ -algebra. Throughout this chapter, B is an operator algebra, usually approximately unital for speciﬁcity. We say that ‘B is a C ∗ -algebra’, or B is selfadjoint, if B is completely isometrically homomorphic to a C ∗ -algebra. Equivalently, B = ∆(B), in the notation of 2.1.2. The sections of this chapter are mostly independent of each other. In fact, our ﬁrst section is largely a continuation of the study of nuclearity and semidiscreteness begun in Chapter 6 (and particularly the last section, Section 6.6). 7.1 OS-NUCLEAR MAPS AND THE WEAK EXPECTATION PROPERTY In Chapter 6, we studied a certain notion of nuclearity for linear mappings valued in an operator algebra. We now consider a related but diﬀerent notion. We also discuss the weak expectation property for operator spaces. 7.1.1 (OS-nuclear maps) Let X and Y be two operator spaces, and suppose that u : X → Y is a bounded operator. We say that u is OS-nuclear if there βt αt Mnt −→ X such that βt αt exist nets of completely contractive maps Y −→ converges to u in the point norm topology. We say that the operator space X is OS-nuclear if the identity mapping IX : X → X is OS-nuclear. Smith showed in [390] that for a C ∗ -algebra, this notion coincides with the usual notion of nuclearity. A generalization of Smith’s result will be given in this section (see 7.1.11 and 7.1.12). It is clear that any OS-nuclear map u : Y → X is completely contractive. 7.1.2 (OS-nuclearity versus 1-nuclearity) Let Y be an operator space, let B be an approximately unital operator algebra and suppose that u : Y → B is a bounded operator. If u is 1-nuclear (in the sense of 6.6.2), then it is OSnuclear, as we observed in 6.6.6. Although we will prove a partial converse later

270

OS-nuclear maps and the weak expectation property

in this section, OS-nuclearity does not imply 1-nuclearity in general. Indeed assume that B is a C ∗ -algebra, and let u : Mn → B be a linear mapping. Then u : Mn → B is OS-nuclear if and only if ucb ≤ 1. On the other hand, u is 1-nuclear if and only if ∆(u) ≤ 1. Indeed assume that u is 1-nuclear and apply 6.6.4 (ii) with Y = Mn and X = Y ∗ = Sn1 . Let j ∈ Sn1 ⊗ Mn be the tensor associated to the identity mapping on Mn . Then jmin = 1, by (1.32). Also, (ISn1 ⊗ u)(j) ∈ Sn1 ⊗ B is the tensor associated to u as in the ﬁrst paragraph of 6.5.1, so that ∆(u) ≤ jmin = 1. The converse direction is clear, from 6.6.4 (iii). Hence u is 1-nuclear if and only if udec ≤ 1, by 6.5.7. However if B is a noninjective W ∗ -algebra, then by Haagerup’s work in [180], there exists a map u as above with 1 = ucb < udec. 7.1.3 (The weak expectation property) Let X be an operator space and let iX : X → X ∗∗ be the canonical embedding. We say that X has the weak expectation property (WEP in short) if there exist a Hilbert space H, and two completely contractive mappings J : X → B(H) and P : B(H) → X ∗∗ , such that iX = P J. In this case, since iX is completely isometric, the mapping J is necessarily completely isometric. Conversely, if X has the WEP, and if J : X → B(H) is an arbitrary complete isometry for some Hilbert space H, then there exists a complete contraction P : B(H) → X ∗∗ such that iX = P J. Indeed this follows from the extension theorem 1.2.10. If X is a dual operator space, then there is a completely contractive idempotent from X ∗∗ onto X (namely the adjoint of iX∗ : X∗ → X ∗ ). It is easy to see from this that X has the WEP if and only if X is injective. Lemma 7.1.4 Let X, Y be operator spaces, with Y ⊂ B(H) completely isometrically, for some Hilbert space H, and let u : Y → X be an OS-nuclear map. Then iX ◦ u : Y → X ∗∗ extends to a completely contractive map P : B(H) → X ∗∗ . Proof By assumption, there exist nets of completely contractive mappings βt

α

t Mnt and Mnt −→ X, such that βt αt converges to u in the point norm Y −→ topology. By the extension theorem 1.2.10, there exists for any t a completely contractive mapping αˆt : B(H) → Mnt extending αt . Since X ∗∗ is a dual op∗ erator space, CB(B(H), X ∗∗ ) = B(H) ⊗ X ∗ is a dual Banach space (see (1.51)). Since the net (iX ◦ βt αˆt )t lies in the unit ball, it has a w∗ -cluster point P : B(H) → X ∗∗ , in that ball. For y ∈ Y and ϕ ∈ X ∗ , we have

limiX ◦ βt αˆt (y), ϕ = limϕ, βt αt (y) = ϕ, u(y). t

t

Hence P (y), ϕ = ϕ, u(y). Thus P|Y = iX ◦ u.

2

Corollary 7.1.5 Any OS-nuclear operator space has the WEP. We proved in 4.2.8 (3) that if an approximately unital operator algebra is injective, then it is selfadjoint (and unital). In Theorem 7.1.7 below we will extend this result to operator algebras with the WEP. We will need the following result,

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which is of independent interest. As usual, ∆(· ) denotes the diagonal C ∗ -algebra of an operator algebra (see 2.1.2). Lemma 7.1.6 Let B be an operator algebra, regarded as a subalgebra of B ∗∗ in the canonical way. (1) If B ⊂ ∆(B ∗∗ ), then B is selfadjoint. (2) If B ∗∗ is selfadjoint, then B is selfadjoint. (3) If the unitization B 1 is selfadjoint (resp. completely isomorphic to a C ∗ algebra), then B is selfadjoint (resp. completely isomorphic to a C ∗ -algebra). Proof For (3), simply use the fact that a two-sided ideal in a C ∗ -algebra is selfadjoint. Clearly (2) follows from (1). For (1), assume that B is a subalgebra of B(H). Then B ∗∗ ⊂ B(H)∗∗ . By A.2.3 (4), B ∗∗ ∩ B(H) = B. Assume that B ⊂ ∆(B ∗∗ ), and let b ∈ B. Then b∗ both belongs to B(H) and B ∗∗ . Hence 2 b∗ ∈ B by the above. Thus B = ∆(B). Theorem 7.1.7 Let B be an approximately unital operator algebra with the WEP. Then B is selfadjoint. Proof Let (et )t be a cai of B. The second dual B ∗∗ is a unital operator algebra. Assume that B ∗∗ is a unital subalgebra of B(H). Thus B ⊂ B ∗∗ ⊂ B(H) completely isometrically, and et → IH strongly (as may be seen via 2.1.9, for example). By 7.1.3 and the WEP assumption, there exists a complete contraction P : B(H) → B(H) taking values in B ∗∗ , such that P (b) = b for every b ∈ B. In particular P (et ) = et for any t. Arguing as in the proof of 4.2.8, we deduce that P (IH ) = IH . Thus P is completely positive, and selfadjoint (see 1.3.3). That is, P (T )∗ = P (T ∗ ), for every T ∈ B(H). Given any b ∈ B, we have b∗ = P (b)∗ = P (b∗ ) ∈ B ∗∗ . This shows that B ⊂ ∆(B ∗∗ ). Hence B is selfadjoint by Lemma 7.1.6. 2 The following is a straightforward consequence of 7.1.5, 7.1.7, and 7.1.2: Corollary 7.1.8 Any OS-nuclear (and hence any 1-nuclear) approximately unital operator algebra is selfadjoint. 7.1.9 (OS-semidiscreteness) Let X be a dual operator space. We say that X αt Mnt is OS-semidiscrete if there exist nets of completely contractive maps X −→ βt

and Mnt −→ X such that βt αt converges to IX in the point w∗ -topology, that w∗

is, βt αt (x) −→ x, for every x ∈ X. It is also possible to deﬁne OS-semidiscrete linear maps but we will not use these here. If a dual operator space X is OSsemidiscrete, then it is injective. To see this, suppose that X1 is a subspace of an operator space X2 , and that u : X1 → X is a complete contraction. Let αt , βt be as above. Since Mn is injective, αt ◦ u has a completely contractive extension α ˆ t : X2 → Mnt . By the argument in 7.1.4, (βt ◦ α ˆ t ) has a w∗ -cluster point u ˆ, say, in CB(X2 , X), and uˆ extends u. Thus X is injective. We shall see later in 8.6.2 that the converse is true too, an injective dual operator space is OS-semidiscrete.

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Let M be a unital dual operator algebra. Since injective unital operator algebras are selfadjoint (see 4.2.8), we deduce that if M is OS-semidiscrete, then M is a W ∗ -algebra. On the other hand, the argument in 7.1.2, combined with Theorem 6.6.5, will show that M is OS-semidiscrete provided that it is 1-semidiscrete. Hence M is a W ∗ -algebra if M is 1-semidiscrete. The next result, concerning uniform algebras, does not extend to general operator algebras. Indeed as we observed in 6.6.2 (2), any ﬁnite-dimensional operator algebra is K-nuclear for some K ≥ 1. Proposition 7.1.10 A uniform algebra B which is K-nuclear for some K ≥ 1, is selfadjoint. More generally, assume that for some K ≥ 1, there exists a net of βt αt pairs of linear mappings B −→ Mnt −→ B, such that βt αt converges strongly to IB , and such that αt cb βt < K, for all t. Then B is selfadjoint. Proof In this proof we shall use (1.10) several times without announcement. Thus B(X, B) = CB(X, B) isometrically for any operator space X. By the argument in 6.6.6, we need only prove the ‘more generally’ part of the statement. Thus we assume the existence of nets (αt )t and (βt )t as above. We will use some Banach space theory which may be found in [324, Section 8.c], for example. We say that a Banach space B has a local unconditional structure if there is a constant C > 0 with the following property: For any ﬁnite-dimensional subspace E if B, there is a Banach space F with an unconditional basis, and two linear mappings w1 : E → F and w2 : F → B, such that w2 w1 is equal to the canonical embedding E → B, and w1 w2 ≤ C. A deep result of Kisliakov [232] asserts that if B ⊂ C(Ω) is a uniform algebra, then B is actually equal to C(Ω) provided that B has a local unconditional structure. It therefore suﬃces to prove that for any ﬁnite-dimensional subspace E of B, there exist an integer N ≥ 1, and two ∞ linear mappings w1 : E → ∞ N and w2 : N → B, such that w2 w1 = jE , the canonical embedding of E into B, and w1 w2 ≤ K + 1. We ﬁx E ⊂ B as above, and let ε > 0. Since E is ﬁnite-dimensional, any linear mapping from E into B factors through a matrix space. More precisely, given any u : E → B, there exist n ≥ 1, and two linear mappings α : E → Mn and β : Mn → B, such that βα = u. Then we may deﬁne a factorization norm γ on the space B(E, B), by letting γ(u) = inf{αcbβ}, where the inﬁmum runs over all integers n ≥ 1, and all possible factorizations as above. It is easy to see that γ is indeed a norm on B(E, B). For example, to see the triangle inequality, let u1 , u2 : E → B, and let c1 , c2 be positive numbers with γ(uj ) < cj , for j = 1, 2. Fix linear mappings αj : E → Mnj and βj : Mnj → B such that αj cb ≤ 1, βj < cj , and uj = βj αj . Let n = n1 + n2 and regard Mn1 ⊕ Mn2 ⊂ Mn in the usual way. Then let P : Mn → Mn1 ⊕ Mn2 be the canonical conditional expectation. We deﬁne a map α : E → Mn by letting α(x) = (α1 (x), α2 (x)) for x ∈ E, and we deﬁne β : Mn → E by letting β(z) = β1 (z1 ) + β2 (z2 ) for z ∈ Mn , if P (z) = (z1 , z2 ). Then βα = u1 + u2 , αcb ≤ 1, and β ≤ β1 + β2 . This shows that γ(u1 + u2 ) < c1 + c2 .

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Since E is ﬁnite-dimensional, γ is clearly equivalent to the usual norm on B(E, B). Thus the topology induced by γ coincides with the point norm topology on B(E, B). By assumption, βt αt |E converges to jE in this topology, and so γ(jE ) = lim γ(βt αt|E ) ≤ lim sup αt|E cbβt ≤ K. t

t

Thus we may write iE = βα, for some α : E → Mn and β : Mn → B satisfying αcb β ≤ K + ε. We now use the well-known elementary fact from Banach space theory, that there exist an integer N ≥ 1, a subspace G of ∞ N , and an isomorphism σ : G → E such that σ −1 σ ≤ 1 + ε. By 1.2.10, the mapping ασ : G → Mn extends to a map w : ∞ N → Mn , with wcb = ασcb ≤ αcb σ. We let w2 = βw : ∞ → B, and we let w1 be the mapping σ −1 regarded as valued N ∞ in N . Then w2 w1 = jE , and moreover, w1 w2 ≤ σ −1 βwcb ≤ σ −1 σαcbβ ≤ (K + ε)(1 + ε). This yields the desired factorization property for jE .

2

Proposition 7.1.11 Let B be a unital operator algebra, and let Y be a unitalsubspace of B. Then the canonical embedding Y → B is 1-nuclear if and only if it is OS-nuclear. Proof We said in 7.1.2 that 1-nuclearity implies OS-nuclearity. The proof of the other direction is a variant of that of Theorem 7.1.7. We let u : Y → B be the canonical embedding and assume that u is OS-nuclear. We let H be a Hilbert space such that B ∗∗ ⊂ B(H) is a unital-subalgebra. We thus have Y ⊂ B ⊂ B ∗∗ ⊂ B(H) completely isometrically, with IH ∈ Y . By Lemma 7.1.4, there is a completely contractive mapping P : B(H) → B(H) taking values in B ∗∗ , and such that P (y) = y, for every y ∈ Y . In particular, P is unital. As in the proof of 7.1.7, P is completely positive, and selfadjoint. Hence P takes values in the diagonal C ∗ -algebra D = ∆(B ∗∗ ). Let A be an approximately unital operator algebra. Since P is completely positive and contractive, we have IA ⊗ P : A ⊗max B(H) −→ A ⊗max D ≤ 1 by Lemma 6.1.10. Since D ⊂ B ∗∗ , the last equation together with 6.1.9 gives IA ⊗ P : A ⊗max B(H) −→ A ⊗max B ∗∗ ≤ 1. (7.1) Let v : Y → B(H) be the canonical embedding. Since u is OS-nuclear, v is OSnuclear too. We claim that the map v is actually 1-nuclear. Indeed for any n ≥ 1, and any linear map β : Mn → B(H), we have βcb = βdec (see the Notes to Section 6.5). Hence our claim follows from 6.6.4. Thus IA ⊗ v : A ⊗min Y −→ A ⊗max B(H) ≤ 1. (7.2)

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Hilbert module characterizations

The canonical map iB : B → B ∗∗ induces a complete isometry from A ⊗max B to A ⊗max B ∗∗ (see 6.1.3). Since P v = iB u, (7.1) and (7.2) gives IA ⊗ u : A ⊗min Y −→ A ⊗max B ≤ 1. Thus u is 1-nuclear.

2

7.1.12 (A remark) It is not hard to extend Proposition 7.1.11 above to approximately unital operator algebras. In that case, the assumption that Y is unital needs to be replaced by the assumption that Y contains a cai of B. The main adjustment one needs to make to the last proof, is that to see that P (IH ) = IH , one uses the argument in 7.1.7. This extension of 7.1.11 applies in particular when B is a C ∗ -algebra and Y = B. Thus we recover Smith’s theorem from [390], which states that B is nuclear if and only if it is OS-nuclear. 7.2 HILBERT MODULE CHARACTERIZATIONS In this section we discuss characterizations of ‘selfadjointness’ in terms of Hilbert modules, or equivalently in terms of the completely contractive representations of B (see 3.1.6 and 3.1.7). 7.2.1 (Complemented submodules) Let π : B → B(H) be a completely contractive representation of an operator algebra B on a Hilbert space H. If we regard H as a Hilbert B-module, then a submodule is simply a closed π(B)-invariant subspace K of H (see 3.2.1). Let p ∈ B(H) be an idempotent map with range equal to K, and let K = Ker(p). Then K is π(B)-invariant if and only if pπ(b) = π(b)p, for every b ∈ B. Indeed assume that K is π(B)-invariant, and let ζ ∈ H. Then π(b)pζ ∈ K, π(b)(1 − p)ζ ∈ K , and π(b)ζ = π(b)pζ + π(b)(1 − p)ζ. Hence pπ(b)ζ = π(b)pζ. Conversely, suppose that p commutes with π(B). If ζ ∈ K , then pζ = 0, hence pπ(b)ζ = π(b)pζ = 0. Thus π(b)ζ ∈ K . In this chapter we will consider direct sum decompositions H = K ⊕ K , for closed subspaces K and K , which are not necessarily mutually orthogonal. We say that a closed submodule K of H is a complemented submodule if H admits such a direct sum decomposition H = K ⊕ K , with the topological complement K also a closed π(B)-invariant subspace. Perhaps one should say topologically complemented submodule here, but for brevity we use the shorter phrase. Reducing submodules (in the sense of 3.2.3) are those submodules for which the latter property holds with K = K ⊥ . By the last paragraph, a submodule K is complemented (resp. reducing) if and only if there exists an idempotent map p in the commutant [π(B)] , whose range is equal to K (resp. the orthogonal projection onto K belongs to [π(B)] ). 7.2.2 (Module complementation property) Let B be an operator algebra, which we assume to be approximately unital for speciﬁcity (the general case is discussed in the Notes). We say that B has the module complementation property if whenever H is a Hilbert space and π : B → B(H) is a nondegenerate completely

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contractive representation, then every π(B)-invariant subspace of H is a complemented submodule. Likewise we say that B has the reducing property if for every nondegenerate completely contractive representation π : B → B(H), every π(B)-invariant subspace is reducing. Any C ∗ -algebra has the reducing property. Assume that B has the module complementation property, and consider a nondegenerate representation π : B → B(H), that we only assume to be completely bounded. If K is a π(B)-invariant subspace of H, then we claim that there is a π(B)-invariant subspace K of H such that H = K ⊕ K . Indeed by Paulsen’s theorem 5.1.2, there is an invertible operator S ∈ B(H) such that πS = S −1 π(· )S is completely contractive. Clearly S −1 (K) is πS (B)-invariant. By the module complementation property, S −1 (K) admits a topological complement V which also is πS (B)-invariant. Then S(V ) is π(B)-invariant, and H = K ⊕ S(V ), which proves the claim. These properties may be viewed as a variant of the property of semisimplicity met in pure algebra. Indeed one characterization of semisimple rings R is that every submodule of every left R-module X is an R-module summand of X (e.g. see [8, Propositions 9.6 and 13.9 (d)]). 7.2.3 (W ∗ -module complementation property) There is a natural ‘w ∗ -topology version’ of the module complementation property. Namely, we will say that a unital dual operator algebra M has the w ∗ -module complementation property, provided that for every unital w ∗ -continuous completely contractive representation π : M → B(H), every π(M )-invariant subspace of H is a complemented submodule. We deﬁne the w∗ -reducing property similarly, by demanding that for any π as above, every π(M )-invariant subspace of H is reducing. 7.2.4 (Passing to the second dual) Let B be an approximately unital operator algebra, and let π : B → B(H) be a completely contractive representation. Let π ˜ : B ∗∗ → B(H) be the unique w∗ -continuous extension of π (see 2.5.5). This is a completely contractive representation, and every unital w ∗ -continuous completely contractive representation of B ∗∗ is of this form. Moreover if K is any π(B)-invariant subspace of H, then K is fairly evidently π ˜ (B ∗∗ )-invariant. ∗∗ Indeed, if η ∈ B , then by A.2.1 there exists a net (bt )t in B converging to η in the w∗ -topology. Thus π(bt ) → π ˜ (η). For ζ1 ∈ K and ζ2 ∈ K ⊥ , we have ˜ π (η)ζ1 , ζ2 = limπ(bt )ζ1 , ζ2 = 0, t

as desired. It follows immediately from this that B has the module complementation property if and only if B ∗∗ has the w∗ -module complementation property. Similarly for the reducing property, which we characterize next. Theorem 7.2.5 (1) An approximately unital operator algebra has the reducing property (see 7.2.2) if and only if it is selfadjoint. (2) A unital dual operator algebra has the w ∗ -reducing property (see 7.2.3) if and only if it is a W ∗ -algebra.

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Proof The ‘if’ parts of both assertions are clear. For the converses, we will in fact prove more than is asserted. Suppose that M is a w ∗ -closed unital-subalgebra of B(H), such that every closed M -submodule of the countable multiple H (∞) of H is reducing. Let L = H (∞) , and view M ⊂ B(L). If ζ ∈ L, then [M ζ] is an M -invariant subspace. By hypothesis, [M ζ]⊥ is an M -invariant subspace too. Equivalently, [M ζ] is an M -invariant subspace. Thus M ζ ⊂ [M ζ]. By A.1.5, M ⊂ M , so that M is selfadjoint, and we have proved (2). For (1), let H be any Hilbert A-module such that the canonically extended representation of A∗∗ on H is completely isometric (see 2.5.5 and 2.5.6). If every closed A-submodule of H (∞) is reducing, then by the argument in 7.2.4, every closed A∗∗ -submodule of H (∞) is reducing. Hence by (1), A∗∗ is selfadjoint. By 7.1.6 (2), A is selfadjoint too. 2 7.2.6 (Stability under isomorphism) Let A and B be approximately unital operator algebras, and assume that B has the module complementation property. It clearly follows from the second paragraph of 7.2.2 that if A is completely isomorphic to B, then A also has the module complementation property. More generally, if there is a completely bounded homomorphism θ : B → A with a dense range, then A has the module complementation property. To see this, let π : A → B(H) be a nondegenerate completely contractive representation, and consider a π(A)-invariant subspace K of H. The composition mapping πθ is a completely bounded representation of B on H, and K is πθ(B)-invariant. Since θ is assumed to have a dense range, πθ is nondegenerate. Thus according to 7.2.2, there is a direct sum decomposition H = K ⊕ K , for some πθ(B)-invariant subspace K . Since θ(B) is dense, K is π(A)-invariant. Since C ∗ -algebras have the reducing property, an approximately unital operator algebra which is completely isomorphic to a C ∗ -algebra, has the module complementation property. It is apparently an open problem whether the converse is true. We next consider some suﬃcient conditions in this direction. Proposition 7.2.7 Let B be a ﬁnite-dimensional unital operator algebra. Then B has the module complementation property if and only if B is (completely) isomorphic to a C ∗ -algebra. Proof We need only discuss the ‘only if’ part. Finite-dimensional spaces may be viewed as Hilbert spaces, and maps between them (and representations of B on them) are necessarily bounded. Hence if B has the module complementation property, then using the second paragraph of 7.2.2 we see that the criterion in the last paragraph of 7.2.2 holds, but for B-modules X of ﬁnite linear dimension. Since B is ﬁnite-dimensional, it follows from basic algebra that B is actually semisimple (e.g. see the paragraph above 13.6 in [8]). Hence by the classical theory of algebras (e.g. see [8, 368] or [106, Theorem 1.5.9]), B is isomorphic to 2 a C ∗ -algebra.

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Lemma 7.2.8 Let C be an approximately unital operator algebra and let B be a subalgebra of C containing a cai of C. Suppose that B has the module complementation property, and that for any nondegenerate completely contractive representation π : C → B(H), we have [π(B)] = [π(C)] . Then B = C. Proof Let B and C be as above, and ﬁx ϕ ∈ C ∗ with B ⊂ Ker(ϕ). We will show that ϕ = 0, so that B = C by the Hahn–Banach theorem. By 1.2.8, for example, there exist a Hilbert space H, a completely contractive homomorphism π : C → B(H) and two vectors ζ, η ∈ H such that ϕ(c) = π(c)ζ, η,

c ∈ C.

(7.3)

Replacing H by [π(C)H], and η by its projection onto [π(C)H], we may clearly assume that π is nondegenerate. Let K = [π(B)ζ]; this is a π(B)-invariant subspace of H, and is therefore a complemented submodule. By 7.2.1, there is an idempotent p ∈ [π(B)] = [π(C)] with range K. By assumption, B contains a net (bt )t which is a cai for C. Let c be an arbitrary element of C. For any t we have π(bt )ζ ∈ K hence π(bt )ζ = pπ(bt )ζ. Thus π(c)π(bt )ζ = π(c)pπ(bt )ζ. Since p commutes with π(c), we deduce that π(c)π(bt )ζ = pπ(c)π(bt )ζ. Since π(c) = limt π(c)π(bt ), we obtain that π(c)ζ = pπ(c)ζ. This shows that 2 π(c)ζ ∈ K. Now recall that ϕ|B = 0, so that η ∈ K ⊥ . Hence ϕ(c) = 0. 7.2.9 (Normal elements) Let B be an approximately unital operator algebra, and let Ce∗ (B) denote its C ∗ -envelope (see 4.3.4). We say that b ∈ B is normal if b is a normal element of Ce∗ (B). By Theorem 4.3.1, if D is another C ∗ -algebra containing B as a subalgebra, and if b ∈ B is normal as an element of D, then b is normal in the above sense. Theorem 7.2.10 (1) Let B be an approximately unital operator algebra and assume that B is generated (as an operator algebra) by its normal elements. Then B is selfadjoint if and only if it has the module complementation property. (2) A uniform algebra is selfadjoint if and only if it has the module complementation property. Proof We ﬁrst prove (1). We assume that B has the module complementation property, we let C = Ce∗ (B), and we will show that B = C. Let N be the set of normal elements of B. By 2.1.7, B contains a cai for C. Hence it suﬃces to check that B and C satisfy the assumption of 7.2.8. Let π : C → B(H) be a completely contractive representation. Then π is a ∗-representation (by 1.2.4). Hence for any N ∈ N , π(N ) is a normal element of B(H). We let T ∈ [π(B)] .

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Then T π(N ) = π(N )T , for every N ∈ N . By Fuglede’s theorem, this implies that T π(N )∗ = π(N )∗ T . Equivalently, T π(N ∗ ) = π(N ∗ )T,

N ∈ N.

Thus T commutes with π(N ) and with π(N ∗ ). Since C is generated by B and B as an operator algebra, our assumption implies that C is generated by N and N . Hence T ∈ [π(C)] , which shows that [π(B)] = [π(C)] . Clearly (2) follows from (1). Indeed if B is a uniform algebra then its C ∗ envelope is commutative, and hence every element of B is normal. 2 In 3.6.3, we showed B(K, H) is ‘injective as an A-B-bimodule’, if A and B are C ∗ -algebras acting on H and K respectively. In our next statement, we show that such results are not generally true for nonselfadjoint operator algebras. Proposition 7.2.11 Let B be an approximately unital operator algebra. Then B is selfadjoint if (and only if ) whenever K is a Hilbert B-module, Y is a left operator B-module, and X is a B-submodule of Y , then any completely contractive B-module map u : X → B(K) admits a completely contractive B-module extension uˆ : Y → B(K). Proof Assume that B satisﬁes the above property. We will apply 7.2.5 to B. Let H be a Hilbert space, let π : B → B(H) be a nondegenerate completely contractive representation, and let K be a π(B)-invariant subspace of H. Thus K is a Hilbert module over B. Let η ∈ K be a unit vector, and let u : K c → B(K) be deﬁned by u(ζ) = ζ ⊗η, for ζ ∈ K. Then u is a complete isometry (see 1.2.23), and u is a B-module map. Let X = K c and Y = H c , equipped with its B-module structure induced by π. By the hypothesis, we obtain a completely contractive map uˆ : H c → B(K), which extends u and satisﬁes u ˆ(π(b)ξ) = π(b)ˆ u(ξ),

b ∈ B, ξ ∈ H.

(7.4)

Let T : H → K be deﬁned by T (ξ) = u ˆ(ξ)η. Then T ≤ 1 and moreover, we have T (ζ) = u(ζ)η = ζ, for every ζ ∈ K. Hence T is the orthogonal projection onto K. It clearly follows from (7.4) that this projection commutes with π(B). According to 7.2.5 (1), this shows that B is selfadjoint. 2 In the following statement we do not require the operator algebras to be approximately unital. Proposition 7.2.12 (1) An operator algebra B is selfadjoint if and only if for any Hilbert space H, and any completely contractive homomorphism π : B → B(H), the commutant algebra [π(B)] is selfadjoint. (2) Let M be a dual operator algebra. Then M is a W ∗ -algebra if and only if for any Hilbert space H, and for any w ∗ -continuous completely contractive homomorphism π : M → B(H), the commutant [π(M )] is selfadjoint.

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279

Proof The ‘only if’ in (1) and (2) is clear from 1.2.4, so we need only prove the ‘if’ parts. If B is nonunital, and if π : B 1 → B(H) is a completely contractive homomorphism on the unitization B 1 , then [π(B 1 )] = [π(B)] ∩ [π(1)] . Moreover π(1) is a selfadjoint projection, by A.1.1, hence [π(1)] is selfadjoint. Thus [π(B 1 )] is selfadjoint if [π(B)] is selfadjoint. Also, B is selfadjoint if B 1 is selfadjoint (by 7.1.6 (3)). Thus, replacing B by B 1 if necessary, we may henceforth suppose that B is unital. For (1), note that if [π(B)] is selfadjoint, then so is [π(B)] . Taking π to be a B-universal representation, we deduce by 3.2.14 that B ∗∗ is selfadjoint. Thus B is selfadjoint, by 7.1.6 (2). A similar argument works for (2), using a dual algebra variant of the double commutant property (see [72]). 2 For some alternative proofs of 7.2.12, see the Notes section. 7.3 TENSOR PRODUCT CHARACTERIZATIONS Corollary 7.1.8 asserts that if B is an approximately unital operator algebra, then B is selfadjoint provided that A ⊗max B = A ⊗min B isometrically for every operator algebra A. Our aim is to extend that result (as well as its dual counterpart at the end of 7.1.9) to obtain a tensor product characterization of selfadjointness not restricted to nuclear or injective objects. This will be achieved in Theorem 7.3.3 below. Proposition 7.3.1 Let B be a unital operator algebra, let Y be an operator space and let u : Y → B be a linear mapping. If u is 1-nuclear and if there exists e ∈ Y such that u(e) = 1 and e = 1, then u(Y ) ⊂ ∆(B). Proof Suppose that B is a unital-subalgebra of B(H), and write 1 for IH . We apply Theorem 6.6.4 to the 1-nuclear mapping u. Using 6.5.1, we obtain a net ut : Y → B of ﬁnite rank operators converging to u point norm, with the following property: For any t, there exist nt ≥ 1, a matrix [ϕtij ] ∈ Mnt (Y ∗ ), and at1 , . . . , atnt , bt1 , . . . , btnt in B, such that ut (y) =

ϕtij (y) ati btj ,

y ∈Y;

(7.5)

1≤i,j≤nt

and

t [ϕij ] ≤ 1,

t [ai ] R

nt (B)

≤ 1,

t (bj ) C

nt (B)

≤ 1.

(7.6)

By assumption, there is a norm one element e of Y such that u(e) = 1. We let ϕtij (e)ati and dti = ϕtij (e)btj , ctj = i

j

for any t, and for 1 ≤ i, j ≤ nt . Then t [c · · · ct ] ≤ [at · · · at ] [ϕt (e)] ≤ 1, 1 nt 1 nt ij

(7.7)

280

Tensor product characterizations

using (7.6). Likewise, we have an estimate t (d ) i Cn (B) ≤ 1.

(7.8)

t

Let θt = ut (e) for each t. It follows from (7.5) that we have θt = ati dti and θt = ctj btj . i

(7.9)

j

We let y ∈ Y , and we will show that u(y)∗ ∈ B. Set and σt = ϕtij (y)ctj dti , zt = ut (y) i,j

for any t. We will show that lim σt − zt∗ = 0.

(7.10)

t

Since ut converges pointwise to u, we have limt zt − u(y) = 0. Thus (7.10) implies that limt u(y)∗ − σt = 0. However σt ∈ B for any t, and so u(y)∗ ∈ B. t∗ It therefore remains to prove (7.10). For any t we have zt∗ = i,j ϕtij (y)bt∗ j ai ∗ by (7.5). Hence σt − zt equals t t t t∗ t∗ = ϕtij (y) ctj dti − bt∗ ϕij (y) ctj − bt∗ ϕtij (y) bt∗ j ai j di + j di − ai . i,j

i,j

i,j

t The ﬁrst of these terms, i,j ϕtij (y) ctj − bt∗ j di , has norm dominated by t t t∗ t [ϕij (y)] (ct1 − bt∗ (di ) Cn (B) 1 ) · · · (cnt − bnt )

≤ y [ctj − bt∗ j ] Rn

t

t (B(H))

,

by (7.6) and (7.8). Similarly we have t t t∗ t∗ ϕtij (y) bt∗ j di − ai ≤ y (di − ai ) Cn i,j

We deduce that σt − zt∗ is less than or equal to t t∗ y [ctj − bt∗ j ] R (B(H)) + (di − ai ) C nt

t (B(H))

.

nt (B(H))

.

(7.11)

Let ξ be an arbitrary element of H. Then we have 2 t t∗ t∗ t (dt − at∗ )ξ 2 = dti ξ2 + at∗ i i i ξ − di ξ, ai ξ − ai ξ, di ξ i

i

=

i

dti ξ2 +

i

2 ∗ at∗ i ξ − θt ξ, ξ − θt ξ, ξ

Selfadjointness criteria

281

by (7.9). We deduce that 2 t 2 ≤ (di ) (dti − at∗ i )ξ Cn i

t (B)

2 + [atj ]Rn

) * ≤ (2 − θt − θt∗ )ξ, ξ

t (B)

ξ2 − θt ξ, ξ − θt∗ ξ, ξ

by (7.6) and (7.8). Taking the supremum over all ξ with ξ ≤ 1, we have t (di − at∗ i ) Cn

t (B(H))

1 ≤ 2 − θt − θt∗ 2 .

The same estimate holds for the ﬁrst term in (7.11). Thus 1

σt − zt∗ ≤ 2y2 − θt − θt∗ 2 . Now recall that θt = ut (e) converges to u(e) = 1 in the norm topology. Thus the sum θt + θt∗ converges to 2, and (7.10) follows. 2 Proposition 7.3.2 Let M be unital dual operator algebra, let Y be an operator space and let u : Y → M be a linear mapping. If u is 1-semidiscrete and if there exists e ∈ Y such that u(e) = 1 and e = 1, then Ran(u) ⊂ ∆(M ). Proof The proof is quite similar to the previous one; we just give an outline. Assume that M is a w∗ -closed unital subalgebra of B(H). By assumption, there exists a net ut : Y → M of ﬁnite rank operators converging in the point w ∗ topology to u, and elements ϕtij , ati , btj , ctj , dti , θt , y, zt and σt as above. Then θt converges to 1 in the w∗ -topology of B(H). Hence for ξ ∈ H, we have ) * lim (2 − θt − θt∗ )ξ, ξ = 0. t

Arguing as in the proof of 7.3.1, we ﬁnd that for any ξ, η ∈ H, we have ) ) * *1 ) *1 (σt − zt∗ )ξ, η ≤ y η (2 − θt − θt∗ )ξ, ξ 2 + ξ (2 − θt − θt∗ )η, η 2 . This shows that limt (σt − zt∗ )ξ, η = 0, for ξ, η ∈ H. For any t, σt ≤ y and zt ≤ y. Hence (σt − zt∗ )t is a bounded net. Thus σt − zt∗ → 0 in the w∗ -topology of B(H), by A.1.4. Moreover, zt = ut (y) converges to u(y), and so zt∗ converges to u(y)∗ in the w∗ -topology of B(H). Hence u(y)∗ = w∗ − limt σt . 2 Since σt ∈ M , we may conclude as before that Ran(u) ⊂ ∆(M ). Theorem 7.3.3 (1) Let B be a unital operator algebra. Then B is selfadjoint if (and only if ) there is a C ∗ -algebra D containing B as a subalgebra, such that for every approximately unital operator algebra A, we have A ⊗max B ⊂ A ⊗max D

isometrically.

(7.12)

282

Amenability and virtual diagonals

(2) Let M be a unital dual operator algebra. Then M is a W ∗ algebra if (and only if ) there is a W ∗ -algebra N containing M as a w ∗ -closed unital-subalgebra, such that for every approximately unital operator algebra A, we have A ⊗nor M ⊂ A ⊗nor N

isometrically.

Proof Let B ⊂ D as in (1), and assume that (7.12) holds for every such A. This property clearly remains true if D is replaced by a smaller C ∗ -algebra containing B, using 6.1.9. Hence we may assume that D is a C ∗ -cover of B (see 2.1.1). We will show that B is selfadjoint by showing that b∗ ∈ B, whenever b ∈ Ball(B). Fix such b ∈ B. As usual, T denotes the unit circle. For any n ∈ Z, let en ∈ C(T) be deﬁned by en (z) = z n , for any z ∈ T. Then there is a (necessarily unique) completely positive map P : C(T) → D satisfying P (en ) = bn for any n ≥ 0 and P (en ) = b∗n for any n ≤ 0. Indeed assume that D acts nondegenerately on some Hilbert space H. By Nagy’s dilation theorem (see 2.4.12), there is a Hilbert space K, an isometry J : H → K, and a unitary U ∈ B(K) such that bn = J ∗ U n J, for any n ≥ 0. Thus if we let π : C(T) → B(K) be the ∗-representation induced by U , then the mapping P = J ∗ π(· )J has the required properties. Since C(T) is a nuclear C ∗ -algebra, it readily follows from Lemma 6.1.10 that P is 1-nuclear. Let Y be the two-dimensional subspace of C(T) spanned by e0 and e1 , and let u : Y → D be the restriction of P to that subspace. Then u is 1-nuclear as well. To see this, note that for any approximately unital operator algebra A, we have A ⊗min Y ⊂ A ⊗min C(T) isometrically, and hence IA ⊗ u : A ⊗min Y −→ A ⊗max D ≤ 1. By (7.12), since u is valued in B, we have IA ⊗ u : A ⊗min Y −→ A ⊗max B ≤ 1. This shows that u : Y → B is 1-nuclear as a B-valued map. Since u(e0 ) = 1 and e0 = 1, Proposition 7.3.1 ensures that b∗ = u(e1 )∗ belongs to B. Assertion (2) can be proved similarly. Indeed, following the proof above, we ﬁnd P such that IA ⊗ P : A ⊗min C(T) → A ⊗max N is a contraction. Since the maximal norm dominates the normal tensor norm (see 6.1.3), we deduce that IA ⊗ P : A ⊗min C(T) → A ⊗nor N is a contraction. We restrict to Y , as before, and see that IA ⊗ u : A ⊗min Y → A ⊗nor N is a contraction. Since u is valued in M , and since A ⊗nor M ⊂ A ⊗nor N isometrically by hypothesis, we deduce that IA ⊗ u : A ⊗min Y → A ⊗nor M is a contraction. Thus u is 1-semidiscrete. Finally, we appeal to 7.3.2. 2 7.4 AMENABILITY AND VIRTUAL DIAGONALS Amenability for Banach algebras was originally introduced by Barry Johnson (see [205]), and is by now a central research area which has impacted much of modern mathematics. In this last section, we investigate amenable operator

Selfadjointness criteria

283

algebras. We will mainly focus on some issues which are related to operator spaces. In particular, we will emphasize the role of the Haagerup tensor product in the study of these objects. This will lead to selfadjointness results which complement those in Section 7.2. 7.4.1 (Amenable Banach algebras) Let B be a Banach algebra and let X be a Banach B-bimodule. A derivation is a bounded linear mapping D : B → X such that D(ab) = aD(b) + D(a)b, for any a, b ∈ B. Given any x ∈ X, it is easy to check that the mapping D : B → X deﬁned by letting D(b) = xb − bx, for b ∈ B, is a derivation. Such derivations are called inner. If X is a Banach B-bimodule, then the dual space X ∗ is a Banach B-bimodule for the dual actions deﬁned by bϕ, x = ϕ, xb

and

ϕb, x = ϕ, bx

for every b ∈ B, x ∈ X, ϕ ∈ X ∗ . In this case, we say that X ∗ is a dual Banach bimodule. Note that the left and right actions B × X ∗ → X ∗ and X ∗ × B → X ∗ are w∗ -continuous in the second and in the ﬁrst variable respectively. By obvious iteration, X ∗∗ also becomes a dual Banach B-bimodule. By deﬁnition a Banach algebra B is amenable if whenever X ∗ is a dual Banach B-bimodule, any derivation from B into X ∗ is inner. 7.4.2 (Facts on amenability) We mention three simple results, for which we refer, for example, to [106] or [304, (1.30)]. First, let A and B be two Banach algebras, and assume that B is amenable. If there is a bounded homomorphism θ : B → A with a dense range, then A is amenable as well. In particular, any Banach algebra isomorphic to an amenable algebra is amenable. Second, any amenable Banach algebra has a bounded approximate identity. Third, a unital Banach algebra B is amenable provided that any derivation from B into X ∗ is inner whenever X is a nondegenerate Banach B-bimodule (i.e. 1x = x = x1 for x ∈ X). For simplicity, we will often restrict our attention to unital algebras. 7.4.3 (Virtual diagonals) Let B be a unital Banach algebra. We may regard ˆ (deﬁned in A.3.3) as a Banach the Banach space projective tensor product B ⊗B B-bimodule by letting c(a ⊗ b) = ca ⊗ b

and

(a ⊗ b)d = a ⊗ bd

(7.13)

for a, b, c, d ∈ B, and then extending by linearity and continuity. We let m be ˆ to B induced by the multiplication the contractive linear mapping from B ⊗B ˆ ∗∗ such on B. By deﬁnition, a virtual diagonal of B is an element u ∈ (B ⊗B) ∗∗ that m (u) = 1 ∈ B, and such that cu = uc for any c ∈ B. In this deﬁnition, ˆ ˆ ∗∗ is equipped with its B-bimodule structure inherited from that on B ⊗B (B ⊗B) (see 7.4.1). Not all Banach algebras admit a virtual diagonal: Lemma 7.4.4 (Johnson) A unital Banach algebra is amenable if and only if it admits a virtual diagonal.

284

Amenability and virtual diagonals

ˆ ∗∗ is a virtual diagonal of B. Let X ∗ be a dual Proof Assume that u ∈ (B ⊗B) Banach B-bimodule, and let D : B → X ∗ be a derivation. By 7.4.2, we may assume that X is nondegenerate. We deﬁne a bounded mapping ˆ −→ X ∗ F : B ⊗B by letting F (a ⊗ b) = D(a)b, for any a, b ∈ B, and then extending by linearity and continuity. According to A.2.2, F admits a w ∗ -continuous extension ˆ ∗∗ −→ X ∗ . We let ϕ = F˜ (u) ∈ X ∗ , and we will show that F˜ : (B ⊗B) D(c) = ϕc − cϕ,

c ∈ B.

(7.14)

ˆ ∗∗ . For Let (ut )t be a net in B ⊗ B converging to u in the w ∗ -topology of (B ⊗B) t t any t, we let (ak )k and (bk )k be ﬁnite families of B such that ut =

atk ⊗ btk .

k

We have ϕc − cϕ = w∗ − lim t

D(atk )btk c −

k

cD(atk )btk .

(7.15)

k

Indeed, ϕ is the w∗ -limit of F (ut ), and so (7.15) follows from the w ∗ -continuity properties of the dual actions (see 7.4.1). On the other hand, cu and uc are the ∗ ˜ w∗ -limits of cut and ut c respectively, and so w -limt (cut − ut c) = 0. Since F is ∗ ∗ w -continuous, this implies that w -limt F (cut ) − F (ut c) = 0, that is, w∗ − lim t

D(catk )btk −

k

D(atk )btk c = 0.

(7.16)

k

Now since D is a derivation, we have D(catk )btk = D(c) atk btk + c D(atk )btk k

k

k

for any t. The ﬁrst term in the right side above equals D(c)m(ut ), and by assumption, m(ut ) → 1 in the weak topology of B. Hence D(c)m(ut ) → D(c) in the weak topology of X ∗ , and therefore also in its w∗ -topology. The identity (7.14) therefore follows by combining the above with (7.15) and (7.16). This shows that D is inner, and hence that B is amenable. ˆ → B is a Assume conversely that B is amenable. The ‘product’ m : B ⊗B ˆ B-bimodule map. Thus its kernel Ker(m) is a B-B-submodule of B ⊗B. Hence Ker(m)∗∗ is a dual Banach B-bimodule (see 7.4.1). Regard Ker(m)∗∗ as a subˆ ∗∗ , and deﬁne D : B → Ker(m)∗∗ by letting space of (B ⊗B) D(b) = 1 ⊗ b − b ⊗ 1,

b ∈ B.

Selfadjointness criteria

285

Clearly, D is a derivation. By amenability, there exists v ∈ Ker(m)∗∗ such that ˆ ∗∗ . Then u is a virtual D(b) = vb − bv for any b ∈ B. Let u = 1 ⊗ 1 − v ∈ (B ⊗B) ∗∗ diagonal of B. Indeed, m (u) = m(1 ⊗ 1) = 1 since m∗∗ (v) = 0. Also, ub − bu = (1 ⊗ 1)b − vb − b(1 ⊗ 1) + bv = 1 ⊗ b − vb − b ⊗ 1 + bv = D(b) − vb + bv = 0, for any b ∈ B.

2

7.4.5 (Amenable C ∗ -algebras) The amenable C ∗ -algebras are exactly the nuclear ones. This follows from the remarkable work of Connes (who showed that amenability implies nuclearity [100]), and Haagerup [179] who showed directly that any nuclear C ∗ -algebra admits a virtual diagonal. Thus any operator algebra which is isomorphic to a nuclear C ∗ -algebra, is amenable by Haagerup’s theorem. It is unknown whether the converse is true. We will present some partial answers in the rest of this section. The simplest nuclear (hence amenable) C ∗ -algebras are the ﬁnite-dimensional ∗ C -algebras. For such algebras, it is easy to describe all (virtual) diagonals. See the Notes to this section. We merely mention here that a diagonal for a ﬁnitedimensional C ∗ -algebra B may be constructed as follows. If U denotes the unitary group of B, and if we let dµ be the Haar measure on this compact group, then ! u=

U

z ⊗ z ∗ dµ(z) ∈ B ⊗ B

(7.17)

is a (virtual) diagonal of B. 7.4.6 (Virtual h-diagonals) We consider a variant of 7.4.3 adapted to operator algebras and the Haagerup tensor product of 1.5.4. Let B be a unital operator algebra. By 3.4.9, B ⊗h B is a Banach B-bimodule for the actions given by (7.13). We let mh : B ⊗h B → B denote the contraction induced by the multiplication on B. We deﬁne a virtual h-diagonal of B to be an element u ∈ (B ⊗h B)∗∗ such that m∗∗ h (u) = 1 ∈ B, and such that cu = uc for any c ∈ B. Since the Banach space projective tensor product dominates the Haagerup tensor product (see 1.5.13), ˆ → B ⊗h B, and we have m = mh ◦ κ. there is a canonical contraction κ : B ⊗B Moreover κ is a B-bimodule map. By the w ∗ -continuity properties of dual actions ˆ ∗∗ → (B ⊗h B)∗∗ is a B-bimodule map. (see 7.4.1), we deduce that κ∗∗ : (B ⊗B) ˆ ∗∗ is a virtual diagonal of B, then u = κ∗∗ (v) ∈ (B ⊗h B)∗∗ is Thus if v ∈ (B ⊗B) a virtual h-diagonal of B. The converse is not true (see 7.4.7 below). However if B is a C ∗ -algebra, it turns out that B admits a virtual diagonal if and only if it admits a virtual h-diagonal (and if and only if it is nuclear, by 7.4.4 and 7.4.5). See 7.4.9 and the Notes to this section. 7.4.7 (A virtual h-diagonal which is not a virtual diagonal) Let B be the unitization of the C ∗ -algebra of compact operators on 2 . That is, B = Span{S ∞ , I2 },

286

Amenability and virtual diagonals

2 the span n in B( ). Let (Eij )i,j≥1 be the matrix units. For any n ≥ 1, we deﬁne un = i=1 Ei1 ⊗ E1i , considered as an element of B ⊗ B. By 1.2.5,

[E11 · · · En1 ] R

n (B)

n n 1 12 ∗ 2 = Ei1 E1i Eii ≤ 1. = i=1

i=1

Likewise, the norm of [E11 · · · E1n ] in Cn (B) is less than or equal to 1. Hence un h ≤ 1. Being a bounded sequencein B ⊗h B, (un )n has a w∗ -cluster point n u ∈ (B ⊗h B)∗∗ . We have mh (un ) = i=1 Eii , and the latter converges to 1 in ∗∗ the weak topology of B. Hence mh (u) = 1. Moreover, we have t

Ejk un = Ej1 ⊗ E1k = un Ejk whenever 1 ≤ j, k ≤ n. Hence Ejk u = uEjk , for j, k ≥ 1. This readily implies that cu = uc for any c ∈ B. Thus u is a virtual h-diagonal of B. ˆ → B ⊗h B be the canonical contraction (see 7.4.6). We claim Let κ : B ⊗B that u does not belong to the range of κ∗∗ , so that, by 7.4.6, u ‘is not’ a virtual diagonal of B. Assume to the contrary that u = κ∗∗ (v) for some v. We will use the ‘trace class’ S 1 = S 1 (2 ), and the duality relations S ∞∗ = S 1 and S 1∗ = B(2 ) (see A.1.2). Consider the natural action of S 1 ⊗ S 1 on B(2 ) ⊗ B(2 ) given by y ⊗ z, a ⊗ b = y, az, b = tr(ya) tr(zb),

y, z ∈ S 1 , a, b ∈ B(2 ). (7.18)

2 ∗ ˆ Since (B(2 )⊗B( )) coincided with bounded bilinear forms on B(2 ) (see A.3.3), it follows from the deﬁnition of the injective tensor product (in A.3.1) that (7.18) 2 ∗ ˆ ˇ 1 → (B(2 )⊗B( )) . Since S 1 = S ∞∗ , induces an isometric embedding S 1 ⊗S the restriction of (7.18) to pairs (a, b) ∈ S ∞ × S ∞ also yields an isometry of ˇ 1 into (S ∞ ⊗S ˆ ∞ )∗ ∼ S 1 ⊗S = B(S ∞ , S 1 ), by a fact towards the end of A.3.3. Since ∞ 2 S ⊂ B ⊂ B( ), we deduce that

ˆ ∗ ˇ 1 → (B ⊗B) S 1 ⊗S

isometrically.

(7.19)

ˆ ∗∗ and S 1 ⊗ S 1 . There is thus a canonical scalar valued pairing between (B ⊗B) 1 1 ∗ Similarly, we can regard S ⊗S as a subspace of (B⊗h B) , and we obtain a scalar ˆ ∗, pairing between (B ⊗h B)∗∗ and S 1 ⊗ S 1 . The map κ∗ : (B ⊗h B)∗ → (B ⊗B) 1 1 is the identity mapping on the copies of S ⊗ S . Thus (7.20) w, u = w, v, w ∈ S 1 ⊗ S 1. m For any m ≥ 1, let wm = i=1 E1i ⊗ Ei1 , regarded as an element of S 1 ⊗ S 1 . For any two integers n ≥ m ≥ 1, we have un , wm =

m

Ei1 , E1i E1i , Ei1 = m.

i=1

Since wm , u is a cluster point of the sequence (wm , un )n , we have wm , u = m for every m ≥ 1. It therefore follows from (7.20) that wm , v = m. Applying (7.19), we deduce that

Selfadjointness criteria

287

m ≤ v(B ⊗B) ∗∗ wm S 1 ⊗S ˇ 1, ˆ

m ≥ 1.

For any a = [aij ] and b = [bij ] in B(2 ), we have m 12 12 ai1 b1i ≤ |ai1 |2 |b1i |2 ≤ a b. |wm , a ⊗ b| = i=1

i

i

According to A.3.1, this shows that wm S 1 ⊗S ∗∗ ≥ m for ˇ 1 ≤ 1. Hence v(B ⊗B) ˆ every integer positive m, a contradiction. 7.4.8 (Normal virtual h-diagonals) We introduce an analogue of 7.4.6 in the dual setting, using the normal Haagerup tensor product (see 1.6.8). Let M be a unital dual operator algebra, and let mh : M ⊗h M → M be the multiplication mapping. Since the product on M is separately w ∗ -continuous (see 2.7.4 (1)), the adjoint mapping m∗h maps M∗ into the subspace (M ⊗h M )∗σ ⊂ (M ⊗h M )∗ of separately w∗ -continuous completely bounded bilinear forms. We let mσ : M ⊗σh M −→ M denote the adjoint of the restriction m∗h|M∗ : M∗ → (M ⊗h M )∗σ . By deﬁnition, mσ is a w∗ -continuous complete contraction extending mh . By 7.4.6, M ⊗h M is an M -bimodule. Taking dual actions (see 7.4.1), (M ⊗h M )∗ is therefore a dual Banach M -bimodule. We observe that (M ⊗h M )∗σ is an M -M -submodule of (M ⊗h M )∗ . Indeed assume that ψ ∈ (M ⊗h M )∗ , and let a, b ∈ M . Then for any c and d in M , we have bψa, c ⊗ d = ψ(ac, db). Since the multiplication on M is separately w ∗ -continuous, bψa ∈ (M ⊗h M )∗σ if ψ ∈ (M ⊗h M )∗σ . Using ∗ the dual bimodule actions from 7.4.1, we may equip M ⊗σh M = (M ⊗h M )∗σ with a Banach M -bimodule structure. One can easily check that the actions on M ⊗σh M extend those on M ⊗h M . By deﬁnition, a normal virtual h-diagonal of M is an element u ∈ M ⊗σh M , such that mσ (u) = 1 and cu = uc for every c ∈ M . It follows from the work of Connes [100], Eﬀros [134], and Eﬀros and Kishimoto [138], that if M is a W ∗ -algebra, then M is injective if and only if it admits a normal virtual h-diagonal. 7.4.9 (Second duals) Let B be a unital operator algebra. By (1.56), (B ⊗ h B)∗∗ and B ∗∗ ⊗σh B ∗∗ coincide as dual operator spaces. Applying 7.4.8, B ∗∗ ⊗σh B ∗∗ is equipped with a canonical B ∗∗ -bimodule structure, and it also has a natural B-bimodule structure. It is easy to check that the latter structure coincides with the bidual B-bimodule structure on (B ⊗h B)∗∗ , under the above identiﬁcation. Furthermore m∗∗ h and mσ are the same under that identiﬁcation. This is because they are both w∗ -continuous, and they coincide on B ⊗ B, which is a w ∗ -dense subspace of B ∗∗ ⊗σh B ∗∗ = (B ⊗h B)∗∗ . According to the above discussion, any normal virtual h-diagonal of B ∗∗ is a virtual h-diagonal of B. One of the key results from [138] is that if B is

288

Amenability and virtual diagonals

selfadjoint, then the converse holds true, and hence normal virtual h-diagonals of B ∗∗ coincide with virtual h-diagonals of B. Thus if a C ∗ -algebra B has a virtual h-diagonal, then B ∗∗ is injective by the last assertion of 7.4.8. Thus B is nuclear, and hence admits a virtual diagonal by Haagerup’s theorem [179]. 7.4.10 We shall relate virtual h-diagonals (and hence amenability) to the module complementation property from 7.2.2. Let M be a unital dual operator algebra. To any unital w∗ -continuous completely contractiverepresentation π : M → B(H), we associate a linear mapping τπ : M ⊗ M → CB B(H), B(H) , deﬁned by letting τπ (a ⊗ b)(T ) = π(a)T π(b),

a, b ∈ M, T ∈ B(H).

Let a1 , . . . , am , b1 , . . . , bm ∈ M , and let u = k ak ⊗ bk ∈ M ⊗ M . We let Q = τπ (u), so that Q(T ) = k π(ak )T π(bk ) for any T ∈ B(H). We deﬁne a representation ρ : B(H) → Mm ⊗B(H) B(2m (H)), by letting ρ(T ) = IMm ⊗T . Deﬁne V, W : H → 2m (H) by V (ζ) = π(b1 )ζ, . . . , π(bm )ζ

and

W (ζ) = π(a1 )∗ ζ, . . . , π(am )∗ ζ ,

for any ζ ∈ H. Then Q = W ∗ ρ(· )V , hence Qcb ≤ V W . We deduce, using the fact that πcb = 1, that Qcb ≤ [π(a1 ) · · · π(am )]Rm (B(H)) [π(b1 ) · · · π(bm )]t Cm (B(H)) ≤ [a1 · · · am ]Rm (M) [b1 · · · bm ]t Cm (M) . By 1.5.4, τπ extends to a contraction from M ⊗h M to CB B(H), B(H) . Lemma 7.4.11 Let π and τπ as above. (1) The above mapping τπ extendsuniquely to a contractive w ∗ -continuous linear map from M ⊗σh M into CB B(H), B(H) (which we still denote by τπ ). (2) For any u ∈ M ⊗σh M , the mapping τπ (u) : B(H) → B(H) is a bimodule map over [π(M )] . That is, τπ (u)(S2 T S1 ) = S2 τπ (u)(T ) S1 ,

S1 , S2 ∈ [π(M )] , T ∈ B(H). (7.21)

(3) Let u ∈ M ⊗σh M and assume that mσ (u) = 1. Then τπ (u) is unital. (4) Let u ∈ M ⊗σh M and c ∈ M , and assume that cu = uc. Then π(c) commutes with τπ (u)(T ), for any T ∈ B(H). Proof For simplicity write τπ as τ below. The uniqueness in (1) is clear, since M ⊗ M is w∗ -dense in M ⊗σh M , as we saw in 1.6.8. According to 7.4.10, we regard τ as a contraction on M ⊗h M . We recall that CB B(H), B(H) is the

dual Banach space of S 1 (H) ⊗ B(H) (by (1.51)), and consider the restriction of

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the adjoint map τ ∗ to this predual. Let ϕ and T be arbitrary elements of S 1 (H) and B(H) respectively. Then for any a, b ∈ M , we have τ ∗ (ϕ ⊗ T ), a ⊗ b = ϕ ⊗ T, τ (a ⊗ b) = ϕ, π(a)T π(b). Since π is w∗ -continuous and the multiplication mapping on B(H) is separately w∗ -continuous, we deduce that τ ∗ (ϕ ⊗ T ) is separately w∗ -continuous, and hence ∗ belongs to M ⊗h M σ . By linearity and continuity, τ ∗ maps S 1 (H) ⊗ B(H) into ∗ ∗ M ⊗h M σ . We let θ : S 1 (H) ⊗ B(H) → M ⊗h M σ be the resulting mapping. The adjoint θ∗ : M ⊗σh M → CB B(H), B(H) is a w∗ -continuous contraction. By construction, θ ∗ (a ⊗ b) = τ (a ⊗ b) for every a, b ∈ M , which proves (1). To prove the last three assertions, we ﬁx an element u ∈ M ⊗σh M , and let Q = τ (u). Let (ut )t be a net in M ⊗ M converging to u in the w ∗ -topology of M ⊗σh M . For any t, we let (atk )k and (btk )k be ﬁnite families of M such that ut =

atk ⊗ btk .

k

Since τ is w∗ -continuous, Q is the w∗ -limit of τ (ut ). Hence

Q(T ) = w∗ − lim t

π(atk )T π(btk ).

(7.22)

k

if T ∈ B(H). If S1 , S2 ∈ [π(M )] and T ∈ B(H), then # " t t t t π(ak )S2 T S1 π(bk ) = S2 π(ak )T π(bk ) S1 , k

k

for any t. Since the multiplication is separately w ∗ -continuous on B(H), (7.21) now follows from (7.22). Assume that mσ (u) = 1. With the above notation, we have for any t π(atk )π(btk ) = π atk btk = π m(ut ) . k

k

Let us apply (7.22) with T = IH . Since 1 = mσ (u) is the w∗ -limit of m(ut ) and π is w∗ -continuous and unital, we deduce that Q(IH ) = IH , which proves (3). Suppose that c ∈ M , with cu = uc, and ﬁx T ∈ B(H). Using (7.22) again, π(c)Q(T ) = w∗ − lim t

whereas Q(T )π(c) = w∗ − lim t

π(catk )T π(btk ),

(7.23)

π(atk )T π(btk c).

(7.24)

k

k

290

Amenability and virtual diagonals

∗ Since cu = uc, we have w∗ -lim t (cut − ut c) = 0 in M ⊗σh M . By the w -continuity ∗ of τ we deduce that w -limt τ (cut ) − τ (ut c) = 0. Therefore we have w∗ − lim τ (cut )(T ) − τ (ut c)(T ) = 0, t

and so w∗ − lim t

"

π(catk )T π(btk ) −

k

# π(atk )T π(btk c) = 0.

k

This, together with (7.23) and (7.24), gives Q(T )π(c) = π(c)Q(T ).

2

Proposition 7.4.12 (1) Let M be a unital dual operator algebra which has a normal virtual h-diagonal u ∈ M ⊗σh M . Then M has the w∗ -module complementation property (see 7.2.3). More precisely, let π : M → B(H) be a unital w ∗ -continuous completely contractive representation. If K is a π(M )-invariant subspace of H, then there exists a idempotent map p in [π(M )] ⊂ B(H), such that Ran(p) = K and p ≤ uM⊗σh M . (2) Let B be a unital operator algebra which possesses a virtual h-diagonal u in (B ⊗h B)∗∗ . Then B has the module complementation property (see 7.2.2). Indeed, if π : B → B(H) is a unital completely contractive representation, and if K is a π(B)-invariant subspace of H, then there exists an idempotent map p in [π(B)] ⊂ B(H), such that Ran(p) = K and p ≤ u(B⊗h B)∗∗ . Proof (1) We consider a unital w ∗ -continuous completely contractive representation π : M → B(H), and we let τπ be the associated mapping considered in 7.4.10 and 7.4.11. We let Q = τπ (u), u being a normal virtual h-diagonal. According to 7.4.11 (3), Q is unital. It therefore follows from (7.21) that Q(S) = S, for any S ∈ [π(M )] . On the other hand, since uc = cu for any c ∈ M , 7.4.11 (4) shows that Q is valued in [π(M )] . Thus Q : B(H) → B(H) is an idempotent, with range equal to [π(M )] . t t As in the proof of Lemma 7.4.11, we let ut = k ak ⊗ bk ∈ M ⊗ M be ∗ such that ut → u in the w -topology of M ⊗σh M . Let K be a π(M )-invariant subspace of H, and let PK ∈ B(H) be the orthogonal projection onto K. Then π(· )PK = PK π(· )PK , and so PK π(atk )PK π(btk ) = π(atk )PK π(btk ), k

and

π(atk )PK π(btk )PK =

k

k

π(atk )π(btk )PK = π

k

atk btk PK ,

k

for any t. Applying (7.22), we deduce that PK Q(PK ) = Q(PK )

and

Q(PK )PK = PK .

(7.25)

Applying (7.21) with S2 = IH , T = PK and S1 = Q(PK ), and using the fact that Q is an idempotent, we deduce that Q(PK )2 = Q(PK ). Now (7.25) shows

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291

that the two idempotents PK and p = Q(PK ) have the same range. Moreover, we have the desired estimate p ≤ QPK = Q ≤ τπ uσh = uσh . (2) Let B be a unital operator algebra with virtual h-diagonal u in (B⊗ h B)∗∗ , and let π : B → B(H) be a completely contractive representation. Let K be any π(B)-invariant subspace of H. Set M = B ∗∗ , and consider the w∗ -continuous extension π ˜ : M → B(H) of π. According to 7.2.4, K also is a π ˜ (M )-invariant subspace. As explained in 7.4.9, we identify (B ⊗h B)∗∗ with M ⊗σh M as dual B-bimodules. Under this identiﬁcation, we have mσ (u) = 1 and cu = uc for π (M )] , (1) above shows that if we let Q = τπ˜ (u), any c ∈ B. Since [π(B)] = [˜ and if we let PK be the orthogonal projection onto K, then p = Q(PK ) is an 2 idempotent whose range equals K, with p ≤ u(B⊗h B)∗∗ . Corollary 7.4.13 (Giﬀord) Let B be a unital operator algebra. If B is amenable then B has the module complementation property. Proof We assume that B is amenable. By Lemma 7.4.4, B has a virtual diagonal, and hence a virtual h-diagonal (see 7.4.6). Now apply 7.4.12 (2). 2 Corollary 7.4.14 Let B be a ﬁnite-dimensional unital operator algebra. Then B is amenable if and only if B is (completely) isomorphic to a C ∗ -algebra. Proof The ‘only if’ part follows from Proposition 7.2.7 and Corollary 7.4.13. The ‘if’ part was discussed in 7.4.5. 2 Corollary 7.4.15 (Curtis and Loy) Let B be a unital operator algebra and assume that B is amenable. If B is generated (as an operator algebra) by its normal elements (see 7.2.9), then B is selfadjoint. Proof Combine Theorem 7.2.10 (1), and Corollary 7.4.13.

2

Corollary 7.4.16 (Sheinberg) A uniform algebra is amenable if and only if it is selfadjoint. Proof The ‘only if’ part follows from Theorem 7.2.10, (2), and Corollary 7.4.13. The ‘if’ part is a special case of 7.4.5. 2 7.4.17 (Norms of virtual diagonals) Let M be a unital dual operator algebra, and assume that u ∈ M ⊗σh M is a normal virtual h-diagonal. Since mσ is a contraction from M ⊗σh M into M taking u to 1, we have uσh ≥ 1. Likewise, if B is a unital operator algebra with a virtual h-diagonal u ∈ (B ⊗ h B)∗∗ , ˆ ∗∗ is a virtual diagonal, then then u(B⊗h B)∗∗ ≥ 1. A fortiori, if u ∈ (B ⊗B) u(B ⊗B) ∗∗ ≥ 1. It turns out these rough estimates are optimal for selfadjoint alˆ gebras. Namely, it follows from Haagerup’s work [179] on amenable C ∗ -algebras, that if B is a nuclear C ∗ -algebra, then B admits a virtual diagonal u such that u(B ⊗B) ∗∗ = u(B⊗h B)∗∗ = 1. Likewise, it follows from [134] that any injective ˆ W ∗ -algebra M admits a normal virtual h-diagonal u such that uσh = 1. It turns out that these properties characterize the amenable C ∗ -algebras among amenable operator algebras:

292

Notes and historical remarks

Theorem 7.4.18 (1) Let M be a unital dual operator algebra. If M possesses a normal virtual h-diagonal of norm 1, then M is selfadjoint (and hence M is an injective W ∗ -algebra, by 7.4.8). (2) A unital operator algebra which admits a virtual h-diagonal of norm 1, is selfadjoint (and hence is a nuclear C ∗ -algebra, by 7.4.6). Proof We only prove (1), the proof of (2) being similar. We apply Theorem 7.2.5 (2). Let π : M → B(H) be a unital w ∗ -continuous completely contractive representation, and let K be a π(M )-invariant subspace of H. According to 7.4.12 (1), there is an idempotent p in [π(M )] such that Ran(p) = K and p ≤ 1. This estimate forces p to be the orthogonal projection onto K. This shows that 2 K is reducing, and hence that M has the w ∗ -reducing property. 7.5 NOTES AND HISTORICAL REMARKS Characterizations of selfadjointness for operator algebras are scattered throughout the literature (e.g. see [354]). We have made no attempt to survey all such results here; no doubt every researcher in this ﬁeld has proved a theorem of this type. We have focused on some results related to operator space theory, although admittedly some of our results hold at the Banach algebra level too, with essentially the same proof. 7.1: The weak expectation property was originally introduced for C ∗ -algebras by Lance [238]. Its deﬁnition for operator spaces appears in [141]; Corollary 7.1.5 is from that paper. Results in 7.1.6 and 7.1.7 may be new. The second assertion of 7.1.8 appeared in Pisier’s book [337]. The WEP is currently attracting much interest from C ∗ -algebraists. For remarkable developments going far beyond the scope of this book we refer the reader to, for example, [136, 181, 228, 296], and the references therein. Proposition 7.1.10 is a new result based on Kisliakov’s remarkable paper [232]. The main result in the latter paper asserts that if B ⊂ C(Ω) is a uniform algebra, and if there is an idempotent P : C(Ω) → C(Ω) with Ran(P ) = B, then B = C(Ω). The facts and arguments in 7.1.11 and 7.1.12 are new, we believe. 7.2: The material here may be viewed as having birth in the famous (and still open) reductive algebra problem of Radjavi and Rosenthal: If M is a weakly closed unital-subalgebra of B(H), and if every M -invariant subspace of H is reducing, then is M selfadjoint? See [354] for details and progress. The assertion is false if ‘weakly closed’ is replaced by ‘w ∗ -closed’, as Loebl and Muhly showed in [256]. The module complementation property deﬁned in 7.2.2 was introduced in a slightly diﬀerent form by Giﬀord in [169]. This remarkable paper was unfortunately never published. The fact that any unital amenable operator algebra has the module complementation property (Corollary 7.4.13) was proved there. Giﬀord studied whether the module complementation property implies selfadjointness for approximately unital operator algebras. He showed that the answer is positive provided that A can be represented as a subalgebra of the compact

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293

operators S ∞ (H) for some Hilbert space H. A special case appears in [430]. The ﬁrst part of Theorem 7.2.5 is from [281, Theorem 3.1]. This theorem may also be deduced from the methods of 7.2.8. In connection with 7.2.2, some other notions of semisimplicity for operator algebras are studied in [200, 223, 281], for example. Note that every ﬁnite-dimensional uniform algebra A is selfadjoint. Indeed, the one-dimensional representations of A separate points, and since A is ﬁnite-dimensional it is easily argued that such an algebra is semisimple in either the algebraic or the Banach algebraic sense. An application of Wedderburn’s theorem as in the proof of 7.2.7, shows that A is isomorphic (and therefore also isometric, by A.5.4) to a commutative C ∗ -algebra. The proof of 7.2.8 was inspired by that of [281, Theorem 3.1]. The result in 7.2.10 (1) is an adaptation of 7.4.15. Proposition 7.2.11 was proved in [50], as a simple consequence of [281, Theorem 3.1]. Those papers contain other related characterizations of C ∗ -algebras. There is probably an ‘isomorphic’ version of 7.2.11. Indeed, we leave it as an exercise using 5.1.2 and 3.6.2, and some diagram chasing, that if B is completely isomorphic to a C ∗ -algebra, and if K, X, Y, and u are as in 7.2.11, then u extends to a completely bounded A-module map from Y to B(K). Conversely, the idea of the proof of 7.2.11 shows that if B(K) has such a ‘completely bounded module map extension property’, for every Hilbert B-module K, then B has the module complementation property. One may also show using the ideas of 3.2.14, that if B has the module complementation property, then every representation of B has the double commutant property 3.2.13. This was ﬁrst noticed by Giﬀord [169]. Also in connection with module map extensions, in [391] Smith exhibits a unital subalgebra B of M6 , a very simple and concrete left B-submodule X of M6 , and a B-module map u : X → M6 which admits no B-module extension from M6 to M6 . Le Merdy proved Proposition 7.2.12 in [252]. An alternative proof using 7.2.5 was given shortly thereafter by Blecher (see [50]). We sketch Le Merdy’s argument, which uses 7.3.3. Let D be the maximal C ∗ -algebra of B (see 2.4.3). Let A be an approximately unital operator algebra, and consider z ∈ A ⊗ B. Let π : A → B(H) and ρ : B → B(H) be two commuting completely contractive representations on some Hilbert space H. By assumption, [ρ(B)] is selfadjoint, and so C = [ρ(B)] is a C ∗ -algebra. Since ρ(B) ⊂ C, there exists a ∗-representation ρ˜ : D → C extending ρ (see 2.4.2). ρ˜ commutes with Now π, since the latter is valued in [ρ(B)] = C . Hence π • ρ(z) = π • ρ˜(z) ≤ zA⊗maxD . We deduce that zA⊗maxB ≤ zA⊗maxD , by taking the supremum over all possible pairs (π, ρ). Hence A ⊗max B ⊂ A ⊗max D isometrically. By Theorem 7.3.3, B is selfadjoint. A similar argument proves 7.2.12 (2). It is easy to extend the module complementation property and reducing property to nonunital operator algebras A; one simply drops the requirement that the representations π considered be nondegenerate. Using 2.1.13, it is clear that these properties are equivalent to the unitization A1 having the corresponding property in 7.2.2. We deduce from 7.2.10 (1) and 7.1.6 (3) that A has the reducing property if and only if A is selfadjoint. Similarly, one may drop the unital

294

Notes and historical remarks

hypothesis in 7.2.7. 7.3: Propositions 7.3.1 and 7.3.2 are due to Le Merdy [252], but their proofs were inspired by Pisier’s proof of 7.1.8 contained in [337]. The ﬁrst part of 7.3.3 is also taken from [252] and its second part is an easy adaptation. A classical related result of Lance [238] asserts that if B is a C ∗ -algebra, then B has the WEP if and only if it satisﬁes the following ‘inclusion’ property: for any C ∗ algebra D containing B as a C ∗ -subalgebra, and for any approximately unital operator algebra A, we have A ⊗max B ⊂ A ⊗max D isometrically. In Lance’s original statement A is a C ∗ -algebra, but it is easy to see that one may allow A to be nonselfadjoint. 7.4: Amenable Banach algebras were introduced by Barry Johnson in [205]. In that memoir, he proved that for a locally compact group G, L1 (G) is an amenable Banach algebra if and only if G is an amenable group. The fact that commutative C ∗ -algebras are amenable (a special case of 7.4.5), as well as Lemma 7.4.4, were also proved in that memoir. For abundant information on amenability, and the history of that subject, we refer the reader to [106, 304, 377], for example. Virtual h-diagonals and normal virtual h-diagonals were implicitly introduced in the selfadjoint setting by Haagerup [179], Eﬀros [134], and Eﬀros and Kishimoto [138]. These fundamental papers are a primary inspiration for the material in this section. The construction in 7.4.10 and Lemma 7.4.11 essentially go back to [138]. If u ∈ M ⊗σh M is a normal virtual h-diagonal, then the mapping Q = τπ (u) constructed in the proof of 7.4.12 is a completely bounded ‘quasiexpectation’ from B(H) onto [π(M )] , in the sense of Bunce and Paschke [79]. It is shown in that paper that if a von Neumann algebra M ⊂ B(H) is the range of a quasi-expectation, then it is injective. In 7.4.9, we mentioned that if a C ∗ -algebra B admits a virtual h-diagonal, then it is nuclear. We sketch a proof of this using quasi-expectations. If u is a virtual h-diagonal in (B ⊗h B)∗∗ , and if π : B → B(H) is a unital ∗-representation, then [π(B)] is the range of a quasi-expectation by the proof of 7.4.12 (2). Thus [π(B)] is injective (by the last paragraph). By the well-known equivalence of semidiscreteness and injectivity for W ∗ -algebras, [π(B)] is semidiscrete. By [66, Proposition 3.7], [π(B)] is semidiscrete. Taking π to be the universal representation shows that B ∗∗ is semidiscrete. By 6.6.8 (1), B is nuclear. A variant of this proof is given in [311]. This paper contains remarkable related results, such as the fact that if a C ∗ -algebra is the range of a quasi-expectation on B(H), then it is injective. See [163] for a beautiful short proof of this. The statements in 7.4.12 and 7.4.18 are new, and 7.4.13 is from [169]. Corollary 7.4.15 is from [105], whereas Sheinberg’s theorem 7.4.16 was proved in [382]. Let B be a unital Banach algebra. A diagonal of B is a virtual diagonal ˆ belonging to B ⊗B. Likewise if B is an operator algebra, then an h-diagonal of B is a virtual h-diagonal belonging to B ⊗h B. Paulsen and Smith proved in [319] that if a unital operator algebra possesses an h-diagonal, then it is ﬁnite-dimensional, and hence isomorphic to a C ∗ -algebra (see 7.4.14). This is a fortiori true if B has a diagonal. However it is unknown whether there may exist

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an inﬁnite-dimensional Banach algebra with a diagonal. See [377, Section 4.1] for more on this topic. Dual algebras with a virtual diagonal in the weak* Haagerup tensor product are discussed in [70]. Ruan introduced operator amenability for matrix normed algebras in [374]. By deﬁnition, a matrix normed algebra B is operator amenable if for every matrix normed B-bimodule X, every completely bounded derivation D : B → X ∗ is inner. For a matrix normed algebra B, let m : B ⊗ B → B be the canonical map induced by the multiplication on B. By deﬁnition, a virtual operator diagonal is ∗∗ an element u ∈ (B ⊗ B)∗∗ such that m (u) = 1, and cu = uc for any c ∈ B. The following analogue of 7.4.4 holds true (with essentially the same proof): A matrix normed algebra B is operator amenable if and only it admits a virtual operator diagonal. Ruan proved in [374] that a locally compact group G is amenable if and only if its Fourier algebra A(G) is operator amenable, if we regard A(G) as the operator space predual of the group von Neumann algebra of G. Moreover, operator amenability does not imply amenability for a matrix normed algebra. Related results appear in [375]. The arguments in the proof of 7.4.4 can also be easily adapted to show that a unital operator algebra B admits a virtual h-diagonal if and only if whenever X is an operator space and a Banach B-bimodule such that X ∗ is an operator B-bimodule, then every completely bounded derivation D : B → X ∗ is inner. It is tempting to call such operator algebras h-amenable. Corollaries 7.4.13–7.4.16 hold true with ‘amenable’ replaced by ‘h-amenable’. Unfortunately, we do not know any example of an h-amenable operator algebra which is not amenable. It is elementary to describe all diagonals of a ﬁnite-dimensional C ∗ -algebra, and it is instructive to look at their norms. Consider for simplicity the case when B = Mn for some integer n ≥ 1. Let (Eij )1≤i,j≤n n be the canonical basis of Mn . Then if [tij ] is an element of Mn such that j=1 tjj = 1, u=

tkj Eij ⊗ Eki ,

(7.26)

1≤i,j,k≤n

is a diagonal of Mn . Moreover all diagonals of Mn ⊗ Mn have this form. The special diagonal deﬁned by (7.17) when B = Mn , corresponds to the case when tij = 0 if j = i and tjj = n1 if 1 ≤ j ≤ n. Let u be given by (7.26), with n t = 1. Then it is not hard to check that u = [t ] h ij S 1 . In particular, j=1 jj uh = 1 if and only if [tij ] is a positive matrix. On the other hand, u∧ = 1 if and only if (7.17) holds. We leave this as an exercise for the reader.

8 C ∗-modules and operator spaces

This chapter only depends on (parts of) Chapters 1–4 of our text. It has several goals. The ﬁrst is to study Hilbert C ∗ -modules (and their W ∗ -algebra variant, W ∗ -modules) as operator modules. We aim to show that the theory of C ∗ -modules ﬁts comfortably into the operator module framework. Indeed the operator space viewpoint will lead us in a streamlined way through several aspects of the theory of C ∗ -modules. Although we will not say much about this here, our methods also permit the generalization of these modules to the nonselfadjoint operator algebra case (see the Notes section for references). In contrast, Banach module methods are not generally compatible with C ∗ -module constructions, and indeed completely break down when one attempts the aforementioned nonselfadjoint generalization. The second goal of our chapter is to consider some important TROs or C ∗ -modules which are associated to every operator space X. In particular, we will discuss further here the noncommutative Shilov boundary T (X) of X. TRO methods, and this Shilov boundary, provide important insights into the structure of X. Third, we will illustrate how C ∗ -module and TRO methods can lead to interesting results about operator spaces. Thus there is a profound two-way interaction between C ∗ -modules and operator spaces, which has attracted much interest in recent years. Because of limitations of space, we cannot reproduce here many of the operator space applications of C ∗ -module theory which appear in the literature. Instead, our more modest goal, and this is the fourth purpose of our chapter, is to lay out, in a systematic way, most of the basic concepts, theory, and connections, which are needed for such applications. There are many diﬀerent ‘pictures’, or ways of looking at, C ∗ -modules, as well as many routes through this theory. We will start from scratch, moving quickly, and later we will begin to add the operator space and TRO perspectives. Our presentation is essentially selfcontained. However, we will concentrate on material not in the standard sources on C ∗ -modules (for example, see [35, 173, 239, 269, 356, 421, 423]). We refer the reader to those texts for topics not covered here. We will mostly state our results for right modules. However, as we shall see, there is a striking ‘left-right symmetry’ to the theory; in particular every right C ∗ -module is also, canonically, a left C ∗ -module over another C ∗ -algebra. When we need to apply to a module which is being viewed as a left module, a result established earlier for right modules, we often refer to the ‘other-handed version’

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of the earlier result. 8.1 HILBERT C ∗ -MODULES—THE BASIC THEORY Throughout this section and the next, A and B are C ∗ -algebras. 8.1.1 (The deﬁnition) A (right) C ∗ -module over A is a right A-module Y , together with a map ·|· : Y × Y → A, which is linear in the second variable, and which also satisﬁes the following conditions: (1) y|y ≥ 0 for all y ∈ Y , (2) y|y = 0 if and only if y = 0, (3) y|za = y|za for all y, z ∈ Y, a ∈ A, (4) y|z∗ = z|y for all y, z ∈ Y , 1 (5) Y is complete in the norm y = y|y 2 . We call ·|· the A-valued inner product on Y . It follows from (3) and (4) that ya|z = a∗ y|z for all y, z ∈ Y, a ∈ A. In (5), the fact that · is a norm follows just as for Hilbert spaces from the following Cauchy–Schwarz inequality: y|z ≤ yz for y, z ∈ Y . This follows from the relation y|zz|y ≤ z2 y|y, which in turn may be proved by using the fact that 0 ≤ y + zb|y + zb, and taking b = −z|y/z2. The last calculation is an easy exercise using the fact that c∗ ac ≤ ac∗ c for any c ∈ A, a ∈ A+ . If the linear span of the range of the inner product ·|· is dense in A, then Y is called full. Left C ∗ -modules are deﬁned analoguously. Here Y is a left module over a C ∗ algebra A, the A-valued inner product is linear in the ﬁrst variable, and condition (3) in the above is replaced by ay|z = ay|z, for y, z ∈ Y, a ∈ A. Note that if A = C, then these are exactly the Hilbert spaces. On the other hand, a right C ∗ -module over C is also a Hilbert space in the usual (mathematical) sense, with the ‘reversed inner product’ ζ × η → η|ζ. If Y is a right C ∗ -module over A, then there is a canonical left C ∗ -module Y over A, which is simply the conjugate vector space of Y with left action a¯ y = ya ∗ , ∗ and inner product ¯ y |¯ z = y|z. This is usually called the conjugate C -module in the literature. We will favour the term adjoint module instead. 8.1.2 (Equivalence bimodules) If Y is an A-B-bimodule, then we say that Y is an equivalence bimodule, if Y is a full right C ∗ -module over B, and a full left C ∗ -module over A, and the two inner products are compatible in the sense that xy|z = [x|y]z, for all x, y, z ∈ Y . Here we have written [·|·] for the A-valued inner product. Equivalence bimodules are also sometimes called imprimitivity bimodules, or strong Morita equivalence A-B-bimodules, or equivalence A-B-bimodules. If there exists such an equivalence bimodule, we say that A and B are strongly Morita equivalent.

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Good examples of equivalence bimodules are furnished by the ternary rings of operators, or TROs, encountered in 4.4.1, and in Example 3.1.2 (6). If Z is a TRO, then we recall that D = Z Z and C = ZZ are C ∗ -algebras. Clearly Z is both a full right C ∗ -module over D, and a full left C ∗ -module over C. Indeed such Z is clearly an equivalence C-D-bimodule. Thus we have one direction of the correspondence between TROs and C ∗ modules. We will return to the other direction later, in 8.1.19 and 8.2.8. 8.1.3 (C ∗ -modules are Banach modules) If Y is a right C ∗ -module over A, then Y is a nondegenerate Banach A-module (see A.6.1). Indeed, ya2 = ya|ya = a∗ y|ya ≤ a2 y2 , for y ∈ Y and a ∈ A. If (et )t is a positive cai for A, then y − yet |y − yet = y|y − et y|y − y|yet + et y|yet −→ 0. Thus yet → y for all y ∈ Y . Hence Y is a nondegenerate Banach A-module. 8.1.4 (The ideal I) Throughout this chapter we reserve the symbol I for the closure of the linear span of the C ∗ -algebra valued inner product on a right C ∗ -module. If Y is a right C ∗ -module over A then it is clear that I is a closed two-sided ideal in A, and that Y is a full right C ∗ -module over I. We make some simple but important remarks: (1) The canonical map I → B(Y ), taking a ∈ I to the operator y → ya, is a linear isometry. Indeed it is clearly contractive. If Y a = 0 then we have that Y |Y a = Y |Y a = 0. This implies that Ia = 0, so that a∗ a = 0. Hence a = 0. Thus the map is one-to-one. It follows from A.5.9 (or from the other-handed version of the later result 8.1.15, for example) that this map is isometric. (2) Since Y is a C ∗ -module, and hence a nondegenerate Banach module, over I, we have Y I = Y by Cohen’s factorization theorem A.6.2. In particular, the linear span of terms of the form xy|z, for x, y, z ∈ Y , is dense in Y . (3) If Z is a Banach A-module then BA (Y, Z) = BI (Y, Z). Indeed, suppose that u ∈ BI (Y, Z), x ∈ Y and a ∈ A. By (2) and A.6.2, we may write x = ya , with a ∈ I, y ∈ Y . Then u(xa) = u(ya a) = u(y)a a = u(ya )a = u(x)a. (4) Conversely to the ﬁrst paragraph of 8.1.4, if Y is a right C ∗ -module over A, and if C is a C ∗ -algebra containing A as an ideal, then Y is also a right C ∗ module over C. The module action here is the canonical one: (ya)c = y(ac), for y ∈ Y, a ∈ A, and c ∈ C. The fact that this action is well deﬁned, and that 8.1.1 (3) holds, follows from the relation z|y(ac) = z|yac = z|yac, for z ∈ Y . Lemma 8.1.5 Let u : Y → Z be a bounded A-module map between right C ∗ modules over A. Then u(y)|u(y) ≤ u2 y|y, for all y ∈ Y . Proof We may suppose that u ≤ 1, and (by 8.1.4 (4)) that A is unital. Then the result follows by a trivial modiﬁcation of the argument for (4.10). 2

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Proposition 8.1.6 Suppose that A is a C ∗ -subalgebra of B(H). The norm and inner product on a right C ∗ -module Y over A are related by the formula: y|yζ, ζ = sup{f (y)ζ2 : f ∈ Ball(BA (Y, A))},

y ∈ Y, ζ ∈ H.

Proof For f ∈ Ball(BA (Y, A)), y ∈ Y, and ζ ∈ H, we have f (y)ζ2 = f (y)∗ f (y)ζ, ζ ≤ y|yζ, ζ, 1

by 8.1.5. Next, set f = zn | ·, where zn = y(y|y + n1 )− 2 . Here n ∈ N. We have 1 1 zn |zn = (y|y + n1 )− 2 y|y(y|y + n1 )− 2 ≤ 1. Hence f ∈ Ball(BA (Y, A)). Note that f (y)∗ f (y) = y|y(y|y + n1 )−1 y|y. By spectral theory, lim y|y(y|y + n

1 −1 ) y|y = y|y. n

Putting these facts together gives the desired equality.

2

Proposition 8.1.6, together with the polarization identity (1.1), shows that the inner product on a C ∗ -module Y is completely determined by, and may be recovered from, the Banach module structure of Y . 8.1.7 (Adjointable maps) If Y, Z are right C ∗ -modules over A, then we write BA (Y, Z) (or simply B(Y, Z)) for the set of adjointable maps from Y to Z, that is, the set of maps T : Y → Z such that there exists a map S : Z → Y with T (y) | z = y | S(z) ,

y ∈ Y, z ∈ Z.

It is easy to see that such an S is unique; it is denoted by T ∗ . It follows immediately from the centered equation that (i) T is a right A-module map, (ii) T is bounded (use the closed graph theorem), (iii) T ∗ is a bounded right A-module map, and (iv) T ∗∗ = T . It is also easy to verify relations such as (T1 T2 )∗ = T2∗ T1∗ . Also, BA (Y, Z) is a norm closed subspace of BA (Y, Z). Indeed, if (Tn )n is Cauchy in BA (Y, Z), then (Tn∗ )n is Cauchy too. The limits of these two sequences evidently are the adjoints of each other. Writing BA (Y ) = BA (Y, Y ), it is easy to verify that BA (Y ) is a C ∗ -algebra with respect to the usual norm of a bounded operator. For example, the more diﬃcult direction of the C ∗ -identity T ∗T = T 2 follows immediately from the fact that if y ∈ Ball(Y ), then T y2 = T (y)|T (y) = T ∗ T (y)|y ≤ T ∗ T . We deﬁne KA (Y, Z) (or simply K(Y, Z)) to be the closure, in B(Y, Z), of the linear span of the ‘rank-one’ operators |zy|, for y ∈ Y, z ∈ Z. We are using ‘braket’ notation here; thus |zy| is the operator which takes an x ∈ Y to zy|x. It is easy to verify familiar relations such as (|zy|)∗ = |yz|, and that the product of ‘rank one’ operators is ‘rank one’: (|yz|)(|y z |) = |y(z|y )z |. Also |ya∗ z| = |yza| if a ∈ A. From these relations, it is easy to see that

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KA (Y ) = KA (Y, Y ) is a C ∗ -subalgebra of BA (Y ). Indeed KA (Y ) is a closed ideal of BA (Y ), as may be seen from the simple relations T (|yz|) = |T (y)z| and (|yz|)S = |yS ∗ z| for T ∈ BA (Y ) and S ∈ BA (Y ). Similarly, KA (Y, Z) is canonically a BA (Z)-BA (Y )-bimodule.

(8.1)

We say that a map u : Y → Z between right C ∗ -modules over A is unitary if u is a surjective A-module map such that uy|uy = y|y, for all y ∈ Y . By the polarization identity (1.1) this is equivalent to: uy|uz = y|z for all y, z ∈ Y . It is clear that any unitary map is adjointable. If there exists such a unitary then we say Y ∼ = Z unitarily. If u is unitary then u∗ = u−1 . If Y = Z then u is unitary if and only if it is a unitary in the C ∗ -algebra BA (Y ). Corollary 8.1.8 An A-module map u : Y → Z between right C ∗ -modules over A is unitary if and only if u is isometric and surjective. Proof The one direction is obvious. The other direction follows from 8.1.5 2 applied to both u and u−1 . 8.1.9 (The direct sum) If {Yi : i ∈ I} is a collection of right C ∗ -modules over A, then we deﬁne the direct sum C ∗ -module by

Yi : yi |yi converges in norm in A . ⊕ci Yi = (yi ) ∈ i∈I

i

With the canonical inner product (yi )|(zi ) = i yi |zi (which converges by the polarization identity (1.1)), and obvious A-module action, ⊕ci Yi is a right C ∗ -module over A. We omit the routine proof. In passing, we remark that there are alternative ways to deﬁne the direct sum. See, for example, 8.2.14. For a cardinal I, CI (Y ) denotes the C ∗ -module direct sum of I copies of Y . We will see in 8.2.3 (4) that there is no conﬂict with earlier operator space notation: viewed as an operator space, CI (Y ) means exactly what it meant before. It is important, and easily seen, that the canonical inclusion and projection maps between ⊕ci Yi and its summands Yi are adjointable. We say that a right C ∗ -module Z is the internal orthogonal direct sum of closed submodules Y and W , if Z = Y + W and Y ⊥ W (i.e. y|w = 0 for all y ∈ Y, w ∈ W ). In this case, Z ∼ = Y ⊕c W unitarily, quite clearly. We say that Y is orthogonally complemented in Z if there exists such a W . It is clear that Y is orthogonally complemented in Z if and only if Y is the range of a projection (i.e. a selfadjoint idempotent) P in the C ∗ -algebra BA (Z). 8.1.10 (A characterization of module maps) Result 8.1.5 may be improved. In fact, a linear map u : Y → Z between C ∗ -modules over B is a contractive B-module map, if and only if u(y)|u(y) ≤ y|y, for all y ∈ Y . This is also equivalent to (u(y), z) ≤ (y, z), for all y ∈ Y and z ∈ Z, where the norms here are those of Z ⊕c Z and Y ⊕c Z respectively. We omit the proofs of these assertions since we shall not need them (see [302,56]). We remark that the latter condition is the analogue for C ∗ -modules of condition 4.5.2 (ii).

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Proposition 8.1.11 Suppose that Y is a right C ∗ -module over A. Then: (1) Y ∼ = KA (A, Y ) isometrically. (2) Y ∼ = KA (Y, A) isometrically. (3) KA (Cn (Y )) ∼ = Mn (KA (Y )) and BA (Cn (Y )) ∼ = Mn (BA (Y )) as C ∗ -algebras. Proof (1) That the canonical map L : Y → BA (A, Y ) is isometric, with range contained inside KA (A, Y ), is a simple modiﬁcation of the proof of 3.5.4 (1). Since that range is a closed vector subspace which contains every ‘rank one’ operator, it must equal KA (A, Y ). y)(z) = y|z, for y, z ∈ Y . Clearly Φ is (2) Deﬁne Φ : Y → BA (Y, A) by Φ(¯ isometric and linear. We show that Φ maps onto KA (Y, A). Indeed, by A.6.2 we y ) = a∗ y |·, the generic may write any y ∈ Y as y a, for y ∈ Y, a ∈ A. Then Φ(¯ ‘rank one’ operator in KA (Y, A). We conclude as in (1). (3) There is a canonical homomorphism θ : Mn (KA (Y )) → BA (Cn (Y )). It is easy to check that θ(a)∗ = θ(a∗ ), so that θ is a ∗-homomorphism into BA (Cn (Y )). Clearly θ is one-to-one. We leave it as an exercise that Ran(θ) = K A (Cn (Y )) (using the (adjointable) canonical inclusion and projection maps between Cn (Y ) and its summands, and (8.1)). The other assertion is similar. 2 Lemma 8.1.12 Let Y be a right C ∗ -module over A, and suppose that T is a linear map on Y . Then T ∈ BA (Y )sa (resp. T ∈ BA (Y )+ ) if and only if T y|y is selfadjoint (resp. T y|y ≥ 0) for all y ∈ Y . Proof We sketch a proof of the diﬃcult implication. If T y|y is selfadjoint for all y ∈ Y , then T y|y = (T y|y)∗ = y|T y. By the polarization identity (1.1), T is adjointable on Y , with T ∗ = T . If, further, T y|y ≥ 0, for y ∈ Y , then T is selfadjoint, by the above. To see that T ≥ 0, suppose that A ⊂ B(H), and apply the following simple general fact about C ∗ -algebras to the collection of positive functionals y| · yζ, ζ on BA (Y ), for y ∈ Ball(Y ), ζ ∈ Ball(H). Namely, let B be a C ∗ -algebra, and let S be a set of positive contractive functionals on B with b = sup{ϕ(b) : ϕ ∈ S},

for all b ∈ B+ .

Then if b ∈ Bsa and if ϕ(b) ≥ 0 for all ϕ ∈ S, then b ∈ B+ . To prove this general fact, suppose that the endpoints of the spectrum of b are α and β, with α ≤ β. The centered equation applied to β1 − b, gives β − α = β1 − b = sup{βϕ(1) − ϕ(b) : ϕ ∈ S} ≤ β. From this it is clear that α ≥ 0. Hence b ≥ 0. Corollary 8.1.13 If a = [aij ] ∈ Mn (A), then an equals

c∗i aij bj : b, c ∈ Ball(Cn (A)) . sup{abCn(A) : b ∈ Ball(Cn (A))} = sup i,j

Moreover, the following are equivalent:

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(i) a is positive in Mn (A), ∗ (ii) i,j bi aij bj ≥ 0 in A, for all b1 , . . . , bn ∈ A, (iii) a is a ﬁnite sum of matrices of the form [a∗i aj ], with a1 , . . . , an ∈ A. Proof Deﬁne θ : Mn (A) → BA (Cn (A)) by the canonical left action of Mn (A) on Cn (A). Clearly θ is a one-to-one ∗-homomorphism into the C ∗ -algebra BA (Cn (A)) of 8.1.7. Then the ﬁrst statement is clear from A.5.8. The equivalence of (i) and (ii) follows from 8.1.12. The equivalence with (iii) is a simple exercise. 2 8.1.14 (C ∗ -modules are equivalence bimodules) Any right C ∗ -module Y over B is canonically also a full left C ∗ -module over KB (Y ), using |··| for the inner product. When checking this, the only nontrivial point is that |yy| ≥ 0, which follows quite easily from 8.1.12. It is now clear that if Y is a right C ∗ -module over B, then Y is also an equivalence KB (Y )-I-bimodule. Here I is as in 8.1.4. Note also that the norm on Y induced by this new inner product corresponding to this left C ∗ -module action, is the same as the old norm. In fact it is evident that |yy| ≤ y2 . The reverse inequality follows from the fact that |yy2 ≥ |yy|(z)2 = z|y y|y y|z,

z ∈ Ball(Y ).

Setting z = y/y yields the desired inequality. The proof of the following result shows, conversely, that given an equivalence A-B-bimodule Y , then A ∼ = KB (Y ) ∗-isomorphically, and via this isomorphism the action of A on Y corresponds exactly to the canonical KB (Y ) action, and the A-valued inner product corresponds to the KB (Y )-valued inner product. Thus we see that full right C ∗ -modules are essentially the same things as strong Morita equivalence bimodules. This gives another ‘picture’ of C ∗ -modules, as the strong Morita equivalence bimodules. For the next result, we write λ for the canonical map from A into B(Y ), for a left Banach A-module Y . Lemma 8.1.15 If Y is an equivalence A-B-bimodule, then λ(A) = KB (Y ). Indeed, A ∼ = KB (Y ) ∗-isomorphically, via the map λ above. Also, the two norms deﬁned on Y via each of the two inner products coincide. Proof Writing [·|·] for the A-valued inner product, we have xay|z = [x|ay]z = [x|y]a∗ z = xy|a∗ z,

x, y, z ∈ Y, a ∈ A.

Thus by 8.1.4 (1), we have ay|z = y|a∗ z. Hence λ(A) ⊂ BB (Y ), and λ is a ∗-homomorphism into BB (Y ). Clearly λ([y|z]) = |yz|, so that it follows by continuity and density that the range of λ is KB (Y ). By the left module version of 8.1.4 (1), λ is one-to-one. Thus λ is an isometry, by A.5.8. For the last assertion, since λ is isometric, and by the second paragraph of 8.1.14, we have [y|y] = |yy| = y|y. 2 Proposition 8.1.16

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(1) If Y is an equivalence A-B-bimodule then the canonical map from the multiplier algebra M (A) to B(Y ) (see 3.1.11) is isometric, and indeed is a ∗-isomorphism onto BB (Y ). This ∗-isomorphism extends the canonical isomorphism between A and KB (Y ) (see 8.1.15). (2) Further, the ∗-isomorphism in (1) extends to an isometric isomorphism between LM (A) and BB (Y ). (3) If Y is a right C ∗ -module over B, then BB (Y ) ∼ = LM (KB (Y )) isometrically (as Banach algebras), and BB (Y ) ∼ = M (KB (Y )) ∗-isomorphically. (4) BB (Y ) is the span of the Hermitian elements (see A.4.2) of BB (Y ). Proof Suppose that Y is an equivalence A-B-bimodule, and that (et )t is a positive cai for A. By 3.1.11, Y is a Banach LM (A)-module, and we have a corresponding contractive homomorphism θ : LM (A) → BB (Y ) given by θ(η)(y) = lim η(et )y = η(a)y , t

y, y ∈ Y, a ∈ A, such that y = ay . (8.2)

It follows from this, and 8.1.4 (1), that θ is a one-to-one homomorphism extending λ. Next ﬁx η ∈ M (A). By (8.2) and 8.1.15, θ(η)y|z = lim η(et )y|z = lim y|η(et )∗ z, t

t

y, z ∈ Y.

Writing z = az for a ∈ A, z ∈ Y , the latter quantity equals lim y|et η ∗ (a)z = y|η ∗ (a)z = y|θ(η ∗ )z, t

using (8.2) again. Thus θ restricts to a ∗-homomorphism from M (A) into B B (Y ). Suppose that Y is simply a right C ∗ -module. We may assume, by 8.1.4 (3), that Y is full, and we apply the above with A = KB (Y ). By (8.1), we may deﬁne a contractive homomorphism ρ : BB (Y ) → LM (KB (Y )) by ρ(S)(T ) = ST , for S ∈ BB (Y ), T ∈ KB (Y ). Here we are viewing LM (A) as the right A-module maps on A. It is easy to see that θ ◦ ρ is the identity map. Hence θ is surjective, and ρ is isometric. Also, by 2.6.8, it is easy to check that ρ takes BB (Y ) into M (KB (Y )). Thus we have proved (3). In the situation of (2), by (3) and 8.1.15 we have isometric isomorphisms LM (A) −→ LM (KB (Y )) −→ BB (Y ). It is easy to check that the composition of these maps is the map θ in (8.2). This proves (2), and (1) is similar. Assertion (4) follows from (3) and 2.6.9. 2 8.1.17 (The linking C ∗ -algebra) Suppose that Y is a right C ∗ -module over B, and write A = KB (Y ). We deﬁne the linking C ∗ -algebra L(Y ) to be the set of 2 × 2 matrices: A Y . L(Y ) = Y B We turn this set into an algebra, using the usual product of 2 × 2 matrices, and using the inner products and module actions. For example, the product y¯ z , of a

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term y from the 1-2-corner, and a term z¯ from the 2-1-corner, is taken to mean |yz| ∈ A = KB (Y ). Or the product z¯a, for a ∈ A, is given by a∗ z. We deﬁne the involution of one of these 2 × 2 matrices in the obvious way. Deﬁne a map π : L(Y ) → B(Y ⊕c B) by the obvious action (i.e. viewing an element of Y ⊕c B as a column with two entries, and formally multiplying a 2 × 2 matrix and such a column. It is easy to check that π(m) ∈ BB (Y ⊕c B) for each matrix m ∈ L(Y ), and moreover that π is a ∗-homomorphism into BB (Y ⊕c B). Also, one can quickly check that π is one-to-one, and that π is an isometry when restricted to each of the four corners of L(Y ). Hence the range of π is closed. We give L(Y ) a norm by pulling back the norm from BB (Y ⊕c B) via π, thus L(Y ) is a C ∗ -algebra ∗-isomorphic to the range of π. Indeed we may regard BB (Y ⊕c B) as a 2 × 2 matrix C ∗ -algebra consisting of matrices t = [tij ] whose four entries are adjointable maps. To see this, note that tij are deﬁned in terms of t and the (adjointable) projection and inclusion maps between Y ⊕c B and its two summands. With this in mind, it is clear from 8.1.11 and (8.1), that KB (Y ⊕c B) is exactly the range of π. Hence the linking C ∗ -algebra may simply be thought of as KB (Y ⊕c B). By the last fact, and 8.1.16 (3), we see that the multiplier algebra M (L(Y )) is BB (Y ⊕c B). We deﬁne the unitized linking C ∗ -algebra L1 (Y ) of Y to be the linear span within M (L(Y )) of KB (Y ⊕c B) and the two diagonal idempotent matrices p = 1 ⊕ 0 and q = 0 ⊕ 1. The last two 1’s may be viewed as the identities of the unitizations of A and B respectively (where we take the unitization of a unital algebra to be itself). Then L1 (Y ) is a unital C ∗ -algebra with identity 1 = p + q. Clearly Y is the 1-2-corner of both L1 (Y ) and L(Y ). In particular, Y ∼ = pL(Y )(1 − p).

(8.3)

If we take a general C ∗ -algebra A, and if Y is an equivalence A-B-bimodule, then we may form L(Y ) as above, but using A instead of KB (Y ). Of course by 8.1.15 this is essentially the same thing; that is, the resulting linking algebras will be ∗-isomorphic. In this case we say that L(Y ) is the Morita linking algebra of Y . We will also use this terminology even when A is not speciﬁed, taking A = KB (Y ), however we insist that Y be full over B in this case. Corollary 8.1.18 If Y is a right C ∗ -module over B, then L(Y ) is strongly Morita equivalent to B (via the equivalence bimodule Y ⊕ c B). Proof Clearly Y ⊕c B is a full B-module. Since KB (Y ⊕c B) ∼ = L(Y ), we have 2 by 8.1.14 that L(Y ) is strongly Morita equivalent to B, via Y ⊕ c B. 8.1.19 (C ∗ -modules and corners) One great advantage of the linking C ∗ algebra of a C ∗ -module Y , is that the inner products and module actions have been replaced by concrete multiplication of elements in a C ∗ -algebra. To see this, we employ the completely isometry in (8.3). This is simply the ‘corner map’ c, taking y ∈ Y to the matrix in L(Y ) with y in the 1-2-corner and zeroes elsewhere. If we identify B with the 2-2-corner in a similar way, then y|z is simply the

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product c(y)∗ c(z) in the C ∗ -algebra L(Y ). Indeed it is convenient, and usually leads to no diﬃculties, to suppress the ‘c’ map and simply write y ∗ z for the last expression above. Similarly, if we write xy ∗ z the reader will have no diﬃculty in seeing that what is meant is the element xy|z, or equivalently |xy|(z). Similar conventions apply to longer such products. Thus, by (8.3), any right C ∗ -module Y is a corner in a C ∗ -algebra, in the sense of 2.6.14. Conversely, any corner pAq in a C ∗ -algebra A, is clearly a right C ∗ -module over qAq. This gives another ‘picture’ of C ∗ -modules, as the corners of C ∗ -algebras. In the language of 4.4.1, any C ∗ -module may be viewed as a TRO, namely, as a subtriple of L(Y ). This, together with the second paragraph of 8.1.2, gives another ‘picture’ of C ∗ -modules, as the TROs. We will tighten up this observation further in 8.2.8. If Y is an equivalence A-B-bimodule, then one can also view LM (A), and its action on Y , in terms of the linking algebra. Indeed, LM (A) ⊂ A∗∗ ⊂ L(Y )∗∗ . The composition of these canonical inclusions is easily seen (since Y = AY B) to have range within LM (L(Y )). That is, we may regard as subalgebras: LM (A) ⊂ LM (L(Y )) ⊂ L(Y )∗∗ . Similar assertions hold for RM (B). We turn to some more corollaries of 8.1.16: Corollary 8.1.20 If A and B are strongly Morita equivalent, then the centers of their multiplier algebras are ∗-isomorphic, via a ∗-isomorphism θ satisfying θ(η)y = yη,

for all y ∈ Y, η ∈ Z(M (B)).

Here Y is the associated equivalence A-B-bimodule. Proof By 3.1.11, Y is a right Z(M (B))-module, with action yη = lim t yη(et ), for y ∈ Y, η ∈ Z(M (B)). Here (et )t is a cai for B. As in A.6.1, this deﬁnes a contractive unital homomorphism π : Z(M (B)) → B(Y ). Clearly π maps into BB (Y ). By 8.1.16 (4), together with A.4.2, π maps Z(M (B))sa , and hence also Z(M (B)), into BB (Y ). Thus by 8.1.16 (1), there exists a unique ν ∈ M (A) such that νy = yη for all y ∈ Y . Clearly this implies that νay = ayν = aνy, if a ∈ A. This, and 8.1.4 (1), implies that ν ∈ Z(M (A)). Moreover, if we deﬁne θ(η) = ν then θ is a homomorphism from Z(M (B)) to Z(M (A)). Now it is easy to see, by symmetry, that θ must be an isomorphism. By the last part of A.5.4, for example, θ is a ∗-isomorphism. 2 Corollary 8.1.21 (1) Let P be a contractive idempotent A-module map on a right C ∗ -module Y over A. Then P is adjointable. Indeed P is an orthogonal projection in the C ∗ -algebra B(Y ), and the range of P is an orthogonally complemented submodule of Y . (2) Suppose that Y and Z are right C ∗ -modules over A, and that α : Y → Z and β : Z → Y are contractive module maps with βα = IY . Then these maps

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306

are adjointable, with β = α∗ . Moreover, α is a unitary module map onto an orthogonally complemented submodule of Z. Proof (1) By 8.1.16 (3), P corresponds to a contractive idempotent in the operator algebra LM (KA (Y )). By 2.1.3, the last idempotent is Hermitian. Thus P is Hermitian, and is adjointable by 8.1.16 (4). The rest follows from 8.1.9. (2) Note that α is an isometry onto the closed submodule W = Ran(αβ) of Z. Since αβ satisﬁes the conditions of (1), it is adjointable. By 8.1.9 we see that W is orthogonally complemented. By 8.1.8 we have that α is unitary, and hence adjointable as a map into W . It is then easy to see that α is adjointable as a map into Z. Indeed, βz|y = (αβ)z|αy = z|(αβ)αy = z|αy, for y ∈ Y, z ∈ Z.

2

From 8.1.21, one may deduce a universal property of the direct sum: Proposition 8.1.22 Suppose that {Yi : i ∈ I} is a collection of right C ∗ modules over A, and that Y is a right C ∗ -module over A, such that there exist contractive module maps i : Yi → Y and Pi : Y → Yi such that Pi ◦ j = δi,j IYi , and such that i i ◦ Pi converges strongly on Y . Then there exists an orthogoc ∼ c nally complemented submodule W of Y such that Y = (⊕i Yi ) ⊕ W unitarily. If i i ◦ Pi converges strongly to IY , then W = 0. Proof By 8.1.21, each i and Pi are adjointable, with Pi = ∗i , and Qi = i ◦ Pi is an adjointable projection on Y . Moreover, the Qi are mutually orthogonal, and of course positive, elements of the C ∗ -algebra B(Y ). Set R(y) = i Qi (y), for y ∈ Y . Then R is a module map on Y , and * ) 0 ≤ R(y)|y = lim Qj y|y ≤ y|y, y ∈ Y, J

j∈J

the limit over ﬁnite subsets J of I. Thus R is contractive and positive (by 8.1.12). Clearly RQi = Qi , which implies that R is idempotent. Hence, by 8.1.21, R and IY − R are adjointable projections on Y . Clearly I − R is orthogonal to each Q i . If W = Ran(I − R) and W ⊥ = Ran(R), then Y ∼ = W ⊥ ⊕c W unitarily. Deﬁne ⊥ c u : W → ⊕i Yi by u(y) = (Qi y). It is easy to check that uy|uy = y|y, so that u is an isometry. Since Ran(u) is norm dense, u is a unitary. 2 8.1.23 (Finite rank approximation) If Y is a right C ∗ -module over B, then we claim that the C ∗ -algebra KB (Y ) has a cai (et )t of the form et =

n(t)

|xtk xtk |,

(8.4)

k=1

for elements xtk in Y . To obtain this, ﬁrst pick a cai (ft )t from the dense ideal of ‘ﬁnite rank’ operators in KB (Y ). Let et = ft∗ ft , then (et )t is also a cai for KB (Y ).

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n If ft = i=1 |yi zi |, then by relations in the second paragraph of 8.1.7 we have n ft∗ ft = i,j=1 |zi aij zj |, where [aij ] = [yi |yj ]. The latter is a positive matrix P with entries in B (this may be seen by the criterion 8.1.13 (ii), for example). Factoring P as the square of its square root, and regrouping, we obtain (8.4). Since KB (Y ) acts nondegenerately on Y (by 8.1.3 and 8.1.14), for any y ∈ Y n(t) we have that k=1 xtk xtk |y → y. From this we deduce the following: Corollary 8.1.24 Let Y be a right C ∗ -module over B. Then there exists a net (n(t))t of positive integers, and contractive B-module maps αt : Y → Cn(t) (B) and βt : Cn(t) (B) → Y , such that βt (αt (y)) → y for every y ∈ Y . Indeed this can be done with α∗t = βt . Proof We use the notation above. For y ∈ Y , deﬁne αt (y) ∈ Cn(t) (B) to have n(t) kth entry xtk |y. Also, deﬁne βt (b) = k=1 xtk bk . Here b has kth entry bk ∈ B. n(t) We have that βt (αt (y)) = k=1 xtk xtk |y = et y → y, as we saw immediately above the corollary. It is easily checked that αt , βt are adjointable, with α∗t = βt , so that they have the same norm. We have: , - αt (y)2 = y|xtk xtk |y = y xtk xtk |y = y|et y ≤ y2 . k

k

2

Thus αt is contractive, and hence so is βt . ∗

8.1.25 (Asymptotic factorization) Thus any right C -module Y over B ‘factors asymptotically’ through spaces of columns over B. In passing, we remark that a simple modiﬁcation of the last proof, using 8.1.14 and the left-handed version of 8.1.23, shows that if Y is a full C ∗ -module over B, then B factors asymptotically (in the sense of the last result) through spaces of the form Cn(t) (Y ). Similarly, one can show that KB (Y ) factors asymptotically (via completely contractive linear maps) through spaces of the form Mn(t) (B). Since we shall not use these, we omit the details (see [46, p. 391]). The converse of 8.1.24 is also true: if Y is a right Banach B-module for which there exist nets of contractive B-module maps as in the lemma, then Y is a right C ∗ -module over B. This gives a characterization of C ∗ -modules among the Banach B-modules. Indeed we have: Theorem 8.1.26 Suppose that B is a C ∗ -algebra and that Y is a right Banach B-module. Suppose further that there is a net (Yt )t of right C ∗ -modules over B, and contractive B-module maps αt : Y → Yt and βt : Yt → Y , such that βt (αt (y)) → y for every y ∈ Y . Then Y is a right C ∗ -module over B, and the norm on Y coincides with the norm induced by the inner product (see 8.1.1 (5)). The inner product on Y is given by the formula y|z = lim αt (y)|αt (z), t

The limit here is in the norm topology of B.

y, z ∈ Y.

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308

A page long proof of this result, which uses only the triangle and Cauchy– Schwarz inequalities, and 8.1.5, may be found in [65, p. 41]. For ﬁnitely generated or countably generated C ∗ -modules one may improve on the last result considerably. By an algebraically ﬁnitely generated B-module, we mean a module Y for which there exists y1 , . . . , yn ∈ Y such that the map f : Cn (B) → Y given by f ((ak )) = k yk ak , is surjective. Theorem 8.1.27 (1) If Y is an algebraically ﬁnitely generated right C ∗ -module over B, then Y is unitarily isomorphic to an orthogonally complemented submodule of Cn (B), for some n ∈ N. (2) A right C ∗ -module Y over B is algebraically ﬁnitely generated if and only if KB (Y ) is unital. In this case, KB (Y, Z) contains all B-module maps from Y to Z, for any C ∗ -module Z over B. Proof (1) Let f be the map above 8.1.27, which is easily seen to be adjointable. By [423] 15.3.8 (or rather the obvious variant of that result to maps between two possibly diﬀerent C ∗ -modules over B), there is a polar decomposition f = u|f |; W = Ran(|f |) is a closed orthogonally complemented submodule of Cn (B); and Y is unitarily isomorphic to W via the partial isometry u. (2) For the ﬁrst part, it is easy to see by A.6.2 that Y is algebraically ﬁnitely generated over B if and only if it is also ﬁnitely generated over B 1 . Thus we may assume that B is unital. If KB (Y ) is unital, then since Y is a nondegenerate KB (Y )-module (by 8.1.3 and 8.1.14), this identity is IY . Given > 0, we can ﬁnd et = k |yk yk | as in 8.1.23, with et − IY < . Hence et is invertible with inverse S say, so that IY = k |S(yk )yk |, from which it is immediate that (S(yk ))k generates Y . Conversely, if Y is algebraically ﬁnitely generated, let u : Cn (B) → Y be the surjective partial isometry in the proof of (1). We have uu∗ = IY . Put yk = u(ek ), where ek has 1B in the kth entry, and is zero elsewhere. Then |yk yk | = u |ek ek | u∗ = uu∗ = IY . k

k

Hence IY ∈ KB (Y ). Finally, any B-module map v : Y → Z into a Banach B-module Z satisﬁes v(y) = v

n k=1

n yk yk |y = v(yk )yk |y,

y ∈ Y.

k=1

Thus v is bounded, and the stated assertion is clear.

2

8.2 C ∗ -MODULES AS OPERATOR SPACES. 8.2.1 (C ∗ -modules are operator spaces) If Y is a right C ∗ -module over B, and if n ∈ N, then Mn (Y ) is a right C ∗ -module over Mn (B), with inner product

C ∗ -modules and operator spaces n [yij ]|[zij ] = yki |zkj ,

309 [yij ], [zij ] ∈ Mn (Y ).

(8.5)

k=1

One way to see this is to identify Y with the 1-2-corner of the linking C ∗ -algebra L(Y ), and B with the 2-2-corner, as in 8.1.19, so that y|z = y ∗ z for y, z ∈ Y . If we do this then, ﬁrst, Mn (Y ) may be identiﬁed with a corner of the C ∗ -algebra canonical inner product inherited from the latter C ∗ -algebra Mn (L(Y )), nand the ∗ ∗ is y z = [ k=1 yki zkj ], for y = [yij ], z = [zij ] ∈ Mn (Y ). This gives (8.5). Second, Y inherits a canonical operator space from the C ∗ -algebra L(Y ). We call this the canonical operator space structure on Y . It is given explicitly by n 21 yki |ykj , [yij ]n =

[yij ] ∈ Mn (Y ),

(8.6)

k=1

as may be seen by using the C ∗ -identity in Mn (L(Y )). When we consider a C ∗ -module as an operator space, it will always be with respect to this structure. The formula (8.6) is also valid for nonsquare matrices. For instance, for a 1 column y = [y1 · · · yn ]t ∈ Cn (Y ) = Mn,1 (Y ), we have y = nk=1 yk |yk 2 . ∗ Viewing Y as a subspace of the linking C -algebra, we also have 1

1

1

y = yy ∗ 2 = [yi yj∗ ] 2 = [|yi yj |] 2 . Note that this shows that Cn (Y ) is isometric to the C ∗ -module direct sum of n copies of Y . Similarly, if [x1 · · · xn ] ∈ Rn (Y ), then n 12 1 |xk xk | . [x1 · · · xn ]Rn (Y ) = [xi |xj ] 2 =

(8.7)

k=1

Proposition 8.2.2 For C ∗ -modules Y and Z over B, every bounded B-module map u : Y → Z is completely bounded, with u = ucb. If u is unitary then it is a complete isometry. Proof This may be seen in many ways. For example, assume that u ≤ 1, and that x1 , . . . , xn ∈ Y, b1 , . . . , bn ∈ B. Set z = i xi bi . Then b∗i (xi |xj − uxi |uxj )bj = z|z − uz|uz ≥ 0, i,j

using 8.1.5. By 8.1.13 (ii) it follows that [xi |xj ]− [uxi |uxj ] ≥ 0. Then the ﬁrst result follows from an easier variant of the proof of the implication ‘(v) implies (ii)’ of Theorem 4.5.2. We leave the second as an exercise. 2 Henceforth, we give BB (Y, Z) = CBB (Y, Z) the operator space structure from 3.5.1. We assign to BB (Y, Z) and KB (Y, Z) the operator space structures which they inherit as subspaces of CBB (Y, Z). We shall see in 8.2.3 (7) below that if Y = Z, the latter operator space structures coincide with their canonical C ∗ -algebra operator space structure.

C ∗ -modules as operator spaces.

310

8.2.3 (Operator space variants of C ∗ -module facts) Most of the results in Section 8.1 have operator space variants. We list the key points below; some of these will be used often in the rest of the chapter. (1) Any right C ∗ -module Y over B is a right operator B-module. Indeed this follows immediately from 3.1.2 (5) and (8.3). By 3.1.11, Y is also a right operator module over M (B), or over RM (B). By symmetry, Y is also a left operator module over KB (Y ), and (using also 8.1.16 (3)) over BB (Y ). Thus Y is an operator BB (Y )-B-bimodule. Similarly, if Y is an equivalence A-B-bimodule, then Y is an operator A-B-bimodule, and an operator M (A)-M (B)-bimodule. (2) Let Y be an equivalence A-B-bimodule. Viewing Y as the 1-2-corner of the linking C ∗ -algebra, and the ‘adjoint module’ Y as the 2-1-corner, one sees the canonical operator space structure on Y , as exactly the adjoint operator space structure Y from 1.2.25. Note that Y is an operator B-A-bimodule (see 3.1.16). (3) If ⊕ci Yi is a direct sum of right C ∗ -modules over B, equipped with its canonical operator space structure, then Mn (⊕ci Yi ) ∼ = ⊕ci Mn (Yi ) unitarily as Mn (B)-modules. We leave this to the reader. (4) For any cardinal I, the right C ∗ -module direct sum of I copies of Y , is completely isometrically isomorphic to the operator space CI (Y ) deﬁned in 1.2.26. To see this, recall the canonical complete isometry c : Y → L(Y ) from 8.1.19. The ampliﬁcation cI,1 of c is a completely isometric embedding from CI (Y ) into CI (L(Y )) (see 1.2.26). It is clear that (yi ) ∈ CI (Y ). c(yi )∗ c(yi ) = yi |yi , cI,1 ((yi ))2 = i

i

This proves the isometric case of our result. The complete isometry may be deduced from the isometric case together with (3) above. (5) There is an ‘operator space version’ of 8.1.26, which we may state as follows. Let Y be an operator space and a right B-module, and suppose that there exist maps αt and βt satisfying all the conditions in 8.1.26. If αt and βt are completely contractive, then in addition to the conclusions of 8.1.26, the given matrix norms on Mn (Y ) coincide with the norm (8.6) induced by the inner product. To prove this, notice that the ampliﬁcations (αt )n and (βt )n are contractive Mn (B)-module maps. Applying 8.1.26 to Mn (Y ), and to these maps, yields the desired assertion. (6) We consider the operator space version of 8.1.16. By (1) above, any right C ∗ -module Y over B, is a left operator A-module, where A = KB (Y ). By 3.1.11 it is also a left operator LM (A)-module. Thus by 3.1.5 (1), the isomorphism θ : LM (A) → BB (Y ) given by (8.2), is a completely contractive homomorphism into CBB (Y ). Also, the map ρ : CBB (Y ) → LM (A) in the proof of 8.1.16, is clearly completely contractive (for example, because CB(Y ) is a matrix normed algebra, as we observed in 2.3.9). Thus LM (A) ∼ = CBB (Y ) completely isometrically isomorphically. Variants of several of the facts in 2.6.6 hold for C ∗ -modules. For example, there is a canonical isometric isomorphism from Mn (CBB (Y )) onto BB (Cn (Y )). Indeed Mn (CBB (Y )) ∼ = Mn (LM (KB (Y ))) by the above, and

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311

Mn (LM (KB (Y ))) ∼ = LM (Mn (KB (Y ))) ∼ = LM (KB (Cn (Y ))) ∼ = BB (Cn (Y )), using 2.6.6 (3), 8.1.11 (3), and 8.1.16 (3). (7) The canonical operator space structure on the C ∗ -algebra BB (Y ) coincides with the inherited operator space structure from CBB (Y ). Indeed, by (6), we have Mn (CBB (Y )) ∼ = BB (Cn (Y )). On the other hand, by 8.1.11 (3), we have Mn (BB (Y )) ∼ = BB (Cn (Y )) ⊂ BB (Cn (Y )). By the discussion in (6) and (7) above, together with 8.1.16 (4), we have: Corollary 8.2.4 If Y is a right C ∗ -module over B, then CBB (Y ) is a unital operator algebra completely isometrically isomorphic to LM (KB (Y )). Moreover, ∆(CBB (Y )) = BB (Y ) (see 2.1.2 for this notation). 8.2.5 (Countably generated modules) We will use operator space column and row notation (see 1.2.26) to lead us through the important ‘stabilization theorems’. We say that a Banach B-module X is countably generated if there is a sequence (xn ) in X such that Span{bxn : b ∈ B, n ∈ N} is dense in X. By a (countable) right quasibasis of a right C ∗ -module Y over B, we mean a row [yk ] ∈ Rw (Y ) (see 1.2.26 for this notation), such that y=

∞

yk yk |y,

y ∈ Y,

(8.8)

k=1

the sum converging in norm. Clearly if there exists a right quasibasis, then Y is countably generated over B. Indeed, clearly if Y has a right quasibasis, then KB (Y ) has a countable approximate identity. By 8.1.15, this is equivalent to A having a countable approximate identity, if Y is an equivalence A-B-bimodule. It is also easy to see that if KB (Y ) has a countable approximate identity, then Y is countably generated over B. We shall not use this fact, but conversely, if Y is a countably generated right C ∗ -module, then KB (Y ) has a ‘strictly positive element’ (see [65, Proof of 7.13]), and hence a countable approximate identity [320, Proposition 3.10.5]; following the proof of [75] Lemmas 2.1–2.3, one sees that Y has a right quasibasis (see e.g. [46, Theorem 8.2]). We claim that if Y has a right quasibasis, then Y is unitarily isomorphic to an orthogonally complemented submodule of C(B). Here of course C(B) = CI (B) (see 8.2.3 (4)) when I = N. This is avariant of Theorem 8.1.27 (1). To prove this, ∞ note that by (8.8) we have y|y = k=1 y|yk yk |y, for all y ∈ Y . This permits us to deﬁne an isometric B-module map α : Y → C(B), by α(y) = (yk |y)k . By a simple calculation analoguous to that in 1.2.27, there is a well deﬁned contractive B-module map β : C(B) → Y given by β((bk )) = k yk bk . Clearly β ◦ α = IY . Our claim then follows from 8.1.21 (2). ∞ A left quasibasis for Y is a column (zk ) ∈ C w (Y ) with y = k=1 y zk |zk for all y ∈ Y . If Y is full, then the latter condition is equivalent to the same condition, but for y ∈ B, since in that case B = Y Y and Y = Y B. Taking adjoints, we see the latter is also equivalent to

C ∗ -modules as operator spaces.

312 ∞

zk |zk b = b,

b ∈ B.

(8.9)

k=1

By the other-handed version of an assertion made in the second last paragraph, (8.9) is also equivalent to B having a countable approximate identity. We shall avoid using this though, since we have not proved it. In the following, to avoid a notational conﬂict, we write K∞ (Y ) for the space we wrote as K(Y ) in 1.2.26. That is, K∞ (Y ) ∼ = K ⊗min Y . Corollary 8.2.6 (Brown–Kasparov stabilization) Suppose that Y is a right C ∗ -module over B. Then (using the notation above): (1) C(B) ⊕c Y ∼ = C(B) unitarily, if Y has a right quasibasis. (2) C(B) ⊕c C(Y ) ∼ = C(B) unitarily, if Y has a right quasibasis. (3) C(B) ⊕c C(Y ) ∼ = C(Y ), if Y is full, and has a left quasibasis. (4) C(B) ∼ = C(Y ), under the hypotheses of both (2) and (3). (5) If Y is an equivalence A-B-bimodule satisfying the hypotheses of both (2) and (3), then K∞ (B) ∼ = K∞ (Y ) ∼ = K∞ (A) linearly completely isometrically. Proof For (1), by the ‘claim’ proved in 8.2.5, we may write C(B) ∼ = Y ⊕c W for a submodule W of C(B). By the ‘associativity’ of the C ∗ -module sum, we may employ the ‘Eilenberg Swindle’: C(B) ∼ = ∼ = ∼ = ∼ =

C(B) ⊕c C(B) ⊕c · · · (Y ⊕c W ) ⊕c (Y ⊕c W ) ⊕c · · · Y ⊕c (W ⊕c Y ) ⊕c (W ⊕c Y ) ⊕c · · · Y ⊕c C(B).

‘Associativity’ of the sum also gives (2). For example, we have using (1), C(B) ∼ = C(C(B)) ⊕c C(Y ) ∼ = C(B) ⊕c C(Y ). = C(C(B)) ∼ = C(C(B) ⊕c Y ) ∼ For (3), suppose that (zk ) is a left quasibasis. Deﬁne a map α : B → C(Y ) by the prescription α(b) = (zj b). That (zj b) ∈ C(Y ) is easily seen from (8.9). Deﬁne β : C(Y ) → B by β((yj )) = j zj |yj . The latter sum converges by the argument in 1.2.27, and indeed this argument shows that β ≤ 1. Then α and β are contractive B-module maps which compose to the identity mapping on B, again by (8.9). Consequently, by 8.1.21, B is unitarily isomorphic to an orthogonally complemented B-submodule of C(Y ). Then (3) follows by an argument similar to that of (1) and (2). Item (4) is clear from (2) and (3). For (5) note that K∞ (Y ) ∼ = R(C(Y )) (this is easily deduced from (1.37), for example). By (4) we deduce that K∞ (Y ) ∼ = R(C(Y )) ∼ = R(C(B)) ∼ = K∞ (B). The assertion about A follows by symmetrical arguments (replacing C(B) above by R(A), and so on). 2

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8.2.7 (The Brown–Green–Rieﬀel stable isomorphism theorem) Since this is in most of the cited C ∗ -module texts, we will be quick here. Suppose that Y is an equivalence A-B-bimodule which has both a left and a right quasibasis (which occurs, as we mentioned in passing in 8.2.5, exactly when A and B both have countable approximate identities). We saw in 8.2.6 (5) that K∞ (A) ∼ = K∞ (B) linearly completely isometrically. Thus by 4.5.13, K∞ (A) ∼ = K∞ (B) ∗-isomorphically. Conversely, if K∞ (A) ∼ = K∞ (B) ∗-isomorphically, then it is easy to see that A and B are strongly Morita equivalent (see the hints in the Notes section). 8.2.8 (Representations of C ∗ -modules) Suppose that Y is an equivalence bimodule over A and B, and that we are given a nondegenerate ∗-representation π : L(Y ) → B(H) of the Morita linking algebra of Y (see 8.1.17). If p is the projection introduced above (8.3), then, using the notation and facts in 2.6.15, q=π ˆ (p) is a projection in B(H), and we may decompose B(H) as a 2 × 2 matrix operator algebra. Indeed the i-j-corner of B(H) is simply B(Hj , Hi ), where Ran(q) = H1 and Ker(q) = H2 . By 2.6.15, π is corner-preserving, and we may decompose π as [πij ]. The maps πij are complete contractions, which are complete isometries if π is faithful. Also π11 and π22 are ∗-representations of A and B on H1 and H2 respectively. In fact π11 and π22 are also nondegenerate. Indeed, if (bβ ) is ˆ (pbβ p), a cai for L(Y ), then qπ(bβ )(qζ) → qζ, for all ζ ∈ H. However qπ(bβ )q = π in the language of 2.6.12 and 2.6.15. Since pbβ p is a cai for A, π11 is nondegenerate. A similar argument applies to π22 . Note also that [π12 (Y )H2 ] = H1 , since we have H1 = [π11 (A)H1 ] = [π12 (Y )π21 (Y )H1 ] ⊂ [π12 (Y )H2 ]. A similar argument shows that [π12 (Y ) H1 ] = H2 . If c is the corner map mentioned in 8.1.19, then we have q ⊥ π(c(y)∗ )qπ(c(z))q ⊥ = q ⊥ π(c(y)∗ )π(c(z))q ⊥ = q ⊥ π(c(y)∗ c(z))q ⊥ . From this, it follows immediately that π12 (y)∗ π12 (z) = π22 (y|z),

y, z ∈ Y.

(8.10)

Similarly, π12 (xy ∗ z) = qπ(c(x)c(y)∗ c(z))q ⊥ = π12 (x)π12 (y)∗ π12 (z),

x, y, z ∈ Y.

Thus π12 is a triple morphism (see 4.4.1). Hence the range of π12 is a TRO inside B(H2 , H1 ). If π is faithful, then we have represented Y completely isometrically as a TRO in B(H2 , H1 ). In fact this must hold even if Y is only a C ∗ -module, since we saw in 8.1.14 that every C ∗ -module Y is an equivalence bimodule. 8.2.9 (Rigged C ∗ -modules) In C ∗ -module theory, the most important tensor product is the so-called interior tensor product. We will discuss this tensor product momentarily; for now we will just say that it is formed from a right C ∗ -module Y over A, and a so-called B-rigged A-module Z. By the latter term, we will mean a right C ∗ -module Z over B together with a ∗-homomorphism θ : A → BB (Z), such that θ is nondegenerate in the sense that Z, considered as a

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left Banach A-module in the canonical way (see A.6.1), is nondegenerate in the sense of A.6.1. Lemma 8.2.10 Suppose that A and B are C ∗ -algebras, and that Z is a right C ∗ -module over B, which is also a left A-module. Then Z is a B-rigged A-module if and only if Z with its canonical operator space structure as a right C ∗ -module (see (8.6)), is also a nondegenerate left operator A-module. Proof If Z is a B-rigged A-module, then by the observations above, Z is certainly a nondegenerate left B-module. By 8.2.3 (1), Z is a left operator module over BB (Z). By 3.1.12, Z is an operator A-module. If Z is a left operator A-module, then since Z is also a right operator module over B, we see by the last assertion in 4.6.7 that Z is an A-B-bimodule. Deﬁne θ : A → CBB (Y ) by θ(a)(y) = ay. Then θ is a contractive homomorphism. Indeed θ is a ∗-homomorphism into ∆(CBB (Y )) = BB (Y ), by the last assertion in 2.1.2 and 8.2.4. The rest is clear. 2 We emphasize that this lemma shows that the bimodules met with in the theory of C ∗ -modules, are operator bimodules. By virtue of the lemma, it makes sense to deﬁne the module Haagerup tensor product (see Section 3.4) of a right C ∗ -module over A and a B-rigged A-module. Theorem 8.2.11 Suppose that Y is a right C ∗ -module over A, and that Z is a B-rigged A-module. Then the module Haagerup tensor product Y ⊗ hA Z is a right C ∗ -module over B, with B-valued inner product determined by the formula y ⊗ z|y ⊗ z = z|y|y z ,

y, y ∈ Y, z, z ∈ Z.

(8.11)

Moreover, the usual operator space structure on Y ⊗hA Z coincides with the canonical operator space structure induced by the inner product as in (8.6). Proof By 8.1.24 and 8.2.2, there exist completely contractive A-module maps αt : Y → Cnt (A) and βt : Cnt (A) → Y such that βt ◦ αt → IY strongly on Y . By the functoriality of the module Haagerup tensor product (see 3.4.5), we obtain contractive B-module maps αt ⊗ IZ : Y ⊗hA Z → Cnt (A) ⊗hA Z and βt ⊗ IZ : Cnt (A) ⊗hA Z → Y ⊗hA Z. By density of the elementary tensors, the net of maps (βt ⊗ IZ ) ◦ (αt ⊗ IZ ) converges strongly to the identity map on Y ⊗hA Z. By 3.4.11, we have Cnt (A) ⊗hA Z ∼ = Cnt (Z), and the latter is a right C ∗ -module over B. Via this isomorphism, it is easily checked that the induced inner product on Cn (A) ⊗hA Z is given by the formula a ⊗ z|a ⊗ z = z|θ(a∗ a )z ,

a, a ∈ Cn (A), z, z ∈ Z.

By 8.1.26, we conclude that Y ⊗hA Z is a right C ∗ -module over B. By 8.1.26 used twice more, the inner product on Y ⊗hA Z is speciﬁed by its value on a pair of rank one tensors y ⊗ z and y ⊗ z , as follows: lim αt (y) ⊗ z|αt (y ) ⊗ z = lim z|θ(αt (y)∗ αt (y ))z = z|y, y z . t

t

This, together with 8.2.3 (5), proves the result.

2

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In the remainder of this section, we will simply write Y ⊗A Z for Y ⊗hA Z. 8.2.12 (Properties of the tensor product) One may view the last result as the assertion that the well-known interior tensor product of Y and Z of C ∗ -modules (see any of the texts cited at the start of this chapter), coincides with the module Haagerup tensor product. This is helpful in many ways, partly because the Haagerup tensor product has many useful properties. For example, one advantage of 8.2.11 is that it gives most of the important properties of the interior tensor product ‘for free’. For example: (1) (Functoriality) If u : Y → Y is a bounded right A-module map between right C ∗ -modules over A, and if v : Z → Z is a bounded A-B-bimodule map between B-rigged A-modules, then u⊗v extends to a bounded right B-module map between the interior tensor products: Y ⊗A Z → Y ⊗A Z (of norm ≤ uv). This follows immediately from the functoriality property of the Haagerup tensor product (see 3.4.5), and 8.2.2. (2) (Associativity) We have (Y ⊗A Z) ⊗B W ∼ = Y ⊗A (Z ⊗B W ) unitarily, if Y is a right C ∗ -module over A, if Z is a B-rigged A-module, and if W is a C-rigged B-module. This follows immediately from the associativity of the Haagerup tensor product (see 3.4.10). (3) (Commutation with the direct sum) We have ⊕ci (Yi ⊗A Z) ∼ = (⊕ci Yi ) ⊗A Z unitarily,

(8.12)

for right C ∗ -modules Yi over A, and a B-rigged A-module Z. Also, ⊕ci (Y ⊗A Zi ) ∼ = Y ⊗A (⊕ci Zi ) for a right C ∗ -module Y over A, and B-rigged A-modules Zi . These relations both follow from the universal property of the direct sum in 8.1.22. To see (8.12), let i and Pi be the canonical inclusion and projection maps between ⊕ci Yi and its summands. Deﬁne maps i = i ⊗ IZ and Pi = Pi ⊗ IZ between (⊕ci Yi ) ⊗A Z and Yi ⊗A Z. These are contractive by (1) above, and are easily checked to satisfy the hypotheses of 8.1.22. The second centered relation above is almost identical, however one ﬁrst should check that ⊕ci Zi is indeed a B-rigged A-module. Observe that the canonical left action of A on ⊕ci Zi is well deﬁned since by 8.1.5, θj (a)zj |θj (a)zj ≤ a2 zj |zj , a ∈ A, zj ∈ Zj . j

j

Here θj : A → BB (Zj ) is the homomorphism associated with the left A-module actions on Zj . Furthermore, it is easy to check that this action of A on ⊕ci Zi is as ‘adjointable operators’, and that the action is nondegenerate. Hence ⊕ ci Zi is a B-rigged A-module. (4) (The adjoint module) By the deﬁnition of the module Haagerup tensor product in Section 3.4, it is easy to see using 8.2.3 (2) that the ‘adjoint C ∗ module’ of Y ⊗A Z is Z ⊗A Y , completely isometrically.

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Corollary 8.2.13 Suppose that Y is a right C ∗ -module over B, and that θ is a nondegenerate ∗-representation of B on a Hilbert space H. Then: (1) Y ⊗B H c is a Hilbert space. (2) If Y is a B-rigged A-module, then Y ⊗B H c is a Hilbert A-module. If θ and the canonical map from A into B(Y ) are both one-to-one, then so is the canonical map from A into B(Y ⊗B H c ). Proof (1) This follows from 8.2.11, which shows that Y ⊗B H c is a right C ∗ module over C. That is, it is a Hilbert space with inner product ζ × η → η|ζ. (2) In this case, Y and Y ⊗B H c are left operator A-modules, by 8.2.10 and 3.4.9. Thus the ﬁrst assertion follows from (1) and 3.1.7. If a(Y ⊗B H c ) = 0 then aY = 0, by the relation a(y ⊗ ζ), y ⊗ η = ζ|ay|y η from (8.11). Here 2 a ∈ A, y, y ∈ Y , and ζ, η ∈ H. 8.2.14 (Avoiding the inner product) As we have already seen, many C ∗ -module constructions can be done, if need be, without explicit reference to the inner product. See 8.1.6, 8.2.12, and 8.4.2, for example. Here we use 8.2.13 to take this thought a little further, omitting full proofs. If Y is a right C ∗ -module over B, and if B is a nondegenerate ∗-subalgebra of B(K), say, then deﬁne a B-module map Φ : Y → B(K, Y ⊗B K c ) by Φ(y)(ζ) = y ⊗ ζ, for y ∈ Y, ζ ∈ K. It is easy to see that Φ(y)∗ Φ(z) = y|z, for y, z ∈ Y (using (8.11)). Also, Φ(Y ) is a C ∗ module over B with the inner product (y, z) → Φ(y)∗ Φ(z), and Φ is a unitary B-module map. Thus the inner product on Y is completely determined by the norm on the space Y ⊗B K c . The latter norm has reformulations avoiding use of the inner product, mentioned at the end of the Notes to Section 8.2. We use the above to give an alternative description of the C ∗ -module direct sum of 8.1.9. For speciﬁcity we discuss the direct sum of two right C ∗ -modules, Y1 and Y2 , over B. Let K be as above, and let Hi = Yi ⊗B K c . We will suppress mention of the map Φ in the last paragraph, and simply write yi ζ for yi ⊗ ζ. Then Y1 ⊕c Y2 may be identiﬁed with the B-submodule W of B(K, H1 ⊕ H2 ) consisting of the maps ζ → (y1 ζ, y2 ζ), for ζ ∈ K, y1 ∈ Y1 , y2 ∈ Y2 . Indeed, the canonical inner product on W , namely S × T → S ∗ T , takes values in B, making W into a C ∗ -module over B which is unitarily B-isomorphic to Y1 ⊕c Y2 . For the next result, we will need to extend the deﬁnition of KB (Y, Z) from 8.1.7, to allow Z to be any right operator B-module. Namely, we deﬁne K B (Y, Z) to be the closure in CB(Y, Z) of the span of the ‘rank one’ operators y → zy |y (here y, y ∈ Y, z ∈ Z). Henceforth, we shall assume that all operator modules are nondegenerate. Corollary 8.2.15 Let Y be a right C ∗ -module over B, and let W be a right C ∗ module over B (or more generally let W be a right operator B-module). Then: (1) W ⊗B Y ∼ = KB (Y, W ) completely isometrically. ess (2) If Y is an equivalence A-B-bimodule, then KB (Y, W ) = CBB (Y, W ), in the notation of the second paragraph of 3.5.2.

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Proof (1) Deﬁne ρ : W × Y → KB (Y, W ) by ρ(w, y¯)(x) = wy|x, for x, y ∈ Y , and w ∈ W . As in 8.1.19, we write y|x as y ∗ x, interpreted as a product in L(Y ). ∗ ] and [xrs ] be matrices with entries in W, Y , and Y respectively. Let [wij ], [zij Since W is a right h-module (see 3.1.3), it is not hard to see that ∗ ∗ wik zkj |xrs ≤ [wij ][zij xrs ] ≤ [wij ][zij ][xrs ], k

using also the fact that L(Y ) is an operator algebra, and hence a matrix normed algebra (see 2.3.9). Thus ρ is completely contractive in the sense of 1.5.4. By 3.4.2, ρ induces a complete contraction from W ⊗B Y to KB (Y, W ), which we will still Given T ∈ KB (Y, W ), write as ρ. Next, take a cai for KB (Y ) of the form in (8.4). ¯tk . From (1.40) and (8.6), deﬁne θt (T ) ∈ W ⊗B Y to be the element k T (xtk ) ⊗ x one may easily check that θt : KB (Y, W ) → W ⊗B Y is completely contractive. However, for w ∈ W, y¯ ∈ Y , θt (ρ(w ⊗ y¯)) =

k

wy|xtk ⊗ x ¯tk = w ⊗

y|xtk ¯ xtk −→ w ⊗ y¯.

k

By density, θt (ρ(u)) → u, for all u ∈ W ⊗B Y . This implies, by a simple modiﬁcation of the principle in 1.2.7, that ρ is a complete isometry. Now it is clear that ρ maps onto KB (Y, W ). (2) If Y is an equivalence A-B-bimodule, then A ∼ = KB (Y ) by 8.1.15. It then ess follows from the argument for (8.1), that CBB (Y, W ) ⊂ KB (Y, W ). Conversely, every ‘rank one’ operator |wy| in KB (Y, W ) is the limit of |wet y| = |wy|et , ess (Y, W ). Here (et )t is as in 8.1.23. It follows that |wy| is in which lie in CBB ess ess CBB (Y, W ), and so KB (Y, W ) ⊂ CBB (Y, W ). 2 8.2.16 (Hom–tensor relations) The tensor product identity in 8.2.15 (1) is a C ∗ -module variant of what is called a Hom–tensor relation in algebra. There are bimodule versions of this particular identity too, namely W ⊗B Y ∼ = KB (Y, W ) as bimodules, if in addition Y and W are operator bimodules. Also, there is a matching left-handed result: if Z is a right C ∗ -module over A, and W is a left operator A-module, then Z ⊗A W ∼ = A K(Z, W ). We leave these assertions to the interested reader. As a sample application of the relation in 8.2.15, we show that for a right C ∗ -module Y over B, KB (C(Y )) is ∗-isomorphic to the minimal tensor product KB (Y ) ⊗min K∞ , where K∞ = K(2 ) again. To see this, note that the adjoint C ∗ -module of C(Y ) is R(Y ). Hence by 8.2.15 used twice, and by basic properties of the Haagerup tensor product from Sections 1.5 and 3.4, we have KB (C(Y )) ∼ = C(Y ) ⊗B R(Y ) ∼ = C ⊗h (Y ⊗B Y ) ⊗h R ∼ = KB (Y ) ⊗min K∞ . Setting Y = B, we may deduce from this and 8.1.14, that B is strongly Morita equivalent to B ⊗min K∞ .

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From 8.2.15 one may deduce further Hom–tensor relations such as: KB (Y ⊗A Z, W ) ∼ = KA (Y, KB (Z, W )) completely isometrically. Here Y is a right C ∗ -module over A, Z is a B-rigged A-module, and W is a right operator module over B. To see the last centered relation, note that by 8.2.15, KB (Y ⊗A Z, W ) ∼ = W ⊗B (Y ⊗A Z). Thus KB (Y ⊗A Z, W ) ∼ = W ⊗B Z ⊗A Y ∼ = KB (Z, W ) ⊗A Y ∼ = KA (Y, KB (Z, W )), using 8.2.15 twice, and 8.2.12 (2) and (4). Indeed, it is also true that for Y, Z, W as above, we have CBB (Y ⊗A Z, W ) ∼ = CBA (Y, CBB (Z, W )) completely isometrically. The proof of this, and other such Hom-tensor relations, may be found in [47]. One may also deﬁne the well-known ‘exterior tensor product’ of right C ∗ modules in operator space terms: Theorem 8.2.17 Let Y and Z be right C ∗ -modules over A and B respectively. Then the minimal tensor product Y ⊗min Z (see 1.5.1) is a right C ∗ -module over A ⊗min B; with inner product determined by y ⊗ z|y ⊗ z = y|y ⊗ z|z ,

y, y ∈ Y, z, z ∈ Z.

We omit the proof of this, which is extremely similar to that of 8.2.11. Theorem 8.2.17 also easily implies results analoguous to those in 8.2.12. Strong Morita equivalence may be restated concisely in the language of operator modules as follows: Theorem 8.2.18 If Y is an equivalence A-B-bimodule, then Y ⊗A Y ∼ = B completely A-A-isometrically, and Y ⊗B Y ∼ = A completely B-B-isometrically. Conversely, if Y and X are respectively operator A-B- and B-A-bimodules, such that X ⊗hA Y ∼ = B completely B-B-isometrically, and also Y ⊗hB X ∼ = A completely A-A-isometrically, then A and B are strongly Morita equivalent, Y is an equivalence A-B-bimodule, and X ∼ = Y completely B-A-isometrically. Proof We will only need the ﬁrst statement later, and will only prove this one here. We refer the reader to [52] Proposition 1.3 in conjunction with [65] Theorem 6.2, for a proof of the second statement. If Y is an equivalence A-B-bimodule. then by 8.2.15 and 8.1.15, we have Y ⊗B Y ∼ = KB (Y ) ∼ = A. Tracing through these identiﬁcations, one sees that the isomorphism holds as A-A-bimodules too. Similarly, Y ⊗A Y ∼ 2 = B. 8.2.19 (Induced representations and Morita equivalence) Let Y be an equivalence A-B-bimodule. If Z is a left operator B-module, then G(Y ) = Y ⊗B Z

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is an operator A-module (see 3.4.9). If u : Z1 → Z2 is a completely contractive B-module map between left operator B-modules, then IY ⊗ u : G(Z1 ) → G(Z2 ) is a completely contractive A-module map (see 3.4.5). That is, G(—) = Y ⊗ B — is a functor from the category B OM OD to the category A OM OD (see 3.5.1). Indeed the map u → G(u) is linear and contractive on B CB(Z1 , Z2 ). It is easy, but tedious, to check that this map is actually completely contractive. Consequently, we call G a completely contractive functor. By 8.2.13, G maps the subcategory B HM OD to the category A HM OD. By 8.2.10 and 8.2.11, G also maps between the subcategories of left C ∗ -modules over B and A respectively. Conversely, if F (—) = Y ⊗A — then F is a completely contractive functor from the category A OM OD to the category B OM OD, and F maps A HM OD to B HM OD, and also maps between the subcategories of left C ∗ -modules. Composing these functors, and using the last theorem, and 3.4.10 and 3.4.6, we have for any left operator A-module W that Y ⊗B (Y ⊗A W ) ∼ = (Y ⊗B Y ) ⊗A W ∼ = A ⊗A W ∼ = W. Thus G ◦ F = I, and similarly F ◦ G = I. Note that from this, and from the fact that G and F are completely contractive, it is easy to see from 1.2.7 that the map u → G(u) above is a complete isometry on B CB(Z1 , Z2 ). Thus if A and B are strongly Morita equivalent, then A OM OD and B OM OD are equivalent as categories (as are also A HM OD and B HM OD, and as also are the categories of left C ∗ -modules over A and B). Of course, a similar argument gives analoguous results for the categories of right modules. The above proves the part we will need later of the following result. Proof of the other parts may be found in [51]. We remark that this theorem is a C ∗ -algebraic version of Morita’s fundamental theorem from pure algebra (e.g. see [8, 368]). Theorem 8.2.20 Two C ∗ -algebras A and B are strongly Morita equivalent if and only if the categories A OM OD and B OM OD are equivalent (via completely contractive functors). Moreover, if F : A OM OD → B OM OD is the equivalence functor, then X = F (A) is a strong Morita equivalence B-A-bimodule. 8.2.21 (The nonselfadjoint algebra case) In the light of 8.2.18, if A and B are approximately unital nonselfadjoint operator algebras, then it is natural to deﬁne A and B to be strongly Morita equivalent if there exists an operator A-B-bimodule Y and an operator B-A-bimodule X, such that X ⊗hA Y ∼ = B completely B-B-isometrically, and Y ⊗hB X ∼ = A completely A-A-isometrically. From this deﬁnition one can then proceed to develop a theory parallel to the C ∗ -algebra case. For example, 8.2.20 generalizes: A and B are strongly Morita equivalent operator algebras if and only if the categories A OM OD and B OM OD are equivalent via completely contractive functors. Similarly, there is a generalization of C ∗ -modules to the nonselfadjoint situation. See the Notes section for references to the literature.

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8.2.22 (Induced representations) Let Y be an equivalence A-B-bimodule. Thinking of modules W in B OM OD as nondegenerate representations of B on W , we saw in 8.2.19 that such representations of B induce representations of A (on Y ⊗B W ), and vice versa. Since L(Y ) is strongly Morita equivalent to B via the equivalence bimodule Y ⊕c B (see 8.1.18), we see that there are one-to-one correspondences between the ‘representations’ of B, the ‘representations’ of A, and the ‘representations’ of L(Y ). In particular, by A.5.8 and the discussion in 8.2.19 about HM OD, there are one-to-one correspondences between the nondegenerate ∗-representations of B on Hilbert space, and those of A, or of L(Y ). In fact we have seen part of this already in 8.2.8, although in a disguised form. There we took a nondegenerate ∗-representation π of L(Y ) on a Hilbert space H, and we saw that there were corresponding ∗-representations of A on a Hilbert space H1 , and B on H2 . At ﬁrst sight the spaces H1 and H2 deﬁned in 8.2.8 look diﬀerent from the ones obtained via the correspondences above. We will now show that they are the same, up to a unitary. First we claim that H1 ∼ = Y ⊗B H2 , and H2 ∼ = Y ⊗ A H1 ,

(8.13)

unitarily. To see this, we will use the notation in 8.2.8, and we deﬁne a module map f : Y ⊗B H2 → H1 by f (y ⊗ ζ) = π12 (y)(ζ). Using (8.10), we have f (y ⊗ ζ), f (y ⊗ η) = π12 (y )∗ π12 (y)ζ, η = π22 (y |yζ, η, for y, y ∈ Y, ζ, η ∈ H. By (8.11), the last number equals y ⊗ ζ, y ⊗ η. It follows from this that f (ξ), f (ξ) = ξ|ξ, for any ‘ﬁnite rank’ tensor ξ ∈ Y ⊗ B H2 . Thus f is a well deﬁned isometry. Also, we checked in 8.2.8 that f has dense range. Hence f is surjective. This yields the ﬁrst relation in (8.13). The second is similar. By (8.12), (8.13), and 3.4.6, we have (Y ⊕c B) ⊗B H2 ∼ = H1 ⊕ c H 2 ∼ = (Y ⊗B H2 ) ⊕c (B ⊗B H2 ) ∼ = H.

(8.14)

This shows that the L(Y )-module induced from the B-module H2 via the equivalence bimodule Y ⊕c B, by the procedure of 8.2.19, is unitarily isomorphic to H. Thus, conversely, the B-module induced from the L(Y )-module H by the procedure of 8.2.19, is unitarily isomorphic to H2 . The argument in and below (8.14) shows that any Hilbert B-module K induces a nondegenerate ∗-representation π of L(Y ) on (Y ⊗B K) ⊕ K. It is easy to check that π is corner-preserving, in the sense of 2.6.15. Its ‘four corners’ consist of a representation of A on B(Y ⊗B K), maps from Y to B(K, Y ⊗B K) and from Y to B(Y ⊗B K, K), and the given representation of B on K. Also, by 8.2.13 (2), π is faithful if the given representation of B on K was faithful. 8.2.23 (Inducing universal representations) Suppose that Y is an equivalence A-B-bimodule, and that K is a B-universal Hilbert module (see 3.2.7), or equivalently, a generator for B HM OD (see 3.2.8). By 8.2.19, Y induces an equivalence

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of categories A HM OD ∼ = B HM OD. By the simplest category theory, the induced representation of A on H = Y ⊗B K is A-universal. One corollary of this: we see from 3.2.12 and the double commutant theorem, that the second commutant of A in B(H), is completely isometrically isomorphic to A∗∗ . This construction is pleasantly functorial, and is therefore quite useful (for example, it is key to the proof of the diﬃcult implication in Theorem 8.2.20). 8.2.24 (Rieﬀel subequivalence) One of the pleasant consequences of purely algebraic Morita equivalence, is that if two rings A and B are Morita equivalent via an A-B-bimodule Y , then there are one-to-one lattice isomorphisms between the following lattices: (1) the two-sided ideals I of A, (2) the two-sided ideals J of B, (3) the A-B-submodules X of Y , and (4) the two-sided ideals D of L(Y ). Rieﬀel showed that similar correspondences hold in the C ∗ -algebra setting, with the word ‘norm-closed’ added. In fact this is quite easy to see: First, we replace the inner products and module actions with concrete multiplication in the linking algebra, as in 8.1.19. Deﬁne a map from lattice (2) to lattice (3) by J → X J = Y J. Conversely, deﬁne a map from lattice (3) to lattice (2) by X → J (X) = Y X, for X in lattice (3). Using A.6.2, we have X = AX = Y Y X. Thus XJ (X) = X. Given J in lattice (2), a similar argument shows that J (XJ ) = J. Thus indeed the lattices (2) and (3) are lattice isomorphic. Similarly for (1) and (3). Note that if X and J are in correspondence as above, then by A.6.2, J = J J = (Y X) Y X = X Y Y X = X X. Since J = X X, X is a full right C ∗ -module over J. By symmetry, if I is the ideal in lattice (1) associated with X, then X is a a full left C ∗ -module over I. Thus X is a equivalence I-J-bimodule, and I and J are strongly Morita equivalent. The linking C ∗ -algebra L(X) of X may be taken to be the ∗-subalgebra D of L(Y ) with corners I, X, X , and J, by the unicity of the C ∗ -norm on a ∗-algebra (see A.5.8). It is easy to check that D is an ideal of L(Y ). Now we shall check that the lattices (2) and (4) are isomorphic, via the correspondence J → D above. By 8.1.18, we may use the lattice isomorphisms we have already veriﬁed, but with Y and A replaced by Y ⊕c B and L(Y ). Thus an ideal J of B corresponds to a L(Y )-B-submodule W of Y ⊕c B, namely the submodule W = (Y ⊕c B)J = X ⊕c J. Also, we obtain a corresponding ideal I = W W of L(Y ). However it is easily checked from facts in the previous paragraph that W W = D. 8.2.25 (Rieﬀel quotient equivalence) Let X be a closed A-B-submodule of an equivalence A-B-bimodule Y . Via the correspondences in 8.2.24, let I be the corresponding ideal in A, and let J be the corresponding ideal in B. Let D be the subset of L(Y ) with four corners I, X, X and J. We saw above that X is an equivalence I-J-bimodule, and D ∼ = L(X) ∗-isomorphically. Moreover, D is a closed ideal in L(Y ). We consider the quotient map π : L(Y ) → L(Y )/D. By 2.6.15, the canonical four corners of L(Y ) induce corners of L(Y )/D, and π is corner-preserving. Write π = [πij ], as in 2.6.15. It is straightforward to check

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that the 1-2-corner W of L(Y )/D is a TRO, and also is an equivalence bimodule over π11 (A) and π22 (B). For example, to see that W W = π22 (B), observe that π22 (z|y) = π21 (¯ z )π12 (y) ∈ W W,

y, z ∈ Y.

It follows that π22 (B) ⊂ W W , and the converse inclusion is clear. Next, note that the ‘1-2-corner’ π12 of π is a complete contraction from Y to W , with kernel X. Since there is a completely contractive projection from L(Y ) onto its 1-2-corner c(Y ), it follows easily that π12 is a complete quotient map. Indeed, to see that it is a quotient map, suppose that we are given an element w ∈ W of norm < 1. Then since π is a quotient map, there exists an element w of L(Y ) of norm < 1 which π maps to w. The 1-2-corner of w is in Y , has norm < 1, and is mapped by π12 to w. Thus W is completely isometrically isomorphic to Y /X, the latter in the sense of 1.2.14. Similarly the 1-1-corner of L(Y )/D is ∗-isomorphic to A/I, and the 2-2-corner of L(Y )/D is ∗-isomorphic to B/J. Thus we see that A/I and B/J are strongly Morita equivalent. The equivalence (A/I)-(B/J)-bimodule may be taken to be Y /X (or equivalently, W ). One easily sees that with respect to these identiﬁcations, the B/J-valued inner product on Y /X, for example, is simply: y + X|y + X = y|y + J,

y, y ∈ Y.

Thus the quotient Y /X has (a) a canonical C ∗ -module structure (discussed above), and (b) a quotient operator space structure (from 1.2.14); and very fortunately these two structures are compatible. That is, the canonical operator space structure on the C ∗ -module Y /X equals its quotient operator space structure. 8.3 TRIPLES, AND THE NONCOMMUTATIVE SHILOV BOUNDARY 8.3.1 (Triple systems) In the literature, the word ternary is often used in place of our usage of the word triple, to avoid confusion with the use of that word in the JB ∗ -triple literature. Since we will not discuss JB ∗ -triples, we will use the word ‘triple’ consistently, with apologies to those for whom it has diﬀerent connotations. In 4.4.1 we discussed the ‘triple product’ [x, y, z] = xy ∗ z on a TRO. It is convenient for us to deﬁne a triple system to be an operator space Y possessing a map [·, ·, ·] : Y × Y × Y → Y (again called a triple product), such that Y is complete isometric to a TRO Z via a linear map θ : Y → Z which is a triple morphism, that is, θ([x, y, z]) = [θ(x), θ(y), θ(z)] for all x, y, z ∈ Y . In fact by 4.4.6, it is clear that an operator space Y can have at most one such triple product, and thus this triple product is uniquely determined by the norms on Mn (Y ), for n ≥ 1. We will therefore write xy ∗ z for this unique triple product on Y , without there being too much danger of confusion. We are not aware of a simple formula for the triple product in terms of the matrix norms; however we mention in passing that there is a remarkable intrinsic characterization of triple systems in terms of these norms due to Neal and Russo [289].

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In any case, by the above, we may simply deﬁne a triple system to be an operator space which is linearly completely isometric to a TRO. By a triple epimorphism we mean a surjective triple morphism, whereas a triple isomorphism is a one-to-one surjective triple morphism. By a subtriple of a triple system, we mean a closed subspace which is closed under the triple product. Thus TROs may be deﬁned to be the subtriples of B(K, H), or of a C ∗ -algebra. Any C ∗ module Y , with its canonical operator space structure, is a triple system. If Y is an equivalence bimodule, then this triple product is just xy|z = [x|y]z; which we simply write xy ∗ z (see also 8.1.19). From the perspective of this section, TROs, C ∗ -modules, equivalence bimodules, and triple systems, are essentially the same thing. That is, we may use these words interchangeably in the statements of most results below. Thus although we often restrict our attention to TROs, such results will immediately imply corresponding results for triple systems or C ∗ -modules. It is clear that if Y is a triple system, then so is Mn (Y ). In particular, if Y = B(K, H), then the triple product on Mn (Y ) simply corresponds to the obvious triple product on B(K (n) , H (n) ). Lemma 8.3.2 (Harris–Kaup) Let θ : Y → W be a triple morphism between TROs. Then: (1) θ is contractive, and indeed completely contractive. (2) θ is completely isometric if and only if it is one-to-one. Proof (1) The ampliﬁcation θn : Mn (Y ) → Mn (W ) is also a triple morphism between TROs (see 8.3.1). Thus it is enough to prove the ﬁrst statement, or equivalently that θ(y)∗ θ(y) ≤ y ∗ y for any y ∈ Y , or equivalently the following containment of spectra: σ(θ(y)∗ θ(y)) ⊂ σ(y ∗ y) ∪ {0}. The latter follows immediately from the following Claim: A nonzero scalar λ is in σ(y ∗ y) if and only if there does not exist a z ∈ Y such that (λI − yy ∗ )z(λI − y ∗ y) = y.

(8.15)

To prove this claim, note that if λ ∈ / σ(y ∗ y) then z = y(λI − y ∗ y)−2 satisﬁes (8.15). However z ∈ Y , since (λI − y ∗ y)−2 is, by spectral theory, a limit of polynomials in y ∗ y. Conversely, suppose that λ ∈ σ(y ∗ y). By basic spectral theory, we can ﬁnd a sequence bn ∈ Y Y with (λI − y ∗ y)bn → 0, but bn itself does not converge to 0. Hence y ∗ ybn , and therefore also ybn , cannot converge to 0. From this, it is easy to see that there can exist no z satisfying (8.15). (2) Again, it suﬃces to prove that if θ in (1) is one-to-one, then it is isometric. This will follow if σ(θ(y)∗ θ(y)) ∪ {0} = σ(y ∗ y) ∪ {0}, in the notation of (1). However if λ ∈ σ(y ∗ y) \ σ(θ(y)∗ θ(y)), with λ = 0, then by Urysohn’s lemma there exists a continuous nonnegative function f on σ(y ∗ y), which is zero on σ(θ(y)∗ θ(y)) and at 0, but is nonzero at λ. Clearly f (y ∗ y) = 0. Since f may be approximated uniformly by polynomials pn with no constant term, we have that f (y ∗ y) ∈ Y Y . For any z ∈ Y and any polynomial p, θ(zp(y ∗ y)) = θ(z)p(θ(y)∗ θ(y)). Replacing

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p by pn and letting n → ∞, we have θ(zf (y ∗ y)) = θ(z)f (θ(y)∗ θ(y)) = 0. Thus zf (y ∗ y) = 0. Since this is true for all z ∈ Y , by 8.1.4 (1) we have f (y ∗ y) = 0. This is a contradiction. 2 We can rephrase part of 8.2.25 in the language of triple systems as follows: Proposition 8.3.3 Suppose that Y is an equivalence A-B-bimodule, and that X is a closed A-B-submodule of Y . Then the quotient operator space Y /X is a triple system. Indeed if q : Y → X is the canonical quotient map, then the triple product on Y /X is given by [q(x), q(y), q(w)] = q(xy ∗ w), for x, y, z ∈ Y . Thus q is a triple epimorphism. If X is a closed linear subspace of a triple system Y , such that xy ∗ z and zy ∗ x are in X for all x ∈ X and y, z ∈ Y , then we say that X is a triple ideal of Y . In this case, the quotient Y /X is a triple system, by 8.3.3. Corollary 8.3.4 Let θ : Y → Z be a triple morphism between TROs. Then: (1) Ker(θ) is a triple ideal in Y . (2) Ran(θ) is closed, and is a subtriple of Z. (3) θ is a complete quotient map onto its range. (4) The induced map θ˜ : Y /Ker(θ) → Z is a completely isometric triple morphism onto Ran(θ). Proof Item (1) is obvious. The induced map θ˜: Y /Ker(θ) → Z is, by 8.3.3, a one-to-one triple morphism onto Ran(θ). By Lemma 8.3.2, θ˜ is completely isometric, which gives (2), (3), and (4). 2 Corollary 8.3.5 (Hamana) A linear map θ : Y → W between full C ∗ -modules or TROs, is a triple morphism if and only if θ is the 1-2-corner of a cornerpreserving ∗-homomorphism π : L(Y ) → L(W ) between the Morita linking C ∗ algebras. In this case, θ is one-to-one (resp. surjective) if and only if π is oneto-one (resp. surjective). Proof The ﬁrst ‘if’ is clear. For the converse, we maysuppose that θ is a n n triple morphism between TROs. Deﬁne ρ( k=1 xk yk∗ ) = k=1 θ(xk )θ(yk )∗ , for x1 , . . . , xn , y1 , . . . , yn ∈ Y . Now θ(Y ) is a TRO in W , and so by 8.1.4 (1) used twice, and by 8.3.4 (3), we have

θ(xk )θ(yk )∗ = sup θ(xk )θ(yk )∗ w : w ∈ Ball(θ(Y )) k

k

= sup θ(xk )θ(yk )∗ θ(z) : z ∈ Ball(Y ) k

xk yk∗ z : z ∈ Ball(Y ) = xk yk∗ . ≤ sup k

k

Hence ρ is well-deﬁned, and extends to a contraction, which we still call ρ, from Y Y to W W . It is easy to see that ρ is a ∗-homomorphism. Similarly one gets a

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canonical ∗-homomorphism σ : Y Y → W W . It is now clear how to deﬁne the corner-preserving ∗-homomorphism π : L(Y ) → L(W ), whose 1-2-corner is θ. It is clear that if π is one-to-one (resp. surjective) then so is θ. If θ is one-toone, then it is isometric. In this case, the one inequality in the centered equations above is also an equality. It follows that ρ is isometric, hence one-to-one. Similarly, σ is one-to-one. Since π is corner-preserving, it is now easy to see that π is also one-to-one. If θ is surjective, then ρ is surjective, since it has dense range 2 in W W . Similarly σ, and hence also π, is surjective. 8.3.6 (Inner products on triple systems) By 4.4.6, a completely isometric linear isomorphism between two equivalence bimodules is a triple morphism. Thus, it is the 1-2-corner of a ∗-isomorphism π between the Morita linking C ∗ -algebras, as in 8.3.5. The 1-1- and 2-2-corners of π are ∗-isomorphisms between the algebras acting on the given bimodules. If Y is a triple system, then Y has a canonical equivalence bimodule structure. Namely, let A be the subspace of B(Y ) densely spanned by the maps z → [x, y, z], and let B be the subspace densely spanned by the maps z → [z, y, x], for x, y ∈ Y . If Y is a TRO, then it is clear that A is just the copy of Y Y in B(Y ). That is, A = KY Y (Y ); and by 8.1.14, Y is a left C ∗ -module over A. Hence it follows that if Y is a triple system, then A is a C ∗ -algebra in the product of B(Y ), and Y is a left C ∗ -module over A in a canonical fashion. Similarly, B is a C ∗ -algebra with the reversed product of B(Y ), and Y is a right C ∗ -module over B. Clearly, Y is an equivalence A-B-bimodule. We will see in 8.4.2 that A is a ∗-subalgebra of Al (Y ) in the notation of 4.5.7, and B ⊂ Ar (Y ). Next we characterize the possible right C ∗ -module actions on a triple system which are compatible with the underlying operator space structure (via equation (8.6)). First we note that if Y is a full right C ∗ -module over a C ∗ -algebra C, and if σ is a faithful ∗-homomorphism from C onto an ideal of a C ∗ -algebra D, then we may make Y into a right C ∗ -module over D as follows. We deﬁne yd = yσ −1 (d) and y|z = σ(y|z), for d ∈ σ(C), and y, z ∈ Y . It is easy to check that Y is a right C ∗ -module over σ(C), and hence by 8.1.4 (4), over D. We denote this C ∗ -module as Yσ . Moreover, the operator space structure on Yσ given by (8.6), is easily seen to coincide with the former one. We claim that every right C ∗ -module action on Y over any C ∗ -algebra D, which is compatible with the given operator space structure, arises in precisely this way. To see this, write Y for Y viewed as a C ∗ -module over D. Thus the identity map I from Y to Y is a complete isometry, and a triple isomorphism. Hence by the ﬁrst paragraph of 8.3.6, there is associated a ∗-isomorphism σ from C onto the ideal I of D. It is easy algebra to check that Y = Yσ (using the fact that σ and I are corners of a ∗-isomorphism between the Morita linking C ∗ -algebras (see 8.3.5)). By the last two paragraphs, it follows that if a triple system Y is a right C ∗ module over a C ∗ -algebra D, such that the norms from (8.6) equal the given matrix norms on Y , then this C ∗ -module equals Yσ for a faithful ∗-homomorphism σ from B onto an ideal of D. Here B is as in the second last paragraph. This gives another ‘picture’ of the C ∗ -modules over a C ∗ -algebra D, as the triple systems

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Y , together with a faithful ∗-homomorphism σ : B → D as above. We may now extend the notation introduced in 8.1.19 to triple systems. Thus products such as y1 y2∗ , y1 y2∗ y3 y4∗ y5 , and so on, make sense for elements (yi ) in a triple system. Indeed y1 y2∗ represents the operator z → [y1 , y2 , z] in the C ∗ algebra A above, whereas y1 y2∗ y3 y4∗ y5 = [[y1 , y2 , y3 ], y4 , y5 ] ∈ Y . As in 8.1.19, any such expressions may be interpreted as products in the Morita linking C ∗ -algebra of Y , where Y is regarded as an equivalence A-B-bimodule as above. 8.3.7 (Nondegenerate triple morphisms) If θ : Y → B(K, H) is a triple morphism deﬁned on a TRO Y , then we say that θ is nondegenerate if (i) θ(Y )K is dense in H, and (ii) θ(Y )∗ H is dense in K. Using the fact that Y is the norm closure of Y Y Y , it is easy to see that (i) is equivalent to saying that the corresponding ∗-homomorphism from Y Y to B(H) given by 8.3.5, is nondegenerate. Similarly, (ii) is equivalent to saying that the corresponding ∗-homomorphism from Y Y to B(K) is nondegenerate. Also, (ii) is equivalent to saying that ∩y∈Y Ker(θ(y)) = {0}. Thus (using 8.2.8 if necessary), we see that θ is nondegenerate if and only if the corresponding ∗-homomorphism from L(Y ) to B(H ⊕ K) from 8.3.5 is nondegenerate. Note that this implies that if, further, θ is one-to-one, then we may view M (L(Y )) ⊂ B(H ⊕K), as in Section 2.6. If θ is not nondegenerate then we may ‘cut it down’ to be nondegenerate, by replacing H with H = [θ(Y )K], and K with K = [θ(Y )∗ H]. 8.3.8 (The noncommutative Shilov boundary) We next discuss the noncommutative Shilov boundary, or ‘triple envelope’, of a nonunital operator space. To a certain extent this will parallel the development in Section 4.3, to which the reader may want to refer back to periodically. Suppose that X is an operator space. If i : X → Y is a linear complete isometry into a triple system Y , such that Y is the smallest subtriple of Y containing i(X), then we say that (Y, i) is a triple extension of X. Notice that in this case, by the argument for (4.7), Y = Span{i(x1 )i(x2 )∗ i(x3 ) · · · i(x2n )∗ i(x2n+1 ) : n ∈ N, x1 , . . . , xn ∈ X} (notation as in 8.3.6). If Y is a full C ∗ -module or equivalence bimodule, then it is easy to see that this is the same as saying that the copy of i(X) supported in the 1-2-corner of the Morita linking algebra L(Y ), generates L(Y ) as a C ∗ -algebra. We say that two triple extensions (Y, i) and (Y , i ) are X-equivalent if there exists a triple isomorphism θ : Y → Y such that θ ◦ i = i . We deﬁne a triple envelope or noncommutative Shilov boundary of X to be any triple extension (Y, i) with the universal property of the next theorem. Theorem 8.3.9 (Hamana) If X is an operator space, then there exists a triple extension (Y, i) of X with the following universal property: Given any triple extension (Z, j) of X there exists a (necessarily unique and surjective) triple morphism θ : Z → Y , such that θ ◦ j = i.

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8.3.10 (Remarks on the universal property) Before we begin the proof, we make a series of important but simple remarks stemming from the universal property of the theorem. These remarks parallel the discussion in 4.3.2, and provide part of the justiﬁcation for the use of the term ‘noncommutative Shilov boundary’. We omit the proofs, which are almost identical to 4.3.2. First, suppose that (Y, i) is a triple extension with the universal property of the theorem. Then there exists no triple ideal W of Y such that qW ◦ i is a completely isometry on X, where qW : Y → Y /W is the canonical quotient triple morphism. This follows by applying the universal property with j = qW ◦ i. The second remark is that the set of triple extensions (Y, i) satisfying the universal property of the theorem, is one entire equivalence class of the relation we called X-equivalence deﬁned above 8.3.9. Third, if (Z, j) is any triple extension of X, and if θ : Z → Y is the triple epimorphism provided by the universal property, and if W = Ker(θ), then (Z/W, qW ◦ j) is clearly X-equivalent to (Y, i). Thus by the second remark, (Z/W, qW ◦ j) may be taken to be a triple envelope of X. Here again, qW : Z → Z/W is the quotient morphism. Putting these remarks together we obtain our fourth remark, namely that the triple envelope of X may be taken to be any triple extension (Y, j) of X for which there exists no closed triple ideal W of Y such that qW ◦ j is completely isometric on X. 8.3.11 (Proof of Theorem 8.3.9) For an operator space X we deﬁne T (X) and J as in 4.4.7. Note that T (X) is simply the subtriple of I(X) generated by J(X). Suppose that (Z, j) is a triple extension of X. We may suppose, by 8.3.6, that Z is an equivalence C-D-bimodule. Let C 1 and D1 be the unitizations of C and D respectively, and let L1 (Z) be the ‘unitized linking C ∗ -algebra’ of 8.1.17: 1 C Z 1 L (Z) = . Z D1 Inside L1 (Z) there is a canonical copy S 1 of the Paulsen system S(X). Moreover, it is easy to see that S 1 generates L1 (Z) as a C ∗ -algebra. Similarly, consider the C ∗ -algebra Ce∗ (S(X)) generated by S(X) in I(S(X)), using the notation of 4.4.2. This is a C ∗ -envelope of S(X) (see the proof of 4.3.1). By 4.3.1, there exists a ∗-homomorphism π : L1 (Z) → Ce∗ (S(X)), such that π extends the canonical map from S 1 to S(X). By 2.6.15, we see that π takes each of the four corners in L 1 (Z) to the matching corner in Ce∗ (S(X)). Let θ be the 1-2-corner of π. As in 8.3.5, θ is a triple morphism, and it is surjective since π is surjective. We will henceforth write (T (X), J) for any triple envelope of X. 8.3.12 (Properties of the triple envelope) (1) As one would expect, if Z is a triple system, then Z is a triple envelope of itself. Indeed, applying the universal property of the theorem to the identity map j : Z → Z, we obtain a triple epimorphism θ : Z → T (Z) with θ ◦ j = i. Thus θ = i, and so θ is a triple isomorphism. (2) If u : X1 → X2 is a surjective linear completely isometry between operator spaces, then one may ‘extend’ u to a triple isomorphism between any triple

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envelopes (T (X1 ), J1 ) and (T (X2 ), J2 ). Indeed a routine ‘diagram chase’ shows that (T (X2 ), J2 ◦ u) is a triple envelope for X1 . (3) The triple envelope shares many of the properties of the injective envelope that we met in Chapter 4. For example, any triple envelope of X is both a rigid and an essential extension of X, in the sense of 4.2.3. The proof is the same as that of 4.3.6. (4) Another useful fact is that T (Mmn (X)) ∼ = Mmn (T (X)) completely isometrically isomorphically (or equivalently, by 4.4.6 and 8.3.2, triple isomorphically) for any operator space X, and for m, n ∈ N. More generally it is true that T (KI,J (X)) ∼ = KI,J (T (X)) for arbitrary cardinals I, J. One may deduce such relations from the analoguous assertion for the injective envelope (see 4.2.10 or 4.6.12), and the deﬁnition of the triple envelope given in the proof of 8.3.9. ∼ ∞ Similarly, although we shall not need this, T (⊕∞ i Xi ) = ⊕i T (Xi ) triple isomorphically. Another way to prove such relations is to use the fourth remark in 8.3.10 (see Appendix A in [53] for details). (5) If A is a unital operator space, or an approximately unital operator algebra, then one can show that any C ∗ -envelope of A is a triple envelope of A. Most of this was observed at the end of 4.4.7. For the rest, see [53]. As a sample application, we give another proof of an earlier result (see 4.5.13): Corollary 8.3.13 (A Banach–Stone theorem) If A and B are unital operator algebras, and if v : A → B is a linear surjective complete isometry, then there exists a unital completely isometric isomorphism π from A onto B, and a unitary U ∈ ∆(B), such that v = U π(·). Proof By (5) above, we may take T (A) = Ce∗ (A), and similarly for B. By 8.3.12 (2), one may ‘extend’ v to a triple isomorphism from Ce∗ (A) to Ce∗ (B), which we still write as v. If v(1) = U and v(a) = 1, then U ∗ U = v(1)∗ v(1) = v(a)v(1)∗ v(1) = v(a) = 1. Similarly U U ∗ = 1, so that U is a unitary in Ce∗ (B). Then π = U −1 v(·) satisﬁes the hypotheses of 1.3.10, and hence it is a ∗-isomorphism. Thus U −1 v(a2 ) = π(a2 ) = π(a)2 = U −1 v(a)U −1 v(a) = U −2 . Hence U ∗ = U −1 = v(a2 ) ∈ B, and so U ∈ ∆(B). Thus the restriction of π to A is a unital completely isometric homomorphism onto B. 2 8.3.14 (The Shilov inner product) It is often convenient to take the triple envelope T (X) of X to be an equivalence bimodule. In this case we call the restriction to X of the associated C ∗ -module inner products on T (X), the Shilov inner products on X. We have already met this concept in 4.4.8. 8.4 C ∗ -MODULE MAPS AND OPERATOR SPACE MULTIPLIERS In the previous section, we saw that the noncommutative Shilov boundary of an operator space may be viewed as a C ∗ -module. This is pleasant, since then we

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may hope to apply C ∗ -module methods directly to the study of operator spaces. For such applications, the multiplier algebras of an operator space considered in Chapter 4, are often useful, since they are intimately connected with C ∗ -module theory, in several ways. The reader might turn to Section 4.5 for the deﬁnitions of Ml (X) and Al (X). We now describe these algebras in C ∗ -module terms. 8.4.1 (Multipliers and the triple envelope) For any operator space X, let (T (X), J) be a triple envelope of X, which we may take to be a right C ∗ -module over a C ∗ -algebra F. The space of bounded right module maps on T (X) is an operator space, as we mentioned after 8.2.2. We temporarily write LM (X) for the subspace consisting of those module maps leaving J(X) invariant. We will now show that Ml (X) ∼ = LM (X) as operator algebras. That is, Ml (X) ∼ = {a ∈ CBF (T (X)) : aJ(X) ⊂ J(X)}

(8.16)

completely isometrically isomorphically. If a ∈ LM (X), then since (iv) implies (i) in 4.5.2, a|X is a left multiplier of X. (A direct proof of this may also be given, using the last paragraph in 8.1.19.) Indeed by 4.5.2, the map a → a|X gives the isometric isomorphism in (8.16). Now Mn (CBF (T (X)) ∼ = BF (Cn (T (X))), by 8.2.3 (6). Thus if a = [aij ] ∈ Mn (CBF (T (X))), then a may be regarded as a module map on Cn (T (X)). By 8.3.12 (4), Cn (T (X)) is a triple envelope of Cn (X). We deduce from the isometric case of (8.16), that Ml (Cn (X)) ∼ = {a ∈ CBF (Cn (T (X))) : aJn,1 (X) ⊂ Jn,1 (X)} ∼ = Mn (LM (X)) isometrically. By (4.11), we now have (8.16) completely isometrically. We next show that Al (X) ∼ = {a ∈ BF (T (X)) : aJ(X) ⊂ J(X), a∗ J(X) ⊂ J(X)}

(8.17)

∗-isomorphically. This follows from (8.16), and the fact that each side of (8.17) is just the diagonal algebra of the matching sides of (8.16) (see 4.5.7 and 8.2.4). Indeed, the diagonal algebra may be deﬁned to be the span of the Hermitian elements. The Hermitian (in this case, selfadjoint) elements of the algebra LM (X) above, clearly are in the set on the right side of (8.17). Conversely, if a is in the ∗ ∗ + i a−a latter set, then writing a = a+a 2 2i , we see that a ∈ ∆(LM (X)). By 8.2.4 and the last paragraph of 8.1.19, we also may regard (as subalgebras) Al (X) ⊂ Ml (X) ⊂ KF (T (X))∗∗ ⊂ L(T (X))∗∗ . Corollary 8.4.2 Suppose that Y is a right C ∗ -module over a C ∗ -algebra B. Then Ml (Y ) = CBB (Y ) and Al (Y ) = BB (Y ). Similar assertions hold for the right multiplier algebras, indeed Mr (Y ) ∼ = RM (I) and Ar (Y ) ∼ = M (I), where I is as in 8.1.4. Proof The ﬁrst two assertions are immediate from the proofs of (8.16) and (8.17) above, and using 8.3.12 (1). The second two also use 8.2.4 and the ‘otherhanded version’ of facts from 8.1.14–8.1.16. 2

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Corollary 8.4.3 Suppose that A is a C ∗ -algebra. Any nondegenerate left operator A-module X is a closed A-submodule, of a B-rigged A-module Z (see 8.2.9), for some C ∗ -algebra B. One may take Z = T (X). Conversely, a closed A-submodule of a B-rigged A-module Z is a left operator A-module. Proof If X is a left operator A-module, then by 4.6.2 (2) there is an associated ∗-homomorphism θ : A → Al (X). Using (8.17), θ may be viewed as a ∗-homomorphism into BF (T (X)). By (4.7) we have θ(et )(z) → z, for z ∈ Z. That is, T (X) is an F-rigged A-module. Clearly X is an A-submodule of T (X). Conversely, by 8.2.10 any B-rigged A-module Z is a nondegenerate left operator A-module; and therefore so is any A-submodule. 2 The following adds to the picture of adjointable multipliers on operator spaces that we began in 4.5.8: Theorem 8.4.4 Let X be an operator space. For a map T : X → X, the following are equivalent: (i) T ∈ Al (X). (ii) There exists a linear complete isometry σ from X into a C ∗ -algebra, and a map R : X → X, such that σ(T (x))∗ σ(y) = σ(x)∗ σ(R(y)), for x, y ∈ X. (iii) There exists a map R : X → X, such that T (x) | y = x | R(y) ,

x, y ∈ X.

The inner product in (iii) is a Shilov inner product for X (see 8.3.14). Proof That (i) implies (ii) follows from 4.5.8 (1), taking R = σ −1 (S ∗ σ(·)). Suppose that T satisﬁes (ii), and that (T (X), J) is a triple envelope of X, with T (X) a full C ∗ -module over a C ∗ -algebra F. The subtriple Y generated by σ(X) is a triple extension of X in the sense of 8.3.8. By 8.3.9, there exists a triple epimorphism θ : Y → T (X) such that θ ◦ σ = J. By the proof of 8.3.5 there is an associated ∗-homomorphism from Y Y to F, taking z1∗ z2 to θ(z1 )|θ(z2 ), for z1 , z2 ∈ Y . Applying this ∗-homomorphism to the equation in (ii), we obtain the equation in (iii). To see that (iii) implies (i), we deﬁne Bl (X) to be the set of maps T on X satisfying (iii). Set T ∗ = R, where R is as in (iii). It is easy to check, just as in 8.1.7, that Bl (X) is a closed subalgebra of B(X), and that ∗ is an isometric involution on Bl (X) satisfying the C ∗ -identity. Thus Bl (X) is a C ∗ -algebra. Next we note that Al (X) is the range in B(X) of the isomorphism in (8.17). Explicitly, this map takes an a in the set on the right of (8.17), to the operator Ta = J −1 (aJ(·)) on X. We have Ta ∈ Bl (X). Indeed a proof similar to the proof that (i) implies (ii) above, shows that a → Ta is a ∗-homomorphism into Bl (X). We therefore will be done if we can show that the range of this faithful ∗-homomorphism equals Bl (X). To do this, it suﬃces to show that if U is a unitary in Bl (X), then U is in the range of the map above (since any unital C ∗ -algebra is spanned by its unitary elements). For such U , we have

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331

U x|U y = U ∗ U x|y = x|y,

x, y ∈ X.

Now T (X) = XF, by the centered equation in 8.3.8. Deﬁne a map taking the n n element y = k=1 xk bk , to k=1 U (xk )bk , for x1 , . . . , xn ∈ X, b1 , . . . , bn ∈ F. To see that this map is well deﬁned and bounded, we compute: n n n , U (xk )bk | U (xk )bk = b∗i xi |xj bj = y|y. k=1

k=1

i,j=1

Hence the map above extends to a map a ∈ BF (T (X)). Clearly Ta = U .

2

∗

The C -algebra Bl (X) in the last proof is another useful description of Al (X). It also shows why the name adjointable is appropriate for these maps. 8.4.5 (Comparisons with C ∗ -module maps) There are very many quite striking parallels between multipliers (resp. adjointable multipliers) on operator spaces, and bounded module maps (resp. adjointable maps) on C ∗ -modules. We have seen some of these already. We mention a few more: for example, from 4.7.4 and 4.7.1 we know that for a dual operator space X, Al (X) is a W ∗ -algebra, and any u ∈ Al (X) is w∗ -continuous. In the next section we shall see that such results play an important role for W ∗ -modules. We shall also brieﬂy mention there some connections with the one-sided M -ideals of Section 4.8. Indeed, a good deal of the results in the noncommutative M -ideal theory follow by applying C ∗ -module techniques. Many more such parallels may be found discussed in [56, 73]. In 8.1.10 we remarked that a linear map u on a right C ∗ -module Y is a contractive module map if and only if the map u ⊕ IY on C2 (Y ) is contractive. This of course is the analogue for C ∗ -modules of condition 4.5.2 (ii), which characterizes operator space multipliers. Beginning from this fact from 8.1.10, and using basic facts about C ∗ -modules met early in this chapter, one may give another development of the theory of operator space multipliers that we saw in Chapter 4, but avoiding many of the technical details about the injective envelope used in Section 4.5. See [56] for details. Such an approach is close to the original development of the operator space multiplier theory (see [53]). For example, we give a quick proof that (ii) implies (iv) in Theorem 4.5.2. Condition (ii) there says that u ⊕ IX : C2 (X) → C2 (X) is completely contractive. By 1.2.11, I(X) is ‘completely complemented’ in B(H) for some H. Thus there is a completely (2) ∼ ) → C2 (I(X)). contractive ∞ 2 -module map projection C2 (B(H)) = B(H, H Hence, by 3.6.2, we can extend u ⊕ IX to a completely contractive ∞ 2 -module map C2 (I(X)) → C2 (I(X)) ⊂ C2 (B(H)). Such a map is necessarily of the form u & ⊕ v. By the ‘rigidity’ property of I(X), v = IX . Since I(X) is a C ∗ -module, by the result at the start of this paragraph, u & is a contractive module map on I(X). By (4.7), it restricts to a contractive module map on T (X). 8.5 W ∗ -MODULES The theory of W ∗ -modules may be thought of as a ‘dual variant’ of the theory of C ∗ -modules. Indeed our development in this section is parallel, to some ex-

W ∗ -modules

332

tent, to the pattern of Sections 8.1 and 8.2 above. However, we shall see that W ∗ -modules are quite a bit simpler than C ∗ -modules. For example, W ∗ -modules behave much more like Hilbert spaces, and there is a very powerful ‘stable isomorphism theorem’ (8.5.28 below) valid for all W ∗ -modules, which is very useful for operator space applications. Throughout this section, M and N are W ∗ -algebras. 8.5.1 (The deﬁnitions) We say that a right C ∗ -module Y over a C ∗ -algebra A is selfdual if every bounded A-module map u : Y → A is of the form u(·) = z|·, for some z ∈ Y . We say that Y is a right W ∗ -module if Y is a selfdual right C ∗ -module over a W ∗ -algebra. If Y is a right C ∗ -module over a W ∗ -algebra M , then we will consistently write I w for the w∗ -closure in M of the span of the range of the M -valued inner product on Y (recall from 8.1.4 that I is the norm closure of this span). It is easy to see, using simple w∗ -approximation arguments (e.g. see 2.7.4 (4)), that I w is a w∗ -closed ideal in M . We say that Y is w ∗ -full if I w = M . In the following remarks, Y is a selfdual right C ∗ -module over A. (1) It follows for example from 8.1.11 (2) and 8.2.15, that CBA (Y, A) ∼ =Y

completely isometrically.

(8.18)

(2) If Z is another right C ∗ -module over A, then BA (Y, Z) = BA (Y, Z), and BA (Y ) = BA (Y ).

(8.19)

Indeed, the fact that any u ∈ BA (Y, Z) is adjointable follows by considering the A-valued map z|u(·), for ﬁxed z ∈ Z. (3) The adjoint C ∗ -module Y (see 8.1.1) is a selfdual left C ∗ -module over A. We leave the details as an exercise. (4) Selfduality is an operator space invariant. That is, if Z is another C ∗ module over a possibly diﬀerent C ∗ -algebra B, say, and if Y and Z are linearly completely isometric, then Z is selfdual as a B-module. To see this, one ﬁrst uses the next result to see that we can assume that Y and Z are both full. By 4.4.6, Y and Z are triple isomorphic. The rest is a pleasant algebraic exercise. Lemma 8.5.2 Let Y be a right C ∗ -module over a C ∗ -algebra A, and let I be as in 8.1.4. Then BA (Y, A) = BC (Y, D) as sets, for any C ∈ {A, A1 , M (A), I} and D ∈ {A, A1 , M (A), I, M (I)}. Hence Y is selfdual as an A-module if and only if Y is selfdual as a D-module, for any D ∈ {A1 , M (A), I, M (I)}. Proof Suppose that u ∈ BI (Y, D). By Cohen’s theorem A.6.2, we may write any y ∈ Y as y = y1 a1 a2 for a1 , a2 ∈ I. Hence u(y) = u(y1 )a1 a2 ∈ AI ⊂ I. Moreover, if c ∈ C then u(yc) = u(y1 a1 (a2 c)) = u(y1 )a1 (a2 c) = u(y)c. Thus BI (Y, D) ⊂ BC (Y, I) ⊂ BI (Y, I) ⊂ BI (Y, D). Thus BC (Y, D) = BI (Y, D) = BI (Y, I), for any C, D as above. The last assertion follows immediately from the ﬁrst.

2

C ∗ -modules and operator spaces

333

Proposition 8.5.3 Suppose that Y is a right C ∗ -module over a W ∗ -algebra M . Then I w is the multiplier algebra of I. Proof Clearly I is a w∗ -dense ideal in I w . Fix a faithful unital w∗ -continuous ∗-representation π of I w on H, say. This is easily seen, using 2.1.9, to be a nondegenerate ∗-representation of I. By Section 2.6, we may identify M (I) with {T ∈ B(H) : T π(I) ⊂ π(I), π(I)T ⊂ π(I)}. In fact the latter algebra clearly contains π(I w ), and is a subalgebra of π(I) by 2.6.5. On the other hand, π(I) = π(I)

w∗

⊂ π(I w ),

by the double commutant theorem. Thus π(I w ) = M (I).

2

Lemma 8.5.4 Let Y be a right C ∗ -module over a W ∗ -algebra M . Then: (1) Y is a W ∗ -module over M if and only if Y has a Banach space predual with respect to which the inner product on Y is separately w ∗ -continuous. If Y is a W ∗ -module, then: (2) Y has a unique Banach space predual with respect to which the inner product on Y is separately w∗ -continuous. (3) With respect to the w ∗ -topology induced by the predual in (2), a bounded net (xt )t converges to x in Y if and only if y|xt → y|x in the w∗ -topology of M , for all y ∈ Y .

(4) Let W = M∗ ⊗M Y (see Section 3.4). Then W is an operator space predual of Y inducing the w∗ -topology in (2) and (3) above. Proof We remark that (4) and its proof, and the proof of (2), are simpler if one replaces M∗ ⊗M Y by the (quite analoguous, but much simpler) Banach module version of the projective tensor product. To understand the argument better, the reader may want to consider this variant ﬁrst. If Y is a W ∗ -module then by 3.5.10 and the other-handed version of (8.18),

(M∗ ⊗M Y )∗ ∼ =

M CB(Y

, M) ∼ = Y,

completely isometrically. Unraveling these isomorphisms yields a completely isometric surjection ρ : Y → W ∗ , given by ρ(x)(ψ ⊗ y¯) = ψ(y|x),

ψ ∈ M∗ , x, y ∈ Y.

By the norm density of the ﬁnite rank tensors from M∗ ⊗ Y in W , it is clear that with respect to the w∗ -topology on Y induced by W , the ‘if and only if’ condition in (3) holds. Thus by A.2.5, the inner product is separately w ∗ -continuous with respect to this topology. Deﬁne θ : W → Y ∗ by θ(ψ ⊗ y¯)(x) = ψ(y|x), for ψ ∈ M∗ and x, y ∈ Y . It is easy to check that θ = (ρ∗ )|W . Suppose that Y∗ is a predual of Y , regarded as a subspace of Y ∗ , for which the inner product on Y is separately w ∗ -continuous.

W ∗ -modules

334

Then θ maps into Y∗ . Viewing θ as a map into Y∗ , θ∗ corresponds to the map ρ above. Hence θ is a (completely) isometric surjection onto Y∗ , and ρ is a homeomorphism for the w∗ -topologies. We have proved (2)–(4). The ‘only if’ in (1) follows from the proof of (4). On the other hand, suppose that Y has a Banach space predual, and that the M -valued inner product is sepBy 8.1.23, arately w∗ -continuous. Let u : Y → M be a bounded M -module map. n we may choose a cai (et )t for KM (Y ), with terms of the form k=1 |xk xk | for some xk ∈ X. For x ∈ X, we have u(et (x)) =

n k=1

u(xk )xk |x =

n ,

xk u(xk )∗ x = wt |x,

(8.20)

k=1

n where wt = k=1 xk u(xk )∗ (which depends on t). It follows from 8.1.11 (2) that wt = u ◦ et ≤ u. Thus (wt )t is a bounded net in Y , and so it has a w∗ convergent subnet, with limit w say. Replace the net with the subnet. By the hypothesis, wt |x → w|x. Since u(et (x)) → u(x) in norm, by (8.20) we have u(x) = w|x, for all x ∈ Y . Thus Y is selfdual over M . 2 We will henceforth use the phrase the w ∗ -topology of a W ∗ -module Y , for the (unique) topology in (2)–(4) above. Corollary 8.5.5 Suppose that Y is a right W ∗ -module over M . Then: (1) BM (Y ) = BM (Y ), and this is a W ∗ -algebra. (2) A bounded net (Ti )i in BM (Y ) converges in the w∗ -topology to T ∈ BM (Y ) if and only if Ti (y) → T (y) in the w∗ -topology of Y , for all y ∈ Y . Indeed, Y ⊗M W is a predual for BM (Y ), where W is as in 8.5.4 (4). Proof By 3.5.10, 8.5.4 (4), and (8.19), we have

(Y ⊗M W )∗ ∼ = CBM (Y, W ∗ ) = CBM (Y ) = BM (Y ). The latter space is a C ∗ -algebra, and it is therefore a W ∗ -algebra by the theorem of Sakai mentioned at the start of Section 2.7. The assertion involving net convergence is proved similarly to the analoguous statements in 8.5.4 (4). 2 There is a stronger variant of the following, which assumes only a Banach space predual. The version here will suﬃce for most operator space applications. Our proof showcases the multipliers from Chapter 4, and has the advantage of generalizing to the nonselfadjoint algebra situation. See the Notes for details. Theorem 8.5.6 (Zettl, Eﬀros–Ozawa–Ruan) Let Y be a full right C ∗ -module over a C ∗ -algebra A, and suppose that Y has an operator space predual. If M is M (A) then M and BA (Y ) are W ∗ -algebras, and Y is a w∗ -full W ∗ -module over M . Moreover, Y has a unique operator space predual, the space in 8.5.4 (4). Proof Let Y∗ be a ﬁxed operator space predual of Y . We will use the fact from 8.4.2 that Al (Y ) = BM (Y ), and Ar (Y ) = M (A) = M . By 4.7.4 we know that

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335

Al (Y ) and Ar (Y ) are W ∗ -algebras, and hence so are BM (Y ) and M . By 4.7.5, the canonical trilinear map BM (Y ) × Y × M → Y is separately w∗ -continuous. We will now check that the inner product on Y is separately w ∗ -continuous, with respect to the w∗ -topology determined by Y∗ . To this end, suppose that we have a bounded net (yt )t converging to y in the w∗ -topology of Y . Fix x, w in Y . By the above, the ‘rank one’ operator |wx| is w ∗ -continuous on Y . Hence wx|yt → wx|y in the w∗ -topology. Let (x|ytµ ) be a w∗ -convergent subnet of the bounded net (x|yt ), converging to b ∈ M say. By the last paragraph, we have that wx|ytµ → wb in the w∗ -topology. Hence wb = wx|y. Since this is true for all w ∈ Y , it follows from 8.1.4 (1) that b = x|y. Hence x|yt → x|y in the w∗ -topology. By A.2.5 (2), the inner product is separately w ∗ -continuous. By 8.5.4 and 8.5.3, Y is a w∗ -full W ∗ -module over M , and we have the other consequences stated in 8.5.4 and 8.5.5. The uniqueness follows easily from 8.5.4 (2), and from basic operator space duality principles (see Section 1.4). 2 Corollary 8.5.7 Let Y be a right C ∗ -module over a W ∗ -algebra M . Then Y is a W ∗ -module if and only if Y has an operator space predual. In this case the operator space predual is unique. Proof The ‘only if’ follows from 8.5.4 (4). For the other direction, by 8.5.3, M (I) = I w . Thus if Y has an operator space predual, then, by 8.5.6, Y is a W ∗ -module over I w . By 8.5.2, Y is also selfdual over M . The uniqueness was proved in 8.5.6. 2 Corollary 8.5.8 A bounded module map u : Y → Z between W ∗ -modules over M , is w∗ -continuous. Proof By (8.19), u is adjointable. We will apply A.2.5 (2). If (yt )t is a bounded net converging to y ∈ Y in the w∗ -topology of Y , then u(yt )|z = yt |u∗ (z) −→ y|u∗ (z) = u(y)|z,

z ∈ Y.

By 8.5.4 (3), u(yt ) → u(y) in the w∗ -topology. Thus u is w∗ -continuous.

2

We separate one other interesting fact, which follows easily from 8.2.3 (1), and the ﬁrst paragraph of the proof of 8.5.6, for example. Corollary 8.5.9 A right W ∗ -module Y over a W ∗ -algebra M is a normal dual operator BM (Y )-M -bimodule in the sense of 3.8.2. 8.5.10 (The linking W ∗ -algebra) If Y is a right W ∗ -module over M , then we deﬁne the linking W ∗ -algebra of Y to be Lw (Y ) = BM (Y ⊕c M ). This equals BM (Y ⊕c M ) by (8.19), since Y ⊕c M is a selfdual M -module, as is easily veriﬁed. As in 8.1.17, by considering the adjointable inclusion and projection maps between Y ⊕c M and its two summands, it is clear that Lw (Y ) may be viewed as a 2 × 2 matrix algebra with corners BM (Y ), Y, Y and M . Thus any W ∗ -module is a corner eN (1 − e), for a W ∗ -algebra N and a projection e ∈ N . The converse is also true, namely that if e is a projection in a W ∗ -algebra N , then Y = eN (1−e)

336

W ∗ -modules

is a right W ∗ -module over (1−e)N (1−e). This may be seen by using 8.5.4 if necessary. Thus we obtain another ‘picture’ of W ∗ -modules, namely as the corners of W ∗ -algebras. This should be compared with 8.1.19. The linking W ∗ -algebra of a W ∗ -module is very useful when it comes to calculations, because the inner product and module actions have been replaced by multiplication in the W ∗ -algebra, just as we saw in 8.1.19. As one illustration of this principle, we invite the reader to check that the proofs in 8.2.24 and 8.2.25 may be adapted in an obvious way to give the analoguous results for w ∗ -closed ideals and quotients. In fact the W ∗ -module versions of these results are simpler, due to the well-known correspondence between two-sided w ∗ -closed ideals in a W ∗ -algebra and central projections. 8.5.11 (W ∗ -closed TROs) Via the linking W ∗ -algebra, one may now view W ∗ modules as the w∗ -closed subtriples of B(K, H), or of a W ∗ -algebra. Indeed, as we just saw, we may write any W ∗ -module Y over M , as a corner eN (1 − e) of a W ∗ -algebra N , with M ∼ = (1 − e)N (1 − e). Suppose that N has been represented faithfully as a von Neumann subalgebra of B(H) say. Then the projection e determines a splitting H = H1 ⊕ H2 say, and it is evident that M corresponds to a von Neumann algebra in B(H2 ), and Y corresponds to a w∗ -closed subtriple, and an M -submodule, of B(H2 , H1 ). Another useful way to represent Y , is to suppose that M is a von Neumann algebra in B(K), and to consider the isometry Φ : Y → B(K, Y ⊗M K) from 8.2.14 satisfying Φ(y)∗ Φ(z) = y|z, for y, z ∈ Y ; with Y unitarily isomorphic to the TRO Φ(Y ). We leave it as an exercise that Φ is a w∗ -homeomorphism onto Φ(Y ), which is w ∗ -closed. (Hint: use A.2.5 and 8.5.4 (3), and simple net arguments of the kind found in 8.5.36, for example.) Conversely, using 8.5.4, for example, it is easy to check that any w ∗ -closed subtriple Z of B(K, H), or of a W ∗ -algebra, is a w∗ -full right W ∗ -module over the w∗ -closure of Z Z. Indeed such a space is a W ∗ -equivalence bimodule in the sense described next: 8.5.12 (Weak Morita equivalence) W ∗ -equivalence M -N -bimodules are deﬁned analoguously to the equivalence bimodules in 8.1.2, with the words ‘C ∗ -module’ replaced by ‘W ∗ -module’, and ‘full’ by ‘w∗ -full’. If there exists such a bimodule over M and N , then we say that M and N are weakly Morita equivalent. It is not hard to see that weak Morita equivalence is an equivalence relation coarser than ∗-isomorphism of W ∗ -algebras (see the Notes to this section). Corollary 8.5.13 If Y is a W ∗ -equivalence M -N -bimodule, then M ∼ = BN (Y ) ∗-isomorphically. ∼ M (KN (Y )). Proof By the ‘left-handed’ version of 8.5.3, and by 8.1.15, M = Now the result follows by 8.1.16 (3), and (8.19). 2 8.5.14 (W ∗ -modules are W ∗ -equivalence bimodules) Analoguously to 8.1.14, any right W ∗ -module Y over a W ∗ -algebra N is a w∗ -full left W ∗ -module over BN (Y ). That it is selfdual as a left module follows, for example, from 8.5.7. To see that it is w∗ -full, suppose that p is the support projection in BN (Y ) for the

C ∗ -modules and operator spaces

337

w∗ -closed ideal generated by the ‘rank one’ operators. For all x, y, z ∈ Y , we have 0 = (1 − p)|yz|(x) = (1 − p)yz|x. By 8.1.4 (2), 1 − p = 0, so that p = 1. Consequently, if Y is a w∗ -full right W ∗ -module over N , then Y is a W ∗ equivalence BN (Y )-N -bimodule. More generally, if Y is a right W ∗ -module over N then Y is a W ∗ -equivalence BN (Y )-I w -bimodule, where I w is as above. Conversely, if Y is a W ∗ -equivalence M -N -bimodule, then by 8.5.3 and the proof of 8.5.13, we have M ∼ = M (I). Since Y is an equiv= M (KN (Y )) and N ∼ alence KN (Y )-I-bimodule (see 8.1.14), it follows from 8.1.20 that M and N have isomorphic centers. Also, as in 8.1.18, we have that if Y is a w ∗ -full right W ∗ -module over N , then N and Lw (Y ) are weakly Morita equivalent. 8.5.15 (W ∗ -summands) If Y is a right W ∗ -module over M then by 3.8.11, CIw (Y ) is a dual operator M -module for any cardinal I. It is easy to see that CIw (Y ) is also a W ∗ -module. For example, if Y is represented, as in 8.5.11, both as an M -submodule, and as a w∗ -closed subtriple, of B(K, H), with M acting as a von Neumann subalgebra of B(K), then (if necessary, by simple arguments of the kind in 3.8.10–3.8.12) CIw (Y ) may be identiﬁed with an M -submodule, and w∗ -closed subtriple, W of B(K, H (I) ). By 8.5.4 (1), W is a W ∗ -module over M , with inner product S × T → S ∗ T . It is then easy to see that the corresponding inner product on CIw (Y ) is given by (yi )|(zi ) =

yi |zi ,

(yi ), (zi ) ∈ CIw (Y ),

i∈I

where the convergence of the sum is in the w ∗ -topology of M . The columns with a ﬁnite number of nonzero entries are w ∗ -dense in CIw (Y ). Indeed if PJ is the projection from CIw (Y ) onto the set of ‘columns supported on CJ (Y )’, for a ﬁnite subset J of I, then (PJ )J converges in the w∗ -topology of BM (CIw (Y )) to the identity map on CIw (Y ), by 8.5.5 (2) and 1.6.3. From this it is not hard to show that BM (CIw (Y )) ∼ = MI (BM (Y )) as W ∗ -algebras. Since we shall not use this fact, we omit the proof. In [73], the interested reader will ﬁnd the proof of an operator space generalization of this fact. Namely, A l (CIw (X)) ∼ = MI (Al (X)) as W ∗ -algebras, for any dual operator space X. Similar assertions hold for RIw (X), for a left W ∗ -module X over M . We say that a submodule Y of a W ∗ -module Z is w∗ -orthogonally complemented in Z, if Y is orthogonally complemented in the sense of 8.1.9, and if Y is a w∗ -closed subspace of Z. In this case Y is a W ∗ -module too, by 8.5.4. Lemma 8.5.16 Suppose that Z is a right W ∗ -module over a W ∗ -algebra M , and that Y is a subspace of Z. The following are equivalent: (i) (ii) (iii) (iv)

Y Y Y Y

is is is is

an orthogonally complemented M -submodule of Z, a w∗ -orthogonally complemented M -submodule of Z, a w∗ -closed M -submodule of Z, a right M -summand of Z, in the sense of 4.8.1.

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Proof Clearly (ii) implies (i) and (iii). If P is a projection in BM (Y ), then P is w∗ -continuous by 8.5.8. Thus Ran(P ) is w ∗ -closed. Therefore (i) implies (ii). By deﬁnition, and by 4.5.15 (iii) and 8.4.2, the right M -summands of Z are precisely the ranges of the adjointable projections on Z. Thus (iv) is equivalent to (i). Finally, given (iii), it follows from 8.5.4 (1) that Y is selfdual as an M -module. Hence the inclusion map from Y into Z is adjointable by (8.19). Its adjoint is easily checked to be an orthogonal projection onto Y , giving (i). 2 Proposition 8.5.17 If Y is a right C ∗ -module over a C ∗ -algebra B, then Y ∗∗ with its canonical B ∗∗ -module action (see 3.8.9), is a W ∗ -module over B ∗∗ . If Y is an equivalence A-B-bimodule, then Y ∗∗ is a W ∗ -equivalence bimodule over A∗∗ and B ∗∗ . Proof First suppose that Y is full over B. We recall that Y ∼ = pL(1 − p), as in (8.3). Here L is the linking C ∗ -algebra of Y , and p is a projection in M (L). By replacing Y with pL(1 − p), we may assume that Y is a subtriple of L, and that B = Y Y ⊂ L. Let Z be the w∗ -closure of Y in pL∗∗ (1 − p). Clearly, Z = pL∗∗ (1 − p), a w∗ -closed subtriple of L∗∗ . By A.2.3, Y ∗∗ ∼ = Z completely isometrically, and w∗ -homeomorphically. Similarly, if N = (1−p)L∗∗ (1−p), then B ∗∗ ∼ = N , as W ∗ -algebras. We have B = Y Y ⊂ Z Z ⊂ N . Thus the w∗ -closure of Z Z contains the w∗ -closure of B in L∗∗ , namely N . Thus the w∗ -closure of Z Z equals N . By 8.5.11, Z is a w∗ -full right W ∗ -module over N . We may transfer these structures, to make Y ∗∗ a w∗ -full right W ∗ -module over B ∗∗ . This B ∗∗ -module action on Y ∗∗ coincides with the canonical second dual action from 3.8.9. This is because the product map Z × N → Z is separately w ∗ -continuous, and extends the canonical map Y × B → B. A similar argument yields the last assertion of the Proposition. If Y is not full over B then we consider the ideal I from 8.1.4. By the above, Y ∗∗ is a right W ∗ -module over I ∗∗ . However I ∗∗ is an ideal in B ∗∗ , and so Y ∗∗ is a W ∗ -module over B ∗∗ , by 8.5.2 and 8.1.4 (4). 2 8.5.18 (Dual triple systems) Appropriate weak* versions of the theory of TROs and triple systems presented in the ﬁrst half of Section 8.3, also go through without diﬃculty. In fact the theory becomes simpler, due to the correspondence between w∗ -closed two-sided ideals in a W ∗ -algebra and central projections. By 8.5.6 and the remarks above it, a TRO Y which has a predual, is a w ∗ -full ∗ W -module over M (Y Y ). Putting this together with 8.5.11 and 8.5.14, we see that such ‘dual TROs’; the triple systems which have a predual; W ∗ -modules; and W ∗ -equivalence bimodules, are essentially the same thing, in a sense similar to the discussion in 8.3.1. A w∗ -continuous triple morphism u : Y → Z between w ∗ -closed TROs, has range which is a w∗ -closed TRO. Indeed, let W = Ker(u), which is a w ∗ -closed triple ideal in Y . By the last several lines of 8.5.10, and the argument for 8.3.3, there is a central projection e such that W = eY . Let W = (1 − e)Y , then uW is a one-to-one w∗ -continuous triple morphism with Ran(u) = Ran(uW ). By 8.3.2, uW is isometric, so that by A.2.5 its range is w ∗ -closed.

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If Y is a TRO, then Y ∗∗ is a TRO and a W ∗ -equivalence bimodule, as in the proof of the last Proposition, for example. Note that if u : Y → B(K, H) is a triple morphism, then by routine w∗ -approximation arguments, the canonical w∗ -continuous extension u˜ : Y ∗∗ → B(K, H) (see A.2.2) is a triple morphism too. As one application of this, we deduce that any subtriple Y of B(K, H) satisﬁes a ‘Kaplansky density theorem’, namely that Ball(Y ) is w ∗ -dense in the unit ball of the w∗ -closure of Y . This follows from the above, 8.3.4 (3), and A.5.10. A completely isometric surjective linear map (or equivalently, a surjective triple isomorphism) between dual TROs, is automatically w ∗ -continuous. This follows from the uniqueness of the predual of a W ∗ -module (see 8.5.7). Theorem 8.5.19 The right M -ideals (see 4.8.1) in a right Hilbert C ∗ -module are exactly the closed right submodules. Proof Let Z be a right C ∗ -module over a C ∗ -algebra B. If Y is a right M -ideal of Z, then the w∗ -closure W of Y in Z ∗∗ is a right M -summand of Z ∗∗ . By 8.5.16, and using 8.5.17, we see that W is a B ∗∗ -submodule of Z ∗∗ . Viewed as subsets of Z ∗∗ , we have Y B ⊂ (W B ∗∗ ) ∩ Z ⊂ Y ⊥⊥ ∩ Z = Y, using A.2.3 (4). Thus Y is a B-submodule of Z. Conversely, if Y is a B-submodule of Z, then its w ∗ -closure W in Z ∗∗ , is a ∗∗ B -submodule. Indeed this follows from the fact that the B ∗∗ -module action on Z ∗∗ is separately w∗ -continuous, and A.2.1. By 8.5.16, W is a right M -summand 2 of Z ∗∗ , so that Y is a right M -ideal of Z. 8.5.20 (C ∗ -modules and M -ideals) In fact one may view the theory of one-sided M -ideals in operator spaces, introduced brieﬂy in Section 4.8, as a generalization of the behaviour of submodules of C ∗ -modules. See [56, 57, 73] for details. In connection with the last result, we note that the classical M -ideals (see 4.8.1) in an equivalence A-B-bimodule are exactly the A-B-submodules. One direction of this is not hard. For example, if Y is an A-B-submodule of an equivalence A-B-bimodule Z, then by 8.5.19, Y is both a left and a right M ideal of Z. By 4.8.4, Y is a complete M -ideal, and hence an M -ideal. The reverse direction seems to be harder. Suppose that Y is an M -ideal in Z. If a ∈ Asa , consider the map T z = az, for z ∈ Z. It is easy to see that T ∈ Her(B(Z)) (see A.4.2), and so by [195, Corollary I.1.25], aY ⊂ Y . Thus Y is a left A-submodule, and similarly it is a right B-submodule. Hence the M -ideals in a TRO Y are the (Y Y )-(Y Y )-submodules. 8.5.21 (Partial isometries in C ∗ -modules) We say that an element u in a right C ∗ -module Y over M is a partial isometry if p = u|u is an orthogonal projection in M . This element p is called the initial projection of u. Note that it follows that up = u (since u − up|u − up = 0). Thus |uu| is an orthogonal projection in the W ∗ -algebra BM (Y ).

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We say that two partial isometries u and v in Y are orthogonal if u|v = 0. In this case the orthogonal projections |uu| and |vv| are mutually orthogonal. Lemma 8.5.22 Suppose that Y is a right W ∗ -module over M , and that y ∈ Y . 1 Then y = u|y|, where |y| = y|y 2 ∈ M , and u is a partial isometry in Y whose initial projection is the range projection of |y| in the von Neumann algebra sense (see 2.2.7 in [320]). Proof We view y as an element of the linking W ∗ -algebra of Y (see 8.5.10), and take its polar decomposition there, as in 2.2.9 of [320]. It is easy to see from the formula given there for u, that u ∈ Y , that u∗ u ∈ M , and that the latter is the range projection of |y| in M . 2 Lemma 8.5.23 (Paschke) Let Y be a right W ∗ -module. Then Y has an orof mututhonormal basis. That is, there exists a set {xi }i∈I in Y consisting ally orthogonal nonzero partial isometries, such that x = x x |x in the i i i |x x | converges in the w∗ -topology of Y , for all x ∈ Y . In particular, i i i∈I w∗ -topology of BM (Y ), to IY . Proof We consider the subsets B of Y consisting of mutually orthogonal nonzero partial isometries in Y , ordered by inclusion. At least one such set exists by 8.5.22, and by Zorn’s lemma we may choose a maximal such set, {xi : i ∈ I} say. We ﬁrst claim that if xi |x = 0 for all i ∈ I, then x = 0. To see this, write x = u|x| for a partial isometry u ∈ Y as in 8.5.22. Then xi |u|x| = 0. If p is the initial projection of u, then u = up, and so xi |up|x| = 0. Since p is the range projection for |x|, we see that xi |up = 0 = xi |u. This contradicts the maximality above, if u = 0. By the remarks before 8.5.22, TJ = i∈J |xi xi | is an orthogonal projection in BM (Y ), for any ﬁnite subset J ⊂ I. We therefore have x|xi xi |x ≤ x|x, x ∈ X. TJ x|x = i∈J

The increasing net (TJ )J converges in the w∗ -topology of B M (Y ) to an operator T say, with 0 ≤ T ≤ I. By 8.5.5 (2), for any x ∈ X the sum i xi xi |x converges in the w∗ -topology of Y , to T (x). Taking the inner product with xj , for a ﬁxed j ∈ I, we have xj |T (x) = xj |x, by the w∗ -continuity of the inner product. Hence T (x) = x by the ﬁrst part of the proof. 2 8.5.24 (The Parseval identity) Suppose that {xi : i ∈ I} is an orthonormal basis for Y , as in 8.5.23. It follows from the proof above, and 8.5.4 (1), that y|xi xi |y = y|y, y ∈ Y, i∈I

the convergence in the w∗ -topology. Since i |xi xi | = I, we see using (8.7) if necessary, that (xi ) is an element of RIw (Y ), and has norm there equal to 1. As a consequence, we obtain another characterization of W ∗ -modules, as exactly the w∗ -closed submodules of CIw (M ):

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Corollary 8.5.25 A Banach module Y over a W ∗ -algebra M is a right W ∗ module over M if and only if Y is isometrically M -isomorphic to an orthogonally complemented submodule of CIw (M ), for some cardinal I. Proof The ‘only if’ follows from 8.5.16 and the remark above it. Conversely, if Y is a right W ∗ -module over M , let {xi : i ∈ I} be an orthonormal basis for Y as in 8.5.23. Deﬁne α : Y → CIw (M ) by α(y) = (xi |y)i , for y ∈ Y . By 8.5.24, α is an isometry. Also, α is w∗ -continuous with respect to the w∗ -topology of Y , by 1.6.3 (2) and 8.5.4 (1), and is an M -module map. By A.2.5, the range of α is 2 w∗ -closed. Now apply 8.5.16. 8.5.26 (The ultraweak direct sum) We deﬁne the ultraweak direct sum ⊕ wc i∈I Yi ∗ ∗ of a family {Y : i ∈ I} of right W -modules over a W -algebra M , to be the set i of (yi )i∈I ∈ i∈I Yi , such that the ﬁnite partial sums of the series i∈I yi |yi are uniformly bounded above. Equivalently, it is the set of (yi )i∈I such that ∗ i∈I yi |yi converges in the w -topology of M . It is easy to check, using the wc polarization identity (1.1), that for (y∗i ) and (zi ) in ⊕i∈I Yi , the ﬁnite partial sums of i∈I yi |zi converge in the w -topology of M . We write (yi )|(zi ) for the w∗ -limit. Most of the conditions in the deﬁnition of a C ∗ -module are easy to check for ⊕wc i∈I Yi , and all will follow from considerations in the next paragraph. Note that if Y is a right W ∗ -module, then the W ∗ -module CIw (Y ) met in 8.5.15 equals the ultraweak direct sum of I copies of Y . Although we shall not use the general ultraweak direct sum much, we mention some of its properties. For these, it is helpful to view Y = ⊕wc i∈I Yi in a slightly diﬀerent way. We begin with a faithful normal representation of M on a Hilbert space K. We suppose that for each i ∈ I, Yi is represented (as in 8.5.11 say) as a w∗ -closed M -submodule, and a subtriple, of B(K, Hi ), for a Hilbert space Hi . Set H = ⊕i Hi , and let Pi be the projection from H onto Hi . We also set W = {T ∈ B(K, H) : Pi T ∈ Yi for all i ∈ I}, and equip Wwith its canonical inner product S × T → S ∗ T . For S, T ∈ W we have S ∗ T = i S ∗ Pi Pi T , which is a w∗ -convergent sum in M . Hence this inner product is valued in M . It is easy to see that W is a w∗ -closed M -submodule, and subtriple, of B(K, H). From 8.5.4 (1), it follows that W is a right W ∗ -module over M . Writing yi = Pi T , for T ∈ W , it becomes evident that W corresponds precisely to the deﬁnition of ⊕wc i∈I Yi above. In other words, W is unitarily M -isomorphic to this sum. It ∗ follows that ⊕wc i∈I Yi is a right W -module over M . One may deduce from the above description, and the associativity property for Hilbert space sums, that the ultraweak direct sum is associative. Thus, for wc wc ∗ ∼ wc ∼ wc example, ⊕wc i∈I (⊕j∈J Yij ) = ⊕j∈J (⊕i∈I Yij ) = ⊕i,j Yij unitarily, for right W modules Yij over M . Also, one can easily see that the set of tuples in an ultraweak direct sum which are zero except in a ﬁnite number of entries, is w ∗ -dense. We shall not use this, but it can be deduced from 8.5.23 that the right W ∗ modules over a W ∗ -algebra M , are exactly the ultraweak direct sums of w ∗ -closed right ideals pM of M . See [302, 421].

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8.5.27 (Second duals of C ∗ -module sums) If (Yi ) is a family of C ∗ -modules over ∗∗ unitarily as B ∗∗ -modules. We merely a C ∗ -algebra B, then (⊕ci Yi )∗∗ ∼ = ⊕wc i Yi c ∗∗ sketch the proof. Let Y = ⊕i Yi , let Z = ⊕wc i Yi , and let L be the ‘augmented c linking algebra’ KB (Y ⊕ B). Let p0 be the projection of Y ⊕c B onto 0 ⊕ B, of Yi . We regard and similarly let pi be the projection of Y ⊕c B onto the copy pi ∈ M (L) ⊂ L∗∗ . In the latter W ∗ -algebra one can show that i pi = (1 − p0 ). As in the proof of 8.5.17, we have Yi ∼ = (1 − p0 )Lp0 , unitarily = pi Lp0 , and Y ∼ as right B-modules. Also as in the proof of 8.5.17, we have Yi∗∗ ∼ = pi L∗∗ p0 and ∗∗ ∼ ∗∗ ∗∗ Y = (1 − p0 )L p0 , unitarily as right B -modules. However it is not hard to check, using basic facts about w ∗ -limits of increasing nets of projections in a W ∗ -algebra, that ∗∗ ∼ ∗∗ ∗∗ ∼ ⊕wc = ⊕wc i Yi i pi L p0 = (1 − p0 )L p0 ,

unitarily as right B ∗∗ -modules. This proves the result. A powerful tool associated with W ∗ -modules, and indeed operator spaces, is the following weak* variant of the stabilization result in 8.2.6. Theorem 8.5.28 Let Y be a w∗ -full right W ∗ -module over a W ∗ -algebra N . Then there exists a cardinal I such that CIw (Y ) ∼ = CIw (N ) unitarily (as right N -modules). Also, MI (Y ) is linearly completely isometrically isomorphic (via a right N -module map) to the W ∗ -algebra MI (N ). Proof We prove this very similarly to our arguments for 8.2.6, which the reader should follow along with (another argument is sketched in the Notes). By 8.5.25 and 8.5.16, there exists a cardinal I and a w ∗ -closed N -submodule W of CIw (N ), such that Y ⊕c W ∼ = CIw (N ) unitarily (as right N -modules). By set theory we may assume that I 2 = I. It follows from this, and (1.59) for example, that CIw (CIw (N )) ∼ = CIw (N ). Using the latter fact, and using ‘associativity’ of the ultraweak sum (see 8.5.26), the ‘Eilenberg swindle’ works, similarly to the proof of 8.2.6 (1), to give: CIw (N ) ∼ = CIw (Y ) ⊕c CIw (W ) ∼ = CIw (Y ) ⊕c CIw (Y ) ⊕c CIw (W ) ∼ = CIw (Y ) ⊕c CIw (N ), unitarily as N -modules. We may assume by 8.5.14 that Y is a W ∗ -equivalence M -N -bimodule. By (the ‘other-handed variant’ of) Lemma 8.5.23, and since N is isomorphic to the algebra of left M -module maps on Y (see 8.5.14), there exists a cardinal J and a subset {zj : j ∈ J} of Y , such that 1N = j zj |zj in the w∗ -topology of N . Thus (zj ) ∈ CJw (Y ), which permits us, exactly as in the proof of 8.2.6 (3) (and in fact a little more easily), to deﬁne maps showing via 8.1.21 that N is unitarily isomorphic to an orthogonally complemented N -submodule of CJw (Y ). Thus by 8.5.16, there exists a w ∗ -closed submodule P of CJw (Y ) such that N ⊕c P ∼ = CJw (Y ), By the argument in the ﬁrst paragraph of this proof, we obtain that CJw (Y ) ∼ = CJw (N ) ⊕c CJw (Y ) unitarily. By set theory, we may suppose that I = J. Thus CJw (Y ) ∼ = CJw (N ).

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The last assertion follows since MI (Y ) ∼ = RIw (CIw (Y )) ∼ = RIw (CIw (N )) ∼ = MI (N ), 2

using (1.20). The following follows at once from 8.5.28 and 8.5.17:

Corollary 8.5.29 If Y is any full right C ∗ -module over a C ∗ -algebra B, then there is a cardinal I such that CIw (Y ∗∗ ) ∼ = CIw (B ∗∗ ) completely B ∗∗ -isometrically. ∗∗ ∗∗ Also, MI (Y ) is completely B -isometric to the W ∗ -algebra MI (B ∗∗ ). 8.5.30 (W ∗ -algebra ‘covers’ of an operator space) In operator space applications one sometimes applies the preceding result, with Y an injective or triple envelope of an operator space X (see Sections 4.4 and 8.3). These are triple systems, and may be taken to be C ∗ -modules, as discussed in those sections. By 8.5.29 we see that for some cardinal I, MI (Y ∗∗ ) is a W ∗ -algebra. Thus for any operator space X, there is a useful and essentially canonical W ∗ -algebra ‘containing’ MI (X). Corollary 8.5.31 (The stable isomorphism theorem for W ∗ -algebras) Two W ∗ -algebras M and N are weakly Morita equivalent if and only if there exists a cardinal I such that MI (M ) ∼ = MI (N ) ∗-isomorphically. Proof The ‘only if’ may be proved similarly to 8.2.7, but replacing the use of 8.2.6 (5) with the last assertion of 8.5.28 (and the ‘other-handed’ variant of that assertion). Another argument is sketched towards the end of the Notes for 8.5. The other direction is easier. For example, one may appeal to the later result 2 8.5.38, and the fact that the commutant of M ⊗ MI is M ⊗ 1. 8.5.32 (A basic construction) Suppose that Y is a full right C ∗ -module over a C ∗ -algebra B. We may suppose that Y is an equivalence A-B-bimodule. Consider the Morita linking C ∗ -algebra L(Y ), and identify Y with a corner pL(Y )(1−p) as usual (see (8.3)). Fix a faithful nondegenerate representation of L(Y ) on a Hilbert space. As we saw in 8.2.8, this Hilbert space may be taken to be H ⊕ K, where H and K are two Hilbert spaces on which respectively A and B are faithfully and nondegenerately represented. Since A = Y Y , we have [Y K] = H. Similarly, [Y H] = K. Using these facts it is easy to explicitly compute the commutant L(Y ) in B(H ⊕ K). Indeed, a simple calculation shows the following facts. First, L(Y ) is a set of diagonal matrices, whose 1-1 entry R is in A , and whose 2-2 entry S is in B . Second, these entries are mutually dependent, and this dependence is given by the equation Ry = yS for all y ∈ Y . This last equation provides a map π : B → A , deﬁned by π(S) = R. That is, π(S) y = y S,

y ∈ Y, S ∈ B .

(8.21)

In fact π is a ∗-isomorphism from B onto A . One (perhaps too slick) way to see this proceeds as follows. Recall from 8.2.19 that the strong Morita equivalence of A and B gives a category isomorphism between the categories of modules over A and B respectively. This isomorphism is implemented by a pair

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of functors F and G, given by F (W ) = Y ⊗B W , and G(V ) = Y ⊗A V . By basic algebra, the map T → F (T ) = IY ⊗ T yields a surjective isomorphism ∼ B B(W ) = A B(F (W )). Indeed this map is clearly contractive (by 8.2.12 (1)), but one may exhibit a contractive inverse via the other functor G, just as in pure algebra (e.g. see [8, Proposition 21.2]). Thus the last isomorphism is also isometric. By 8.2.22, F (K) = Y ⊗B K is unitarily A-isomorphic to H above. Setting W = K and appealing to the above facts, we have B B(K) ∼ = A B(H) isometrically isomorphically. This says precisely that B ∼ = A ; and it may be veriﬁed that this isomorphism is exactly our map π above. We have by 1.2.4 that π is a ∗-isomorphism. We remark in passing that this proves that strongly Morita equivalent C ∗ -algebras naturally have Hilbert space representations in which their commutants are ∗-isomorphic. Summarizing, we saw that the commutant of L(Y ) is the set of matrices π(S) 0 , S ∈ B. 0 S It is now easy to see that the second commutant L(Y ) is the set of matrices c e , c ∈ A ; d ∈ B ; e, f ∈ E, f∗ d where E = {T ∈ B(K, H) : T S = π(S)T for all S ∈ B }. In other words, E = B B(K, H), where we are viewing H as a B -module via π. Also, we have E = pL(Y ) (1 − p). Thus E is a subtriple of B(K, H), and E has a B -valued inner product deﬁned by T1 |T2 = T1∗ T2 , for T1 , T2 ∈ B(K, H). Indeed we have B = Y Y ⊂ E E ⊂ B , the last inclusion following from the deﬁnition of E and a direct calculation. Taking w∗ -closures in B(K) in the last equation, and using the double commutant theorem, we deduce that E E is w∗ -dense in B . From 8.5.4 (1), we deduce that E is a w∗ -full right W ∗ -module over B . Hence, by symmetry, E is a W ∗ -equivalence A -B -bimodule. By the double commutant theorem, L(Y ) is w ∗ -dense in L(Y ) . It is clear from this, and from the fact that E = pL(Y ) (1 − p), that Y is a w∗ -dense A-B-submodule of E. We will see some applications of this momentarily, and in the Notes section. 8.5.33 (Universal representations) We mention in passing that if we begin the construction in 8.5.32 by choosing a ‘universal representation’ (see 3.2.7) of L(Y ) (or equivalently, by 8.2.23 and 8.1.18, a representation of L(Y ) induced from a universal representation of B, say), then that construction allows one to recover 8.5.17. In fact 8.5.32 yields much more information in this case, such as the fact that Y ∗∗ ∼ = B B(K, H), for appropriate Hilbert modules H and K. It also allows us to treat representations of B ∗∗ , or Y ∗∗ , in a functorial way that is often important in applications.

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8.5.34 (Normal rigged W ∗ -modules) We deﬁne a normal N -rigged M -module to be a right W ∗ -module Z over N for which there exists a (unital) normal ∗-homomorphism θ : M → BN (Z). This name is due to Rieﬀel, who does not however insist on all of the conditions above. In the literature, they are often called M -N -correspondences. The following shows again that the ‘interesting objects’ in the theory fall within the operator module framework: Proposition 8.5.35 A W ∗ -module over N is a normal N -rigged M -module if and only if it is a normal dual operator M -N -bimodule in the sense of 3.8.2. Proof If Z is a normal N -rigged M -module, then by 8.5.9 it is a normal dual operator BN (Z)-N -bimodule. Since the left action comes from a normal ∗-homomorphism θ : M → BN (Z), Z is a normal dual M -N -bimodule. Conversely, if Z is a normal dual operator M -N -bimodule, then as in the proof of 4.7.6, there is a normal ∗-homomorphism θ : M → Al (Z). However 2 Al (Z) = BN (Z) by 8.4.2. 8.5.36 (Inducing normal representations) Suppose that H is a Hilbert space on which M is normally represented on. If Y is a right W ∗ -module over M , then it follows from 8.2.13 that Y ⊗M H is a Hilbert space. Note, further, that if Y is a normal M -rigged N -module, then Y ⊗M H is a normal Hilbert N -module (in the sense of 3.8.5). To see this, suppose that (at )t is a bounded net in N converging in the w∗ -topology to an a ∈ N . Then by 8.5.35, at y → ay in the w∗ -topology of Y , for any y ∈ Y . By 8.5.4 (3), if ζ, η ∈ H and y, z ∈ Y , then at y ⊗ η, z ⊗ ζ = z|at yη, ζ −→ z|ayη, ζ = ay ⊗ η, z ⊗ ζ. Since ﬁnite sums of ‘rank one tensors’ are norm dense in Y ⊗M H, it is evident that at y ⊗η → ay ⊗η weakly in the Hilbert space Y ⊗hM H. For the same reason, we may now deduce that at ξ → aξ weakly, for any ξ ∈ Y ⊗M H. Thus by A.2.5 (2), Y ⊗M H is a normal Hilbert N -module. By 8.2.13, if, further, M is faithfully represented on H, and if the canonical map from N into BM (Y ) is one-to-one, then N is faithfully represented on the Hilbert space Y ⊗M H. We shall not use this, but it follows easily from A.6.2 that if I is as in 8.1.4, then Y ⊗M H = Y ⊗I H. Corollary 8.5.37 (Rieﬀel) Suppose that π : M → B(K) is a faithful normal representation, and let R = π(M ) , the commutant in B(K). (1) If Y is a W ∗ -module over M , then there exists a Hilbert space H on which R is normally represented (namely Y ⊗M K), such that Y ∼ = R B(K, H) completely isometrically, and w ∗ -homeomorphically. (2) Conversely, if H is a Hilbert space on which R is normally represented, then the w∗ -closed right π(M )-submodule R B(K, H) of B(K, H), with its canonical B(K)-valued inner product, is a right W ∗ -module over π(M ). In fact the isomorphism in (1) is a unitary M -module map.

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Proof (2) This is clear by a direct computation, using the double commutant theorem, and 8.5.4 (1). (1) We will use 8.5.32 (Rieﬀel’s proof is sketched in the Notes section). First assume that Y is w∗ -full over M . Since Y is a W ∗ -module, the normal representation of M on K induces, by 8.5.36, a faithful normal representation of BM (Y ) on H = Y ⊗M K. Similarly, using also the deﬁnition of Lw (Y ) in 8.5.10, we have a faithful normal ∗-representation of Lw (Y ) = BM (Y ⊕c M ) on the Hilbert space (Y ⊕c M ) ⊗M K. The latter space, as in (8.14), is unitarily equivalent to (Y ⊗M K) ⊕ K = H ⊕ K. Just as in the last paragraph of 8.2.22, the associated normal representation of Lw (Y ) on H ⊕ K is one-to-one (and hence completely isometric, by 1.2.4) and corner-preserving, and its ‘1-2-corner’ is a map from Y to B(K, H). Clearly the latter map is completely isometric and w∗ -continuous, and hence (using also A.2.5 (3)) Y may be regarded as a w∗ -closed subspace of B(K, H). If I is as in 8.1.4 then, as we saw in the proof of 8.5.3, I acts nondegenerately on K, and M (I) = π(I) = π(M ). By 8.2.13 (2), A = KM (Y ) acts nondegenerately on H. Now we are in a position to apply the arguments in 8.5.32. By the last facts in 8.5.32, we have E = Y . Since Y is an M -submodule of E, the last assertion of the Corollary is also clear in this case. In the general case, we use the fact that a w ∗ -closed ideal in a W ∗ -algebra is of the form pM for a central projection p. Thus I w = pM for such a projection p. Let K = π(p)K, and apply the previous case to the canonical representation of I w on K . Write θ for this last representation, and let N = θ(I w ) . We obtain a Hilbert space H on which N is normally represented, such that Y ∼ = N B(K , H). However, there is a canonical normal ∗-homomorphism r → π(p)r, from R onto N . Thus K and H may be viewed as R-modules, and it is easily checked that ∼ 2 R B(K, H) = N B(K , H) unitarily as M -modules, via the map T → T|K . Corollary 8.5.38 (Connes) W ∗ -algebras M and N are weakly Morita equivalent if and only if there exist faithful normal representations π : M → B(K) and ρ : N → B(H), with π(M ) ∼ = ρ(N ) ∗-isomorphically. Moreover, in this case, writing R for π(M ) and for ρ(N ) , the TRO R B(K, H) in 8.5.37 (2) is a W ∗ -equivalence N -M -bimodule. Proof If Y is a W ∗ -equivalence N -M -bimodule, and if π is as in 8.5.37, then the proof of (1) of that result, together with the paragraph after (8.21), shows that if ρ is the induced representation of N on Y ⊗M K, then π(M ) ∼ = ρ(N ) . A quick proof of the converse is given in the Notes, however it does not yield the ﬁnal assertion. So instead, suppose that π, ρ, R are as in the statement of 8.5.38. If Y = R B(K, H) then, as in 8.5.37 (2), it is easy to see that Y is a TRO, which is a right and a left W ∗ -module over π(M ) and ρ(N ) respectively. We need to show that these W ∗ -modules are full. The w∗ -closure I w of Y Y is a w∗ -closed ideal in π(M ). Thus there is a central projection e in π(M ) with I w = eπ(M ). Since Y = Y I w (e.g. see 8.1.4 (2)), we have Y = Y e. Since e ∈ π(M ), (1 − e)K is an R-module. Let P be 1 − e, viewed as a map from K onto (1 − e)K. Since e ∈ R , P is an R-module map. Since H is a cogenerator of R N HM OD (see 3.8.6), if

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(1 − e)K = (0) then there exists a nonzero R-module map T : (1 − e)K → H. Thus T P ∈ Y , and so T P = T P e, which is absurd. Thus e = 1, and Y is w ∗ -full 2 on the right. A similar argument shows that Y is w ∗ -full on the left. 8.5.39 (Correspondences) Another important ‘picture’ of W ∗ -modules is related to the standard form L2 (M ) of a W ∗ -algebra M (see 3.8.5, or for full details, see, for example, [175] or [408, Chapters VIII and IX]). One reason why the standard form is of importance here, is that it is a normal faithful Hilbert space representation of M such that M ∼ = M op . Thus in Corollary 8.5.37 we may replace op R = M by M . We view a left M op -module action on a Hilbert space, as a right M -module action of M . In particular, by 8.5.37 (1), if Y is a right W ∗ -module over M , then there exists a Hilbert space H = Y ⊗M L2 (M ) on which M op is normally ∗-represented, or equivalently on which M is normally represented on the right of H, such that Y ∼ = BM (L2 (M ), H). Conversely, by 8.5.37 (2), any Hilbert space H on which M is normally represented as a right action (that is, any normal ∗-representation of M op ), gives rise to a right W ∗ -module over M , namely BM (L2 (M ), H). One may show that BM (L2 (M ), H) ⊗M L2 (M ) ∼ = H unitarily. Indeed it is easy to see that the canonical map from BM (L2 (M ), H) ⊗M L2 (M ) to H is isometric. That it has dense range, and is thus surjective, follows from modular theory (see [27, Theorem 2.2], and references therein). Thus we see that there is a bijective correspondence between such Hilbert spaces H, and right W ∗ -modules over M . It is easily seen that the bijection above restricts to a bijective correspondence between the Hilbert spaces H on which M is normally represented on the right, and on which another W ∗ -algebra N is normally represented on the left; and the class of normal M -rigged N -modules (see 8.5.34). A correspondence between N and M is a Hilbert space H which is a normal N -M -bimodule as above. Thus, such Hilbert spaces are in a bijective relation with the normal M -rigged N -modules. We do not have space to even touch on this extremely important topic in detail here, but refer the reader to [101, Section V.B], [408, Chapters VIII and IX], and [346, 6, 27, 119], for example, and references therein. 8.5.40 (The W ∗ -module tensor product) This is a ‘W ∗ -module version’ of ¯ M Z, the interior tensor product discussed in Section 8.2. We write it as Y ⊗ ∗ for a right W -module Y over M , and a normal N -rigged M -module Z. Just as the C ∗ -module interior tensor product is just the module Haagerup tensor ¯ M coincides with the module weak* product, the W ∗ -module tensor product ⊗ ∗ Haagerup tensor product ⊗w hM which we discussed brieﬂy in 3.8.14. Some of the beneﬁts of knowing that these tensor products coincide are that, ﬁrst, we getuseful expressions for elements in this tensor product as w ∗ -convergent sums i∈I xi ⊗ yi , with a convenient description of the norm of such a sum. This facilitates easy computations. Second, as in 8.2.12, we can appeal to the useful properties of this tensor product to show that this product is functorial, associative, commutes with the ultraweak sum, and so on. One may deduce that, for example, analoguously to 8.2.15, we have

348

A sample application to operator spaces ¯MZ ∼ W⊗ = CBM (Z, W )

completely isometrically.

We omit the proof of this result, which is similar to the proof of 8.2.15, and is a variant of a result originally from [119]. See [48] for a complete discussion of this tensor product, which parallels our earlier development of the C ∗ -module tensor product. Indeed there are precisely analoguous W ∗ -module versions of results 8.2.11–8.2.19. We refer the reader to [361, 48] for a development of this theory which parallels our earlier discussion. To avoid this already lengthy chapter becoming completely unwieldy we have omitted these results. However, the reader who has followed the discussion till now, should at least have no diﬃculty stating appropriate W ∗ -module versions of 8.2.11–8.2.19. 8.6 A SAMPLE APPLICATION TO OPERATOR SPACES Because of space limitations, we will only list one of the very many applications of the preceding theory to operator spaces. See the Notes for references to the literature for other applications. 8.6.1 (Injectivity and semidiscreteness) We return to the notions of ‘OSnuclearity’, ‘OS-semidiscreteness, and the ‘WEP’, introduced in Section 7.1. For a general operator space, the relationships between these concepts, and other properties such as ‘injectivity’, are quite interesting. Some of these are not difﬁcult, such as the result from 7.1.5 that any OS-nuclear operator space X has the WEP, or the fact from 7.1.9 that an OS-semidiscrete dual operator space is injective. In fact, for a ﬁnite-dimensional operator space X, all of the following properties are equivalent: injectivity, OS-nuclearity, OS-semidiscreteness, and the WEP. These are also equivalent to X being a triple system, and also equivalent to saying that for some n ∈ N, X is a completely contractively complemented subspace of Mn (that is, there exists a complete isometry from X onto a subspace W of Mn , and a completely contractive projection from Mn onto W ). Most of these equivalences are quite trivial to see. Indeed, if X is injective then it is a TRO by 4.4.2. For any ﬁnite-dimensional TRO X, the linking C ∗ -algebra L(X) is ﬁnite-dimensional. By (8.3), X is completely contractively complemented in L(X). However any ﬁnite-dimensional C ∗ -algebra is completely contractively complemented in some Mn , so that X is completely contractively complemented in Mn too. We leave the remaining implications to the reader. Theorem 8.6.2 (Eﬀros, Ozawa, and Ruan) If X is a dual operator space then the following are equivalent: (i) (ii) (iii) (iv)

X is injective, X is OS-semidiscrete, X has the WEP, X is completely isometrically isomorphic and w ∗ -homeomorphic to a ‘corner’ pM (1 − p), for an injective W ∗ -algebra M , and a projection p ∈ M .

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Proof We saw in 7.1.3 that (i) is equivalent to (iii). We saw in 7.1.9 that (ii) implies (i). Using the fact that an injective W ∗ -algebra is OS-semidiscrete (see [99, 419]), it is clear that (iv) implies (ii). Finally, if X is an injective dual operator space then by 4.4.2, X may be regarded as an TRO, and hence an equivalence bimodule. By 8.5.6, X is a W ∗ -equivalence M -N -bimodule, over W ∗ -algebras M and N say. By the proof of 4.2.10, MI (X) is injective for any cardinal I. By 8.5.28, we deduce that MI (N ) is injective. Let L be the Morita linking W ∗ -algebra of X (see 8.5.10). By the last remark in 8.5.14, N and L are weakly Morita equivalent. By 8.5.31, we deduce that M I (L), and consequently also L, is injective. Since X is the 1-2-corner of L, we have proved (iv). 2 8.6.3 It is known (see [149]) that an operator space X is OS-nuclear if and only if it is locally reﬂexive (see 6.6.7 for one deﬁnition of this term), and X ∗∗ is OS-semidiscrete. This result should be compared with 6.6.8. Putting it together with 8.6.2, we see that an operator space X is OS-nuclear if and only if X is locally reﬂexive and X ∗∗ is injective. In light of the ﬁnite-dimensional case discussed in 8.6.1, and the fact that any OS-semidiscrete operator space is injective, and hence is a triple system, it is tempting to suppose that any OS-nuclear locally reﬂexive operator space is a triple system. We will argue below that this is not the case. 8.6.4 (The commutative case) It is interesting to consider the ‘commutative case’ of certain topics in this section, and chapter. If Y is a triple system with the property that xy ∗ z = zy ∗ x for all x, y, z ∈ Y , then we say that Y is a commutative triple system. These objects have been throughly analyzed in the JB ∗ -triple literature (e.g. see [225, 165, 379] and references therein). In fact, the following turns out to be essentially the only example of a commutative triple system. Suppose that Ω is a compact space, and that we have a continuous action of the circle T on Ω. Then H = {f ∈ C(Ω) : f (α · w) = αf (w) for all w ∈ Ω, α ∈ T}, is a triple subsystem of C(Ω), which is commutative in the sense above. Such a space H is called a Cσ -space. Conversely, every commutative triple system is completely isometric to a Cσ -space (this may be deduced, for example, from a fact about minimal operator spaces mentioned in the Notes for Section 8.3). There are a host of characterizations of Cσ -spaces scattered throughout the Banach space and JB ∗ -triple literature. We can add the following to the known list: Proposition 8.6.5 Let Y be a triple system. Then the following are equivalent: (i) Y is a commutative triple system, (ii) Y is a minimal operator space (see 1.2.21), (iii) Y = Y op (see 1.2.25 for this notation), (iv) Y ∗∗ is completely isometrically isomorphic, and w ∗ -homeomorphic, to an L∞ space, (v) Y Y and Y Y are commutative algebras.

350

Notes and historical remarks

Proof It is clear that (ii) implies (iii). Suppose (iii). If Y is a subtriple of a C ∗ -algebra A, we may view Y op as a subtriple of Aop . Since the identity map from Y to Y op is a complete isometry, it follows from 4.4.6 that we have (i). Supposing (i), then (v) follows by simple algebra. Also, we saw in 8.5.17 that Y ∗∗ is a triple system. It is easy to check by routine w ∗ -density arguments that Z = Y ∗∗ is a commutative triple system. It is well known that a commutative triple system Z with a predual is isometric to an L∞ space. The usual way that this is proved, is to show ﬁrst that any extreme point u of Ball(Z) is a unitary. That is, uu∗ z = zu∗ u = z, for any z ∈ Z. Then the map z → u∗ z is a complete isometry from Z onto Z Z. Thus the commutative C ∗ -algebra Z Z has a predual, and is therefore an L∞ space. This yields (iv). It is obvious that (iv) implies (ii). Finally, given (v), we view Y as an equivalence A-B-bimodule in the usual way, with A = Y Y and B = Y Y . If y, x1 , x2 , x3 , x4 ∈ Y , then (using the notation of 8.1.19) yx∗1 x2 x∗3 x4 = x2 x∗3 yx∗1 x4 = x2 x∗1 x4 x∗3 y = x4 x∗3 x2 x∗1 y. Hence if θ is as in 8.1.20, then θ(x∗1 x2 x∗3 x4 ) = x4 x∗3 x2 x∗1 ∈ A. Thus for any x ∈ Y we have θ(x∗ x)2 = (xx∗ )2 , and so θ(x∗ x) = xx∗ . By the polarization identity (1.1), θ(x∗ y) = yx∗ , for all x, y ∈ Y . This yields (i). 2 That there exist OS-nuclear operator spaces which are not triple systems, may now be seen as follows. We will use the well-known fact from Banach space theory that every Banach space, and hence every minimal operator space (see 1.2.21), is locally reﬂexive. Consequently, by 8.6.3, and using the fact from 1.4.12 that Min(E)∗∗ = Min(E ∗∗ ), saying that a minimal operator space X = Min(E) is OS-nuclear is the same as saying that X ∗∗ = Min(E ∗∗ ) is injective as an operator space. By (4.5), this is equivalent to E ∗∗ being an injective Banach space. However the injective Banach spaces are commutative C ∗ -algebras (see 4.2.11). Thus a minimal operator space X is OS-nuclear if and only if X ∗∗ ∼ = L∞ . ∗ 1 Using 1.4.12, the latter is equivalent to X being an ‘L -space’. By 8.6.5 and the remark above it, if E is an ‘L1 -predual’ which is not a Cσ -space, then Min(E) is a OS-nuclear operator space which is not a triple system. In fact there are extensive studies, in the Banach space literature, of interesting L1 -preduals which are not Cσ -spaces (e.g. see [242] and references therein). 8.7 NOTES AND HISTORICAL REMARKS C ∗ -modules over commutative C ∗ -algebras were introduced by Kaplansky [220]. The general case surfaced in the early 1970s in the work of Paschke (e.g. see [300, 302, 303]) and Rieﬀel (e.g. see [359–361, 363]). Sometimes they are called ‘Hilbert C ∗ -modules’, or simply ‘Hilbert modules’ (the latter term of course has a quite diﬀerent meaning in our book). Their use has become quite ubiquitous in C ∗ -algebra theory. In the 1980s, Connes, Kasparov, and many others, used C ∗ -modules as a tool at quite a sophisticated level. Thus some of the results in

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this chapter are essentially folklore, well known to the experts for a long time, but hard to date or attribute precisely. For more on the history of the subject, and for more detailed citations, see the texts cited at the start of this chapter. Frank has compiled an extensive listing of papers related to C ∗ -modules in [161]. Wittstock was the ﬁrst to explicitly consider the canonical operator space structure on a C ∗ -module [432]. In the late 1980s and early 1990s, this connection was developed by Hamana and Kirchberg from the TRO viewpoint. Indeed Kirchberg used the operator space structure of TROs in his amazing work from that period (see his papers cited in the reference list below, and references therein). The next contributions to the ‘operator space view’ of C ∗ -modules came in the early 1990s with the project [65], which led to the papers [46–52,64] and others. At much the same time, and independently, Magajna and Tsui became interested in C ∗ -modules from this perspective (see [264, 412], and references therein). Around 1999, interest in TROs picked up with the important paper [141]. As evidenced by the number of recent papers using them, it seems that TRO and C ∗ -module methods are playing an increasingly central role in operator space theory at the present time. Many of these papers are listed below. See also, for example, an amazing series of papers recently undertaken by Junge (e.g. see [207]), which among other things apply W ∗ -module techniques and free probability to study the operator Hilbert space and related topics. 8.1: Most of the results in Section 8.1 not attributed below to others, are due to Paschke or Rieﬀel. Our presentation here and in some later sections, beneﬁts from a frequent use of Cohen’s factorization theorem, and the linking algebra. Item (4) in 8.1.1 follows from (1)–(3), and the polarization identity (1.1). The deﬁnition given in 8.1.2 of a strong Morita equivalence is not quite Rieﬀel’s original one; 8.1.2 is essentially the formulation in [77]. Sometimes strong Morita equivalence is called Morita–Rieﬀel equivalence. It is easy to check that strong Morita equivalence is an equivalence relation (using, for example, 8.2.18 together with 8.2.12 (2) and (4)), and that ∗-isomorphic C ∗ -algebras are strongly Morita equivalent (using the Bσ construction in 8.3.6). In connection with 8.1.4 (2), if y ∈ Y then there is a z ∈ Y with y = zz|z (see [38, 387]). Result 8.1.5 is due to Paschke [302]. The observation 8.1.8 seems to appear ﬁrst in [239]. It seems likely that simple results such as 8.1.6, 8.1.21, and 8.1.22, were well known to a few experts well before they appeared in [47]. Frank noticed independently a variant of the proof of 8.1.6 at about the date of [47], but phrases it in terms of quasimultipliers [160]. We are grateful to Paschke for communicating to us the argument in 8.1.12, based on [302, Lemma 4.1]. We are not sure exactly when 8.1.12 dates to. The result 8.1.13 was certainly known at the beginning of the 1970s (see the cited papers of Paschke), although some of this is no doubt much older. Kasparov and Lin established 8.1.16 (3) in [221,254]. If one applies this result in the case that Y is a C ∗ -algebra B, and use the fact from 8.1.11 that KB (B) ∼ = B, then we deduce that BB (B) ∼ = M (B). This relation is more or less the well-known ‘double centralizer’ description of the multiplier C ∗ -algebra. In connection with 8.1.10, there are some other characterizations of

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Notes and historical remarks

bounded module maps in [56, 383]. The linking C ∗ -algebra is due to Brown et al. [76]; see also [77]. Strongly Morita equivalent commutative C ∗ -algebras are isomorphic: indeed the map θ in 8.1.20 restricts to a ∗-isomorphism between A and B (see the last part of the proof of 8.6.5). From 8.1.22, it follows easily that C ∗ -module direct sums are commutative and associative. Corollary 8.1.24 is a restatement of a common tool. Theorem 8.1.26 appeared ﬁrst in [47]; a simpler proof communicated to us by Kirchberg appears in [65, p. 41]. Theorem 8.1.27 appears in the standard sources on C ∗ -modules, and is quite useful for operator spaces. For example, it may be applied in the study of ﬁnitely generated operator modules (e.g. see [53, 54]). The following folklore results are often found together with the result 8.1.27: (a) Any algebraically ﬁnitely generated C ∗ -module is projective in the sense of pure algebra (see [203, Section IV.3]); (b) any two algebraically ﬁnitely generated C ∗ -modules which are algebraically A-isomorphic, are unitarily isomorphic; (c) any algebraically ﬁnitely generated and projective module Y over a C ∗ -algebra A, possesses an A-valued C ∗ -module inner product ·|·; and (d) if Y is as in (c), and if we write Z for Y with the inner product in (c), then any other such inner product on Y may be written as u(·)|u(·), for a bicontinuous module automorphism u of Z (in fact u and u−1 may be chosen in BA (Y )+ ). We give quick proofs of these results (a)–(d): For (a), note that as in the proof of 8.1.27 (2), Y is algebraically ﬁnitely generated over A if and only if it is also ﬁnitely generated over A1 . Thus by 8.1.27 (1) we have that Y is a complemented A1 -submodule of Cn (A1 ), from which it follows by algebra that it is projective as an A-module (in the sense of the algebra texts). For (b), we may suppose that A is unital, and, by 8.1.27 (1), that Y and Z are orthogonally complemented submodules of Cn (A) and Cm (A) respectively, and that g : Y → Z is a surjective one-to-one A-module map. We obtain an induced A-module map on Cn (A) which factors through g. However any A-module map from Cn (A) to Cm (A) may be viewed as left multiplication by an m × n matrix of A-module maps on A, and is therefore bounded. Thus g is bounded. Applying [423] 15.3.8 again, we obtain a unitary from Y onto Z. For (c), note that a ﬁnitely generated projective A-module, in the purely algebraic sense, may be taken to be simply a submodule Y of Cn (A), which is the range of an idempotent module map P : Cn (A) → Cn (A). Any such P may be viewed as left multiplication by a matrix of A-module maps on A, and therefore P is bounded by A.6.3. So Y is a closed submodule of Cn (A). Therefore Y may inherit the inner product from Cn (A). Finally, for (d), by (b) any other such inner product on Y equals u(·)|u(·), with u and u−1 continuous by 8.1.27 (2). Also, u∗ u(·)|· = |u|(·)||u|(·). By basic spectral theory in the C ∗ -subalgebra of B(Z) generated by u∗ u, we may ensure that |u| and |u|−1 are positive. See, for example, [162,355] for extensions of some parts of the last mentioned results. We remark that if two C ∗ -modules are A-isomorphic via a map f with f, f −1 adjointable, then they are unitarily isomorphic. This follows easily from the polar decomposition [423]. A pre-C ∗ -module is a right (say) A-module Y with an A-valued inner product

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satisfying (1)–(4) in 8.1.1. The completion of a pre-C ∗ -module in the associated norm is a C ∗ -module. We omit the easy details. We did not need this construction in our presentation of the theory, although it is an important tool. There is a bimodule version of the multiplier algebra, which has been studied by Echterhoﬀ and Raeburn [132], and others. This is often useful in generalizing results that are valid for modules over unital algebras. Recall that if Y is a full right C ∗ -module over B, or an equivalence A-B-bimodule, and if L(Y ) is the linking C ∗ -algebra, then we observed in 8.1.17 that M (L(Y )) may be taken to be BB (Y ⊕c B). We deﬁne the multiplier bimodule M (Y ) to be the 1-2-corner of this C ∗ -algebra. Clearly M (Y ) may be identiﬁed with BB (B, Y ) (recall that by 8.1.11 and 8.1.17, Y is identiﬁed canonically with K(B, Y )). Also M (Y ) is a right C ∗ -module over BB (B) = M (B), and similarly M (Y ) is a left C ∗ -module over B(Y ) ∼ = M (A)). One sees that Y is an M (A)-M (B)-submodule of M (Y ). It can be also shown M (Y ) ∼ = BA (Y¯ , A), and that if L(Y ) is represented nondegenerately on H ⊕ K as in 8.2.8, then M (Y ) may be viewed as a subset of the weak operator topology closure of Y in B(K, H) (which also equals the w ∗ -closure of Y ). Many facts about C ∗ -modules which we develop in this chapter, may be easily adapted to its multiplier bimodule. E.g. see [132, 355]. 8.2: The tensor products here are originally due to Rieﬀel. The ﬁrst part of 8.2.2 dates to Wittstock’s early study [432] of C ∗ -modules as operator spaces. The papers [46–52, 64, 65], particularly [47] and [65], are the source of the operator space aspects of most other results in this section. Some of these papers develop analogues for nonselfadjoint algebras of C ∗ -modules and strong Morita equivalence, as mentioned in 8.2.21, and they use decisively most of the theory in the ﬁrst half of the present text. Unfortunately we are not able to discuss this topic further here. We refer the interested reader to these papers for further results and references. See, for example, [166] for an application of the operator space viewpoint of the interior tensor product. A version of 8.2.4 was ﬁrst proved by Paulsen, in connection with [65]. See [65, 46, 355] for some generalizations of the material in 8.2.5–8.2.7. These led to our presentation here, which is not very novel. Parts of 8.2.6 are due to Kasparov; all of it is intimately related to the advances in [75, 76]. This result is closely tied to the subject of frames [162, 355]. Indeed frames have been used in C ∗ -module theory since the beginning (e.g. see [75]), and are usually called quasibases. Note that any C ∗ -algebra A is strongly Morita equivalent to K∞ (A) (via the equivalence bimodule and TRO C(A), for example). Thus the converse assertion in 8.2.7 follows from the facts, mentioned in the Notes for 8.1, that strong Morita equivalence is an equivalence relation coarser than ∗-isomorphism. One may avoid the use of 4.5.13 in 8.2.7 by using the ∗-isomorphisms K∞ (A) ∼ = K∞ (KB (Y )) ∼ = KB (C(Y )) ∼ = KB (C(B)) ∼ = K∞ (B), (using 8.1.15, 8.2.6 (4), and the second paragraph of 8.2.16). We believe that this argument is due to Blackadar. Similar remarks apply to the proof of 8.5.31. We could only ﬁnd 8.2.12 (3) in [47] in this generality. The operator space insights in

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Notes and historical remarks

8.2.25 were certainly known in the 1980s to Hamana and others. See also [285]. The correspondences in 8.2.24 and 8.2.25 may also be viewed as a special case of the ‘inducing procedure’ discussed in 8.2.19 and 8.2.22. For example, a closed ideal J of B may be regarded as an object in B OM OD, and the induced left A-module Y ⊗B J can be shown to be unitarily A-B-isomorphic to XJ = Y J. We may deduce from this, for example, that Y ⊗B (B/J) ∼ = Y /XJ . For much more on induced representations, particularly in settings involving groups, see, for example, [362, 356], and references therein. Strong Morita equivalence may also be characterized in terms of a functorial equivalence between the categories of C ∗ -modules. A proof may be found in [47]; another was shown to us by Skandalis. Beer showed in [30] that unital C ∗ algebras are strongly Morita equivalent if and only if they are Morita equivalent in the sense of algebra. See also [11, 51, 52]. As we mentioned in 8.2.14, if Y is a right C ∗ -module over a nondegenerate ∗ C -subalgebra B of B(K), then the structure of Y is closely linked to the norm on Y ⊗B K c . We give formulae for this norm which avoid mention of the inner product. Namely, if z ∈ Y ⊗ K c , then 1 zY ⊗B K c = sup ζk 2 2 β(y1 , . . . , yn ) : z = yk ⊗ ζk , k

k

the supremum taken over ﬁnite sums z =

k

yk ⊗ ζk , for yk ∈ Y, ζk ∈ K c , where

1 u(yk )∗ u(yk ) 2 = sup yk bk }, β(y1 , . . . , yn ) = sup k

k

the last two suprema being taken, respectively, over all u in Ball(BB (Y, B)), and (bk ) in Ball(Cn (A)). See [47] for more details. 8.3: The main source of results in this section are the papers of Hamana (e.g. see [189–191, 193]), and the papers of Kirchberg in the reference list below. See also [154]. Triple systems are particular examples of JB ∗ -triples, itself a vast ﬁeld in its own right. See the survey [379], for example. Various parts of 8.3.2 are in [197,198,225]. In connection with 8.3.5, Solel has proved that a surjective linear isometry between TROs extends to an isometry between the linking algebras, one of a tractable form [395]. Note that if π : A → B(H) is a ∗-homomorphism and if U is a unitary, then θ = U π(·) is a triple morphism. From 8.3.5 it is easy to prove a converse to this, and characterize triple morphisms from a C ∗ -algebra into a TRO (or into a C ∗ -algebra, or into B(K, H)). See, for example, [58, 226] for variants on this. From such a result, and 4.4.6, one may recover quickly some of the Banach–Stone theorems we have encountered in this text. See also the notes to Section 6.7 in [158]. Some of the unattributed material in this section, for example parts of 8.3.12, is from [53]. See that paper for more on the triple envelope and its properties. For example, it is shown there that if X is a Banach space then the triple envelope of Min(X) is triple isomorphic to the Cσ -space (see Section 8.6)

C ∗ -modules and operator spaces

355

{h ∈ C(S) : h(αψ) = αh(ψ) for all ψ ∈ S, α ∈ T}, where S is the w∗ -closure of the extreme points in Ball(X ∗ ). In this case, Theorem 8.3.9 may be rephrased in terms of the existence of a ‘Shilov boundary line bundle’ for any Banach space. See also [445] for the ﬁnite-dimensional case. Most of 8.3.12 (5) also comes from the latter paper. For a subspace X of M m,n , with m, n ﬁnite, the triple envelope (which coincides in this case with the injective envelope) is easy to describe. Indeed, analoguously to 4.3.7 (2), the subtriple of Mm,n generated by X is triple isomorphic to a ﬁnite direct sum of ‘rectangular matrix blocks’ (e.g. see [392]). Some of these blocks are redundant, just as in 4.3.7 (2). If one eliminates such blocks, then the remaining direct sum of blocks is the triple envelope of X. As we said earlier, [141] sparked much recent interest in TROs (e.g. see [222, 226, 289, 290, 293, 295, 376, 395], and [154], which was earlier). In many of these papers, facts about TROs are deduced via the linking C ∗ -algebra. The noncommutative Shilov boundary, or triple envelope, has been used extensively in [55, 56, 59], for example. As an illustration of the use of the techniques in this chapter in operator space theory, we brieﬂy discuss OS-nuclearity (see Section 7.1) of TROs. Most of what we say next is due to Kirchberg, and Kaur and Ruan (see [226]), although we have changed the proofs. We ﬁrst give an illustrative proof of an old result from [30], namely that Morita equivalence preserves nuclearity of C ∗ -algebras. We use ideas from [46, 131]. Suppose that Y is an equivalence A-B-bimodule, and that B is an OS-nuclear C ∗ -algebra. Hence B factors through ﬁnite-dimensional matrix algebras, as in 7.1.1. By 8.1.24, we obtain a nets of maps factoring Y through ﬁnite-dimensional matrix algebras over B. It follows from these facts, and using a doubly-indexed net, that Y is OS-nuclear. These nets easily yield completely contractive nets of maps factoring A ∼ = KB (Y ) through ﬁnite-dimensional matrix algebras over B (using, for example, 8.2.15 (1), and 8.2.12 (1), as in [46, p. 391]). It follows that A is OS-nuclear. By the same principle, and by 8.1.18, the linking algebra L(Y ) is OS-nuclear. Conversely, it is clear that if L(Y ) is OS-nuclear then so is Y . Finally, suppose that Y is OS-nuclear. We remarked in 8.1.25 that B factors through ﬁnite-dimensional matrix algebras over Y . Hence B is nuclear by the same arguments. 8.4: This material from [53] was part of the original development of the operator space multiplier theory (see the Notes to Section 4.5). One may vary the deﬁnition of Ml (X) in 4.5.1 by considering the set of linear maps u : X → X, such that there exists a linear complete isometry σ : X → Z into a right C ∗ module Z over a C ∗ -algebra B, and an a ∈ BB (Z) such that σ(u(x)) = aσ(x), for all x ∈ X. This new set of maps coincides precisely with the set in 4.5.1. Indeed the new set clearly contains all the maps considered in 4.5.1. Conversely, given such σ : X → Z and a ∈ BB (Z), we may view a as an element u (supported on the ‘1-1-corner’) of BB (Z ⊕c B). The latter, by 8.1.16 (3) and 8.1.17, is just LM (L(Z)). There is a canonical copy of X in (the 1-2-corner of) L(Z). Note

356

Notes and historical remarks

that u is a left multiplier of L(Z), and hence is in Ml (X), by 4.5.9. The operator space centralizer algebra is the set Al (X) ∩ Ar (X) of operators on X. This is a commutative C ∗ -algebra which is discussed thoroughly in [66, Section 7]. Multipliers between two diﬀerent operator spaces are discussed in [56]. 8.5: Selfdual modules over a W ∗ -algebra were ﬁrst deﬁned and investigated by Paschke [302]. Unfortunately, some of their theory has been relegated to folklore, which is a pity because they are a very powerful tool. The main sources for this section are [302, 303, 361], from which most of the results in this section come, which are not speciﬁcally attributed below to others. Our approach here uses the linking W ∗ -algebra extensively, and the operator space framework investigated in [48]. See the latter paper for some additional information. For further applications of the linking W ∗ -algebra to operator space theory, e.g. see [226]. Much of the proof in 8.5.4, 8.5.5, and 8.5.6 is unchanged if the module tensor product here is replaced by the Banach module projective tensor product. The latter was used in Paschke’s original approach [302]. Additional ideas for the proofs of these results come from [48, 54, 383, 444]. Results 8.5.6 and 8.5.7 are a weaker variant of a result of Zettl and Eﬀros et al. [444, 141]; see also the paper in preparation cited in the Notes to 4.7. The latter paper reverses the order, and uses the ingredients of these results to prove deep facts about operator space multipliers, and also to generalize W ∗ -modules to nonselfadjoint algebras. Note that 8.5.6 quickly follows from 4.4.10 (see [141, Theorem 2.6]). Weak Morita equivalence was called ‘Morita equivalence’ by Rieﬀel; some of the postulates in his deﬁnition have been dropped over time. See, for example, [30] for more on this topic. One can see easily, using 8.5.38 for example, that weak Morita equivalence is an equivalence relation coarser than ∗-isomorphism of W ∗ algebras. Indeed, suppose that Y is a W ∗ -equivalence M2 -M1 -bimodule. If K is a faithful Hilbert M1 -module, then by the ﬁrst paragraph of the proof of 8.5.38, the induced representation of M2 on Y ⊗M1 K is a faithful M2 -module in which the commutant of M2 is isomorphic to the commutant of M1 in B(K). By the same principle, if W is a W ∗ -equivalence M3 -M2 -bimodule, then W ⊗M2 Y ⊗M1 K is a faithful Hilbert M3 -module for which the commutant of M3 is isomorphic to the commutant of M2 , and hence also to the commutant of M1 in B(K). Now apply 8.5.38 to see that M1 is weakly Morita equivalent to M3 . For more on dual TROs, or 8.5.18, see for example [198, 226, 376]. The ‘Kaplansky density theorem’ in 8.5.18 is due to Harris [198]. By considering the linking algebra, it is clear that the w ∗ -closure of a subtriple of B(K, H) coincides with its WOT-closure [226]. We imagine that parts of 8.5.16, 8.5.17, 8.5.28, 8.5.29, and 8.5.31, may be described as folklore. That is, they do not appear in the older literature as far as we know, but nonetheless seem to be well known to some experts. For example, we could not ﬁnd 8.5.17 explicitly in the C ∗ -module literature, but is somewhat implicit in [77]. It also corresponds to a well known result in the JB ∗ -triple literature (see [379, 28], and references therein); and appears in a more explicit form in [53] (see also [141]). Item 8.5.19 is from [57]; and 8.5.20 should be compared with [28]. The ultraweak direct sum, and 8.5.22–

C ∗ -modules and operator spaces

357

8.5.25, are due to Paschke. A special case of 8.5.27 may be found in [56]. Results 8.5.28 and 8.5.29 are sketched in [53, Lemma 5.8]. Also, 8.5.31 is mentioned without proof in various places in the literature; in [303] Paschke proved the related result that if M and N are weakly Morita equivalent, then there is a cardinal I and projection e in MI (N ) with central cover 1, such that M∼ = eMI (N )e ∗-isomorphically. A more recent exposition of the separable case of 8.5.31 may be found in [376]. The W ∗ -algebra ‘cover’ in 8.5.30 has hitherto been useful because it can often help reduce a general operator space problem to a von Neumann algebra problem. More concretely, the following kind of principle is sometimes useful. Suppose that P and Q are properties that an operator space X (resp. an operator on X) may or may not have. Suppose that if a W ∗ -algebra (resp. a map on a W ∗ -algebra) has property P then it has property Q. Suppose that property Q descends to subspaces (resp. invariant subspaces). Finally, suppose that P ‘lifts’ to injective (or triple) envelopes, second duals, and ‘ampliﬁcations’ MI (X), for any cardinal I. Then it follows from the construction in 8.5.30 that if X (resp. a map on X) has property P then it has property Q. This technique was used in [53, 57]. The construction in 8.5.32 is adapted from Magajna’s explanation from [264] of some results from [361]. It follows from 8.5.32, taking A to be a W ∗ -algebra M , that every C ∗ -module Y over M is a w∗ -dense M -submodule of a W ∗ -module E over M . This latter W ∗ -module is called the selfdual completion of Y (see [302, 361]). A variation on the proof of 8.5.37 shows that E ∼ = M B(Y , M ) unitarily. We sketch a slight variant of Rieﬀel’s proof of 8.5.37 (1) (see [361, Section 6]): Deﬁne Φ : Y → B(K, Y ⊗M K) as in 8.5.11, so that Φ(y)∗ Φ(z) = y|z, for y, z ∈ Y , and Φ is w∗ -continuous. Since I w is a w∗ -closed ideal in M , there is a central projection e in M , and therefore also in R = M , such that I w = eM . Let K = eK, then the canonical map from B(K) to B(K ) restricts to a normal ∗-homomorphism from M onto the commutant of the copy of I w in B(K ). As in 8.5.32, the latter commutant is ∗-isomorphic to a certain commutant in B(H), where H = Y ⊗M K = Y ⊗I w K . Composing, we obtain a normal representation of R on H. By 8.5.37 (2), W = R B(K, H) is a right W ∗ -module over M . It is easily checked that Ran(Φ) is an M -submodule of W , which is w ∗ closed by A.2.5. If Ran(Φ) = W , then by 8.5.16, Ran(Φ) = pW , for a nontrivial orthogonal projection p. One may view p ∈ B(H), leading to the contradiction that [Φ(Y )K] = H. So Φ is a unitary M -module map onto W . Facts in 8.5.35 and 8.5.40 are from [48]. See [226] for other applications. The W ∗ -module tensor product is due to Rieﬀel (e.g. see [361]), as is 8.5.36. The argument there shows that any normal M -rigged N -module Y gives rise to a functor M N HM OD → N N HM OD, just as in 8.2.19. If Y is a W ∗ -equivalence N -M -bimodule, then it is easy to see that we obtain an equivalence of categories ∼ M N HM OD = N N HM OD. Conversely, Rieﬀel shows that if these categories are equivalent, then M is weakly Morita equivalent to N . Observation 8.5.38 is due to Connes (unpublished), as is the theory of correspondences (see 8.5.39). As is the case in much of this chapter, there are several alternative routes

358

Notes and historical remarks

through many of the results in this section. We have presented the route which we guess would be most useful for operator space applications. We sketch another attractive alternative route through some main results in this section: Suppose that Y is a w∗ -full right W ∗ -module over a W ∗ -algebra N ; or equivalently, suppose that Y is a W ∗ -equivalence M -N -bimodule. By 8.5.37 and its proof, we may take Y = R B(K, H) for faithful normal Hilbert modules K and H over N and M respectively. Here the commutants of N and M , in B(K) and B(H) respectively, are ∗-isomorphic (which may be proved in many ways), and we have written R for these commutants. Since R acts faithfully, by 3.8.6 and the remark after it, H and K are both ‘universal for NHMOD’ in the sense of Chapter 3. By a simple variant of 3.2.11 (4), there is a unitary R-module map U : K (J) → H (J) , for some cardinal J. Since the commutant of MJ (M ) is I ⊗ R, the map taking x ∈ MJ (M ) to U ∗ xU is a ∗-homomorphism from MJ (M ) into (I ⊗ R) = MJ (N ) = MJ (N ). By symmetry of the construction, this is a ∗-isomorphism MJ (M ) ∼ = MJ (N ). This yields 8.5.31, and also easily gives one direction of 8.5.38. We are indebted to Weaver for communicating this argument to us. Also, CJw (Y ) ∼ =

R B(K, H

(J)

)∼ =

R B(K, K

(J)

)∼ = CJw (R B(K, K)) = CJw (R ) = CJw (N ),

giving 8.5.28, and the Corollary preceding it. For a sample of the literature on W ∗ -modules and correspondences, see [101], [6,27,119,168,287,346], and references therein. Anantharaman-Delaroche, Kraus, Magajna, and Pop, have worked on questions concerning both operator spaces and correspondences. E.g. see [6,234,268,345], and references therein. There is a notion of Morita equivalence of ‘correspondences’ (see e.g. the recent papers of Muhly and Solel), and many results generalize to such contexts. Much of the theory of W ∗ -modules generalizes to selfdual C ∗ -modules over monotone complete C ∗ -algebras, as has been notably developed in [192]. This will be useful in operator space theory because injective C ∗ -algebras (and in particular the algebra I22 (X) studied in Sections 4.4 and 4.5) are monotone complete but are not usually W ∗ -algebras. 8.6: Theorem 8.6.2 is from [141], but the proof here highlights W ∗ -module results. A proof avoiding ‘stabilization’ may be found in [54]. In [68] it is shown that any injective operator space is a selfdual C ∗ -module. It follows that if Y is a right C ∗ -module over a C ∗ -algebra B, equipped with its canonical operator space structure, and if Y is injective as an operator space, then Y is selfdual over B, and BB (Y ) is an injective C ∗ -algebra. These results are related to results of Hamana and Lin [192, 255]. Equivalence bimodules over two commutative C ∗ algebras are well understood (e.g. see [356], and references therein). The last part of the proof of 8.6.5 we have seen in [387]. With a little more work, one may remove the word ‘completely’ in (iv) of 8.6.5. Preduals of L1 -spaces are called Lindenstrauss spaces, and they have an extensive literature (e.g. see [237], and references therein). The example of an OS-nuclear operator space which is not a TRO is due to Rosenthal (private communication).

Appendix

A.1

OPERATORS ON HILBERT SPACE

We begin by reviewing a few basic facts about operators on Hilbert space, which may be found in almost any book on functional analysis. A.1.1 If S, T are contractive linear operators between Hilbert spaces such that ST = I, then it follows that S = T ∗ , and T is an isometry and S a coisometry. If in addition T S = I, then T is a unitary. If P is an idempotent operator on a Hilbert space then P is a projection (i.e. P = P ∗ ) if and only if P ≤ 1. A.1.2 If H is a Hilbert space then the space S ∞ (H) of compact operators is a norm closed (two-sided) ideal in B(H). We write S 1 (H) for the usual trace class, a (two-sided) ideal in S ∞ (H) (and also in B(H)), and which is a Banach space with respect to the trace class norm T 1 = tr|T |. The trace tr is a contractive functional on S 1 (H), and via the dual pairing (S, T ) → tr(ST ) it is well-known that S ∞ (H)∗ ∼ = S 1 (H) and S 1 (H)∗ ∼ = B(H) isometrically. From this it is evident that the product on B(H), viewed as a map from B(H) × B(H) to B(H), is separately w ∗ -continuous. That is, if St → S in the w∗ -topology on B(H), then St T → ST and T St → T S in the w∗ -topology too. The w∗ -topology on B(H) is also called the σ-weak topology. A linear ∞ functional on B(H) is σ-weakly continuous if and only if it is of the form k=1 · ζk , ηk , ∞ 2 2 ζ and η ﬁnite. By such considerations, for ζk , ηk ∈ H with ∞ k k k=1 k=1 the involution ∗ on B(H) may also be seen to be w ∗ -continuous. A.1.3 More generally if H, K are Hilbert spaces, we let S ∞ (H, K) denote the compact operators from H into K. For any 1 ≤ p < ∞, we let S p (H, K) denote the Schatten p-class of compact operators T : H → K such that |T |p belongs 1 to S 1 (H). This is a Banach space for the norm T p = tr|T |p p . The following ideal property holds: for any T ∈ S p (H, K), V1 ∈ B(K), V2 ∈ B(H), the operator V1 T V2 belongs to S p (H, K) and V1 T V2 p ≤ V1 T pV2 . A.1.4 We write WOT for the weak operator topology. This topology makes the map T → T ζ, η continuous on B(H), for all ζ, η ∈ H. On bounded sets the WOT and σ-weak topologies coincide. Thus a bounded net in B(H) converges in the WOT topology if and only if it converges in the w ∗ -topology. A.1.5

A subspace X ⊂ B(H) is said to be reﬂexive if X = {T ∈ B(H) : T ζ ∈ [Xζ] for all ζ ∈ H}.

360

Duality of Banach spaces

We write T ∞ for the operator T ⊕ T ⊕ · · · on H (∞) , and X (∞) = {T ∞ : T ∈ X}. If W is a w∗ -closed subspace of B(H), then W (∞) is reﬂexive in B(H (∞) ). To see this, suppose that T ζ ∈ [W (∞) ζ] ⊂ [B(H)(∞) ζ], for all ζ ∈ H (∞) . By ﬁrst setting ζ = (0, · · · , 0, η, 0, · · · ), and then ζ = (η, · · · , η, 0, · · · ), it is easy to argue that T = S ∞ for some S ∈ B(H). Let ϕ ∈ W⊥ . By A.1.2, there exist vectors ζ, η ∈ H (∞) such that ϕ(R) = R∞ ζ, η, R ∈ B(H). Then ϕ(S) = T ζ, η = 0, since T ζ ∈ [W (∞) ζ], and ϕ ∈ W⊥ . Thus S ∈ (W⊥ )⊥ = W , so that T ∈ W (∞) . (E.g. see [354, 102, 108] for more on this topic.) A.2

DUALITY OF BANACH SPACES

In this section and the next, E and F are Banach spaces. We write iE : E → E ∗∗ for the canonical embedding. However we often suppress this map and simply consider E as a subspace of E ∗∗ . Lemma A.2.1 (Goldstine) Ball(E) is w ∗ -dense in Ball(E ∗∗ ). Lemma A.2.2 Let u : E → F ∗ be a bounded linear map. Then there exists a unique w∗ -continuous u ˜ : E ∗∗ → F ∗ extending u. Moreover ˜ u = u. Proof Set u ˜ = i∗F ◦ u∗∗ where iF : F → F ∗∗ is the canonical isometry.

2

Lemma A.2.3 Let E be a closed linear subspace of a Banach space F . w∗ (1) As subsets of F ∗∗ we have E = E ⊥⊥ . (2) The second dual of the inclusion map E → F is an isometry from E ∗∗ onto E ⊥⊥ . Thus E ∗∗ ∼ = E ⊥⊥ isometrically, via this canonical isometry. ∗∗ ∼ ∗∗ (3) (F/E) = F /E ⊥⊥ isometrically, and w∗ -w∗ -homeomorphically, via the ‘transpose’ of the canonical isomorphism (F/E)∗ → E ⊥ . This is the same as the map obtained from q ∗∗ , where q : F → F/E is the canonical quotient map, by factoring out Ker(q ∗∗ ) = E ⊥⊥ . (4) F ∩ E ⊥⊥ = E. Proof Items (1), (2), and (3) are in the standard sources. For (4), notice that / E, choose ϕ ∈ F ∗ such that ϕ(E) = 0, but E ⊂ E ⊥⊥ ⊂ F ∗∗ . If y ∈ F but y ∈ ⊥⊥ there exists by (1) a net (xt )t in E with xt → y in the ϕ(y) = 0. If y ∈ E 2 w∗ -topology of F ∗∗ . Hence 0 = ϕ(xt ) → ϕ(y) = 0, a contradiction. We will also (silently) use the following simple principle many times: Lemma A.2.4 If T : E → F is a w∗ -continuous map between dual Banach spaces, and if W is a w∗ -closed subspace of Ker(T ), then the induced map from E/W to F is w∗ -continuous. Theorem A.2.5 (Krein–Smulian) (1) Let E be a dual Banach space with predual E∗ , and let F be a linear subspace of E. Then F is w∗ -closed in E if and only if Ball(F ) is closed in the w ∗ topology on E. In this case F is also a dual Banach space, with predual E∗ /F⊥ , and the inclusion of F in E is w∗ -continuous.

Appendix

361

(2) If u ∈ B(E, F ), where E and F are dual Banach spaces, then u is w ∗ continuous if and only if whenever xt → x is a bounded net converging in the w∗ -topology in E, then u(xt ) → u(x) in the w∗ -topology. (3) Let E and F be as in (2), and u : E → F a w ∗ -continuous isometry. Then u has w∗ -closed range, and u is a w ∗ -w∗ -homeomorphism onto Ran(u). Proof Items (1) and (2) may be found in the standard texts; (2) is often stated for functionals ϕ but the result as stated here follows from this by considering ϕ ◦ u. For (3), note that it is easy to check using (1) that Ran(u) is w ∗ -closed in F . Thus the restriction of u to Ball(E) takes w ∗ -closed (and thus w∗ -compact) sets to w∗ -compact (and thus w∗ -closed) sets in Ran(u). Thus the inverse of u 2 restricted to the ball is w∗ -continuous, so u−1 is w∗ -continuous by (2). A.3

TENSOR PRODUCTS OF BANACH SPACES

We review a few facts about tensor products of Banach spaces E and F , whose proofs may be found in many texts (see [118, 121, 324, 407], for example). A.3.1 If (xk )k and (yk )k are ﬁnite families in E and F respectively, then one may deﬁne for z = k xk ⊗yk in the algebraic tensor product E ⊗F , the quantity

xk ⊗ yk = sup ϕ(xk )ψ(yk ) : ϕ ∈ Ball(E ∗ ), ψ ∈ Ball(F ∗ ) . k

∨

k

This is a norm on E ⊗ F . The completion of E ⊗ F in this norm is called the ˇ . This tensor norm gets its name injective tensor product and written as E ⊗F from the fact that it has the injective property. Namely, if ui : Ei → Fi are ˇ 2 → F1 ⊗F ˇ 2 isometries for i = 1, 2, then the corresponding map u1 ⊗ u2 : E1 ⊗E is an isometry too. More generally if u1 , u2 are contractive then so is u1 ⊗ u2 . We remark that the deﬁnition and facts in the last paragraph have obvious ˇ · · · ⊗X ˇ N of any N -tuple of Banach variants for the N -fold injective product X1 ⊗ spaces. There is an associativity law: for example, if N = 3 then we have that ˇ X3 = X1 ⊗ ˇ X2 ⊗ ˇ X3 = (X1 ⊗ ˇ X2 ) ⊗ ˇ ˇ X1 ⊗ (X2 ⊗ X3 ). We may identify any element z = k xk ⊗ yk as above with a bounded operator u : F ∗ → E, namely u(ψ) = k ψ(yk )xk for any ψ ∈ F ∗ . We say that u is associated with z. Under this identiﬁcation, E ⊗ F coincides with the space of all ﬁnite rank and w∗ -to-norm continuous operators from F ∗ into E. Clearly ˇ → B(F ∗ , E) E ⊗F

isometrically.

(A.1)

ˇ ⊂ B(E ∗ , F ) isometrically too. Likewise if F is a dual space with Of course E ⊗F predual F∗ , we may identify E ⊗ F with ﬁnite rank operators from F∗ into E ˇ → B(F∗ , E) isometrically. and we have E ⊗F A.3.2 Let Ω be a compact space. We let C(Ω; E) denote the Banach space of all continuous functions f : Ω → E. Equip C(Ω; E) with the supremum norm, that is, f = sup{f (t)E : t ∈ Ω}. We simply write C(Ω) for C(Ω; C). We may

362

Tensor products of Banach spaces

identify C(Ω) ⊗ E with a subspace of C(Ω; E) by regarding any f = k gk ⊗ xk (with gk ∈ C(Ω) and xk ∈ E) as afunction, f (t) = k gk (t)xk . Then the norm of f in C(Ω; E) is equal to sup{| k gk (t)ϕ(xk )| : t ∈ Ω, ϕ ∈ Ball(E ∗ )}, hence is equal to its injective tensor norm by (A.1). Moreover C(Ω) ⊗ E is dense in C(Ω; E) (e.g. see [407, IV; 7.3] for a proof) hence we have ˇ ∼ C(Ω)⊗E = C(Ω; E)

isometrically.

(A.2)

Similarly if Ω is a locally compact space, we let C0 (Ω; E) denote the Banach space of all continuous functions from Ω to X vanishing at ∞, equipped with ˇ ∼ the supremum norm. Then (A.2) extends to the relation C0 (Ω)⊗E = C0 (Ω; E). A.3.3 A bounded bilinear map T : E × F → Z is a bilinear map for which there is a constant C such that T (x, y) ≤ Cxy, for all x ∈ E, y ∈ F . The least such C is written as T . We say that T is contractive if T ≤ 1. ˆ is the completion of the The Banach space projective tensor product E ⊗F algebraic tensor product E ⊗ F in a certain norm. We do not need to explicitly write down this norm, instead we will simply state the universal property ˆ , namely that it linearizes bounded bilinear maps. More precisely, the of E ⊗F ˆ is a contractive bilinear map, and for any canonical map ⊗ : E × F → E ⊗F bounded bilinear T : E × F → Z, the associated linear map E ⊗ F → Z is continuous with respect to the just mentioned norm, and extends to a bounded ˆ → Z with T˜ = T . From this it is easy to see that linear map T˜ : E ⊗F ˆ Z) ∼ B(E ⊗F, = B(E, B(F, Z)) ∼ = B(F, B(E, Z))

isometrically.

(A.3)

In particular, via the obvious isomorphisms, ˆ )∗ ∼ (E ⊗F = B(F, E ∗ ) = B(E, F ∗ ) ∼

isometrically.

(A.4)

A.3.4 A bounded operator u : E → F is said to be 2-summing if I2 ⊗ u extends ˇ into 2 (F ). We set to a bounded operator from 2 ⊗E ˇ −→ 2 (F ). π2 (u) = I2 ⊗ u : 2 ⊗E It is not hard to check that π2 (· ) is a (complete) norm on the space Π2 (E, F ) of 2-summing operators from E into F . We also note that π2 (u) is the supremum ˇ → 2n (F ), for n ∈ N. of the norms of the mappings I2n ⊗ u : 2n ⊗E A.3.5 We review three tensor products related to Hilbert space factorization. Let (ek )k denote the canonical basis of 2 . For z ∈ E ⊗ F deﬁne n γ2 (z) = inf ek ⊗ xk k=1

ˇ 2 ⊗E

n ek ⊗ yk k=1

ˇ 2 ⊗F

(A.5)

where inﬁmum is over all ﬁnite families (xk )nk in E and (yk )nk in F such that the n z = k=1 xk ⊗ yk . The quantity γ2 is a norm on E ⊗ F , and we let E ⊗γ2 F denote the resulting completion.

Appendix

363

n ∗ 2 Consider k=1 ek ⊗ xk for x1 , . . . , xn ∈ E as above, and let u : E → be n ⊗x is equal to the associated linear map. According to (A.1), ˇ k=1 e kn k 2 ⊗E the usual operator norm of u. Analoguously, we deﬁne k=1 ek ⊗ xk Π2 (E ∗ ,2 ) to be π2 (u). Then for z ∈ E ⊗ F we deﬁne n ek ⊗ xk g2 (z) = inf

Π2 (E ∗ ,2 )

k=1

n ek ⊗ yk

(A.6)

ˇ 2 ⊗F

k=1

n where the inﬁmum is over ways to write z = k=1 xk ⊗ yk , with xk ∈ E, yk ∈ F . As above, g2 is a norm, and we let E ⊗g2 F denote the resulting completion. Next we let n ek ⊗ xk γ2∗ (z) = inf k=1

Π2 (E ∗ ,2 )

n ek ⊗ yk k=1

Π2 (F ∗ ,2 )

,

(A.7)

n where again the inﬁmum over all ways to write z = k=1 xk ⊗ yk in E ⊗ F . Again, γ2∗ is a norm and we let E ⊗γ2∗ F denote the resulting completion. It is not hard to see that · ∨ ≤ γ2 (·) ≤ g2 (·) ≤ γ2∗ (·) ≤ · ∧ on E ⊗ F . ¯ of ﬁnite rank operators A.3.6 Let H, K be Hilbert spaces. Then the space K ⊗ H ∞ 1 from H to K is dense both in S (H, K) and in S (H, K). It is clear from (A.1) ¯ and it turns out that ˇ H, that S ∞ (H, K) ∼ = K⊗ ¯ ¯ ∼ ˆH S 1 (H, K) ∼ = K⊗ = K ⊗γ2∗ H. A.4

(A.8)

BANACH ALGEBRAS

We refer the reader in this section to [74, 106, 297] for any omitted proofs of assertions below, or for more background. A.4.1 A C-Banach algebra is a Banach space A which is also an algebra such that ab ≤ Cab for all a, b ∈ A. If C = 1 then we simply say Banach algebra. We say that a Banach algebra A is unital if it has a unit (i.e. identity) of norm 1. A bounded approximate identity is a bounded net (et )t with et a → a and aet → a. This is a contractive approximate identity (cai) if moreover e t ≤ 1 for all t. If A possesses a cai we say that A is approximately unital. A.4.2 Let A be a unital Banach algebra. A state on A is a contractive unital functional on A. An element h ∈ A is said to be Hermitian if ϕ(h) ∈ R for every state ϕ on A. Note that by the Hahn–Banach theorem we may replace states on A here by contractive unital functionals on Span {1, h}. Equivalently, h is Hermitian if exp(ith) ≤ 1 for all t ∈ R. We write Her(A) for the set of Hermitian elements of A. By the ﬁrst deﬁnition of Hermitians above, it is evident that if u : A → B is a unital contractive linear map between unital Banach algebras, then u(Her(A)) ⊂ Her(B).

C ∗ -algebras

364

It is well-known that if ϕ(h) = 0 for all states ϕ on A, then h = 0. Also, if A is a unital C ∗ -algebra, then Her(A) is exactly the set Asa of selfadjoint elements. A nonzero homomorphism on a unital Banach algebra is a state, and is called a character. The maximal ideal space MA of a commutative unital Banach algebra A is the set of characters of A, together with the w ∗ -topology inherited from A∗ . A.4.3 Suppose that A is an approximately unital Banach algebra. We may deﬁne a unitization of A by considering the canonical ‘left regular representation’ λ : A → B(A), and identifying A + C1 with the span of λ(A) + CIA , which is easy to see is a unital Banach subalgebra of B(A). Thus if a ∈ A and α ∈ C then a + α1 = sup ac + αc : c ∈ A, c ≤ 1 . (A.9) We write this unitization as A1 if A is nonunital. It is occasionally useful that there are some other equivalent expressions for the quantity above. For example, if (et )t is a cai for A then a + α1 = lim aet + αet = sup aet + αet . t

(A.10)

t

To see that this limit exists and that these quantities are the same, let β be the quantity on the right-hand side of (A.9), let > 0 be given, and choose c ∈ Ball(A) with ac + αc > β − . Then aet c + αet c → ac + αc, so that there is a t0 with aet c + αet c > β − for t ≥ t0 . Then β − < aet c + αet c ≤ aet + αet ≤ β. This proves what we asserted. One can see that one does not change the quantities in (A.9) and (A.10) by considering expressions ca + αc or et a + αet . As a consequence of (A.10), if π : A → B is a contractive homomorphism between Banach algebras, such that (π(et ))t is a cai for B for some cai (et )t for A, then π extends uniquely to a contractive unital homomorphism π ˜ between the unitizations. To see this deﬁne π ˜ (a + α1) = π(a) + α1, for a ∈ A and α ∈ C, and appeal to formula (A.10) twice to see that π ˜ is contractive too. If, further, π is isometric, then by (A.10) it follows that π ˜ is isometric too. A.5

C ∗ -ALGEBRAS

We refer the reader in this section to any book on C ∗ -algebras for any omitted proofs of assertions below, or for more background. A.5.1 A concrete C ∗ -algebra is a closed ∗-subalgebra of B(H) for a Hilbert space H. A von Neumann algebra is a concrete C ∗ -algebra which is closed in the w∗ topology on B(H), and which contains IH . By A.1.2 it follows that the product on a von Neumann algebra is separately w ∗ -continuous in each variable, and that the involution is also w∗ -continuous. An (abstract) C ∗ -algebra is a Banach algebra A with a conjugate linear involution ∗ : A → A such that (a∗ )∗ = a and (ab)∗ = b∗ a∗ , for all a, b ∈ A, which also satisﬁes the C ∗ -identity: a∗ a = a2 , for a ∈ A. A C ∗ -subalgebra of a C ∗ -algebra is a closed selfadjoint subalgebra.

Appendix

365

A.5.2 An element a in a C ∗ -algebra A is positive if a = b∗ b for some b ∈ A. We write A+ for the set of such elements, and write a ≤ b if b − a ∈ A+ , and if a and b are selfadjoint. We will assume familiarity with the basic properties of this ordering and the continuous functional calculus for normal operators. A C ∗ -algebra is approximately unital, and indeed has a positive increasing cai. The unitization A1 in A.4.3 of a C ∗ -algebra A is a C ∗ -algebra. A functional ϕ ∈ A∗ is called a state if it is ‘positive’ (i.e. ϕ(a) ≥ 0 if a ≥ 0) and has norm 1. This is equivalent to other deﬁnitions of states elsewhere in this book. Indeed there is a host of equivalent deﬁnitions of states, or indeed of elements in A+ . For example, a≥0

⇔

ϕ(a) ≥ 0

for all states ϕ of A (or of A1 ).

(A.11)

A ∗-homomorphism (resp. ∗-isomorphism) is a homomorphism (resp. isomorphism) satisfying π(a∗ ) = π(a)∗ for all a. Such maps are automatically positive. A.5.3 A primary result in the subject of operator algebras is the Gelfand– Naimark theorem, which states that every abstract C ∗ -algebra A is ∗-isomorphic to a concrete C ∗ -algebra. A major part of the proof of this result is the Gelfand– Naimark–Segal (GNS) construction, which shows that the positive functionals ϕ on a C ∗ -algebra A are the functions of the form π(·)ζ, ζ, for a ∗-homomorphism π : A → B(H), and a vector ζ ∈ H, such that [π(A)ζ] = H. If ϕ is a state then we can take ζ = 1. A W ∗ -algebra is an abstract C ∗ -algebra with a Banach space predual, such that there exists a w∗ -continuous ∗-isomorphism from A onto a von Neumann algebra. By a theorem of Sakai, the last part of this deﬁnition is redundant, but we will avoid using this deeper fact here. A.5.4 The commutative C ∗ -algebras, are by a theorem of Gelfand, exactly the spaces C0 (Ω) mentioned in A.3.2. We will assume that the reader is familiar with the basic correspondences between constructions in the category of compact spaces K, and the category of commutative unital C ∗ -algebras. For example, the correspondences between closed subsets of K, and ideals of C ∗ -algebras and their quotients. Or, more generally, the correspondence between continuous (resp. and one-to-one, and surjective) functions between compact spaces, and unital (resp. and surjective, and one-to-one ) ∗-homomorphisms between commutative unital C ∗ -algebras. These correspondences follow easily from Gelfand’s theory. Recall that algebraic isomorphisms between C(K)-spaces (resp. between closed subalgebras of C(K)-spaces containing constant functions), are ∗-isomorphisms (resp. isometric). A.5.5 The universal representation πu : A → B(Hu ) of a C ∗ -algebra A is constructed by taking a direct sum of ∗-homomorphisms associated with all the states on A by the GNS construction (see A.5.3 above). Thus πu has the property that for any state ϕ of A, there exists a unit vector ξ ∈ Hu such that ϕ = πu (·)ξ, ξ.

C ∗ -algebras

366

Every nondegenerate ∗-homomorphism from A to B(H), for any Hilbert space H, is unitarily equivalent to the restriction to an invariant subspace of a direct sum of suﬃciently many copies of πu . Theorem A.5.6 The second dual of a C ∗ -algebra A is a W ∗ -algebra. Indeed A∗∗ is linearly isometric, via a w ∗ -continuous map, to a von Neumann algebra. Proof Let πu : A → B(Hu ) be the universal representation of A (see A.5.5). Let π u : A∗∗ → B(Hu ) be the unique (contractive) w ∗ -continuous extension of u = i∗ ◦ πu∗∗ , where i : S 1 (Hu ) = B(Hu )∗ → B(Hu )∗ is the πu as in A.2.2 (thus π canonical injection). Let ψ be a linear functional on A of norm 1. It is well-known that we may write ψ = π(·)ξ, ξ , for unit vectors ξ, ξ and a ∗-homomorphism π (see 1.2.8 for an easy proof of this fact; also Zsido has recently shown us a beautiful simple proof that will be in [401]). For a ∈ A we have that |ψ(a)|2 ≤ π(a)ξ2 = π(a∗ a)ξ, ξ, from which we see that the positive map ϕ = π(·)ξ, ξ has norm 1, and is therefore a state on A. Thus by A.5.5, ϕ = πu (·)η, η for a unit vector η ∈ Hu . We now have |ψ(a)|2 ≤ πu (a∗ a)η, η = πu (a)η2 . Thus the functional πu (a)η → ψ(a) on [πu (A)η] is well deﬁned and contractive. By the Riesz representation theorem, there exists a vector ζ ∈ Hu such that ψ = πu (·)η, ζ. Hence, since A πu (ν)η, ζ for ν ∈ A∗∗ . We deduce that is w∗ -dense in A∗∗ , we have ν, ψ = u is an isometry. Since A is w∗ -dense |ν, ψ| ≤ πu (ν), which implies that π in A∗∗ , we have π u (A∗∗ ) ⊂ πu (A) ∗

w∗

. By the Krein–Smulian theorem A.2.5, the

range of π u is w -closed. Hence πu (A)

w∗

⊂π u (A∗∗ ), and so πu (A)

w∗

Thus A∗∗ is linearly isometric to the von Neumann algebra πu (A)

w∗

=π u (A∗∗ ). .

2

A.5.7 By the last result together with an observation in A.5.1, we see that if A is a C ∗ -algebra then A∗∗ possesses a product extending that of A, which is separately w∗ -continuous in each variable, and with respect to which A∗∗ is a W ∗ -algebra. By Goldstine’s lemma A.2.1 such a product on A∗∗ must be unique. We call this the canonical W ∗ -algebra structure on A∗∗ . Proposition A.5.8 Let π : A → B be a homomorphism between C ∗ -algebras. Then π is contractive if and only if π is a ∗-homomorphism. If these hold then ﬁrst, π has closed range, and induces a ∗-isomorphism between the C ∗ -algebras A/Ker(π) and π(A). Second, π is isometric if and only if π is one-to-one. Proof Most of these may be found in any book on C ∗ -algebras (and will be generalized in 8.3.2). We merely sketch the part that cannot be so easily found, namely that a contractive homomorphism π is a ∗-homomorphism. If A is unital then we can assume that B is unital and that π(1) = 1 (otherwise replace B by π(1)Bπ(1)). If a ∈ A+ and if ϕ is a state on B, then ϕ ◦ π is a state on A. Using (A.11) twice shows that ϕ(π(a)) ≥ 0, and that π(a) ≥ 0. Thus π is positive, and is therefore also a ∗-homomorphism. In the nonunital case extend π

Appendix

367

to a contractive homomorphism between the unitization C ∗ -algebras (or simply consider π ∗∗ : A∗∗ → B ∗∗ , and use the results on second duals above), and then apply the ‘unital case’ above to obtain the result. 2 Theorem A.5.9 [409, 39] Let A be a C ∗ -algebra, B a Banach algebra, and π : A → B a contractive homomorphism. Then π(A) is norm closed, and it possesses an involution with respect to which it is a C ∗ -algebra. Moreover, π is then a ∗-homomorphism into this C ∗ -algebra. If π is one-to-one then π is an isometry. Proof We may assume that B is the closure of π(A). If A is unital then B is unital, and π(1A ) = 1B . Since π(Her(A)) ⊂ Her(B) (see A.4.2), π(A) = π(Her(A)) + iπ(Her(A)) ⊂ Her(B) + iHer(B) ⊂ B. Hence Her(B)+ iHer(B) is dense in B. By the Vidav–Palmer theorem [74,298] B is a C ∗ -algebra (in fact Her(B) + iHer(B) is always norm closed [297, Theorem 2.6.7]). Thus we may use A.5.8 if necessary to obtain the stated conclusions. If A is not unital, we may conclude as in the proof of A.5.8 (here one may use A.4.3, for example, to extend π to suitable unitizations). 2 A.5.10 The following simple principle is useful in proving the Kaplansky density theorem, and variants of it. Namely, suppose that E is a closed subspace of a dual space F ∗ , and that we wish to prove that Ball(E) is w ∗ -dense in the unit ball ˜ : E ∗∗ → F ∗ of its w∗ -closure. By A.2.2, there is a w ∗ -continuous contraction u ∗ extending the inclusion map u : E → F . Suppose that u ˜ takes the open unit ball u). This is often automatic, as in the case of E ∗∗ onto the open unit ball of Ran(˜ that E is a ∗-subalgebra of a von Neumann algebra F ∗ (this follows by A.5.8; in this case u˜ is a homomorphism by a standard w ∗ -density argument using the separate w∗ -continuity in A.5.7 and A.5.1). It is easy to check, by A.2.5 (1), that w∗

u). Indeed, if z ∈ Ran(˜ u), with z < 1, Ran(˜ u) is w∗ -closed. Thus E = Ran(˜ ˜(η) = z and η < 1. By A.2.1, there is a net then there exists η ∈ E ∗∗ with u (et )t in Ball(E), converging in the w∗ -topology to η. Thus et = u(et ) → u ˜(η) = z in the w∗ -topology of F ∗ . It is quite obvious that this implies that Ball(E) is w∗ -dense in the unit ball of E A.6

w∗

.

MODULES AND COHEN’S FACTORIZATION THEOREM

Again, see [106,297,321] for omitted proofs, complementary results, and history. A.6.1 Reﬂecting on the diﬀerence between algebras and rings, it is natural that this diﬀerence should be reﬂected in the modules over each. For us, a (left) module over an algebra A will always be a vector space X over C, which is a (left) module in the traditional sense, but we insist also that (αa)x = a(αx) = α(ax),

a ∈ A, α ∈ C, x ∈ X.

There is then a one-to-one correspondence between left A-modules, and representations of A on vector spaces (that is, homomorphisms π from A into the

368

Modules and Cohen’s factorization theorem

algebra of linear maps on X, for a vector space X). This correspondence is given by the formula π(a)(x) = ax, a ∈ A, x ∈ X. We call π the canonical homomorphism associated with the module action. Now suppose that X is a (left, say) A-module over a Banach algebra A. We say that X is a normed (resp. Banach) A-module if X is a normed (resp. Banach) space, and the module action A × X → X is a contractive bilinear map. This is the same as saying that the associated homomorphism π : A → B(X) in the last paragraph, is contractive. We shall not do so here, but for many results (such as those in Chapter 5) one may want to weaken the last condition to allow bounded module actions. An A-B-bimodule is a left A-module which is also a right B-module, such that the two actions commute. That is, a(xb) = (ax)b for a ∈ A, b ∈ B, x ∈ X. A Banach A-B-bimodule is a Banach space and a bimodule, which is both a left and a right Banach module. We say that a (left) Banach A-module X is a nondegenerate A-module, or that A acts nondegenerately on X, if X equals the norm closure of the linear span of the products ax for a ∈ A and x ∈ X. If A has a bounded approximate identity (et )t then this is equivalent to saying that et x → x for all x ∈ X, and we will see some other equivalent conditions in A.6.4 below. A bimodule will be called nondegenerate if it is nondegenerate both as a left and a right module. Theorem A.6.2 (Cohen’s factorization) Suppose that A is a Banach algebra with a bounded approximate identity, and that X is a left Banach A-module. Then X is a nondegenerate A-module if and only if any x ∈ X may be written in the form x = ay for some a ∈ A, y ∈ X. In this case, if further A has a cai, and if x has norm < 1, then we may also choose a and y with norm < 1. Corollary A.6.3 Let A, X be as in A.6.2. Then every A-module map f : A → X is bounded. Proof Take a sequence an → 0 in A. Let Y = c0 (A), the set of sequences converging to 0 with terms in A. This is a right Banach A-module, which is not hard to see is nondegenerate. Applying Cohen’s theorem we may write an = bn b, 2 for b, bn ∈ A with bn → 0. Thus f (an ) = bn f (b) → 0. A.6.4 If X is a (not necessarily nondegenerate) left Banach A-module over a Banach algebra A then we deﬁne the essential part of X to be the norm closure of the linear span of the products ax for a ∈ A and x ∈ X. If A has a bounded approximate identity (et )t then this set is clearly exactly the set of x ∈ X such that et x → x. Clearly in this case the essential part is a nondegenerate A-module. Thus in this case (i.e. if A has a bounded approximate identity), it follows from Cohen’s theorem applied to the essential part, n that the essential part of X equals {ax : a ∈ A, x ∈ X}, and also equals { k=1 ak xk : n ∈ N, ak ∈ A, xk ∈ X}. Thus our use of the notation AX in this book (see Section 1.1), at least in this case, is less ambiguous than it may seem.

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Index

2-summing, 362 Adjoint module, 297 Adjoint operator space, X , 12, 60, 109 Adjointable maps, B(Y, Z), K(Y, Z), 299 Adjointable multipliers, Al (X), 172, 329 Algebraically ﬁnitely generated, 308 Amenability, 283, 285 Ampliﬁcation, 4, 13 Approximate identity, 363 Approximately unital, 49, 363 Arens product, 78 Arens regular, 79 Balanced bilinear map, 119 Banach module (or bimodule), 368 Banach space injective tensor product, ˇ , 361 E ⊗F Banach space projective tensor product, ˆ , 362 E ⊗F Banach–Stone theorems, 173, 191, 328 Bimodule Paulsen system, 128 Boundary representation, 150 Cσ -space, 349 C*-algebra, 5, 364 C*-cover, 50 C*-envelope, Ce∗ (·), 156 C*-extension, 150 C*-module, 297 Cai (contractive approximate identity), 49, 363 Centralizer algebra, Z(E), 131, 172 CES-representation, 117 Choquet boundary, Ch(E), 148 Cogenerator, 111 Commutative triple system, 349 Complemented submodule, 274 Complete order isomorphism or injection, 17 Completely bounded bimodule maps, A CBB (X, Y ), 124 Completely bounded maps, CB(X, Y ), 9 Completely bounded module maps, A CB(X, Y ), CBB (X, Y ), 124 Completely bounded multilinear map, 30

Completely bounded norm, cb , 4 Completely positive, 17 Conditional expectation, 155, 166 Corner of a map, 87 Corner of an algebra, 87 Corner-preserving, 87 Correspondence, 345, 347 Countably generated module, 311 Decomposable maps, DEC(A, B), 256 Decomposable norm, dec , 257 Delta norm, δ, 248 Delta tensor product, X ⊗δ B, 249 Diagonal, ∆(·), 19, 50 Direct limit, 67 Direct sum, 3, 8, 26, 57, 105, 106, 110, 139 Direct sum C ∗ -module, ⊕ci Yi , 300 Dirichlet algebra, 159 Disc algebra, A(D), 59 Double commutant property, 114 Dual operator algebra, 88, 92 Dual operator module (or bimodule), 136, 183 Dual operator space, 10, 22, 39 Enveloping operator algebra, O(B), 71 Equivalence bimodule, 297, 336 Essential extension, 153 Essential ideal, 85 Essential module maps, A CB ess (X, Y ), ess (X, Y ), 124 CBB Essential part of a module, 368 Extension of an operator space, 153 Extremal module actions, 177 Free product, A ∗ B, 75 Full C ∗ -module, 297 Function module, 131, 135 Function multiplier algebra, M(E), 131, 172 Function-extension, 148 Generator, 111 h-module (or bimodule), 103 Haagerup tensor product, X ⊗h Y , 30

386

Index

Hardy space, H ∞ (D), 91 Hermitian, 363 Hilbert column space, H c , 11 Hilbert module, HM OD, 105, 109, 110 Hilbert row space, H r , 11 Hilbert space factorization, 362 I11 , I22 , I(S(X)), 162 Ideal, 66 Induced representation, 319, 320, 345 Injective bimodule, 129 Injective envelope for bimodules, 179 Injective envelope, I(X), 153, 154 Injective operator space, 7 Interior tensor product, Y ⊗A Z, 315 Interpolation, 15, 66 Jointly completely bounded, 35 Left multiplier algebra, LM (A), 82, 84 Left multipliers of an operator space, Ml (X), 168, 171, 329 Linking algebra, L(Y ), 303, 335 Locally reﬂexive, 264 Logmodular algebra, 159 M-ideal (or summand), 184 M-projection, 134 Matrix normed algebra, 68 Matrix normed module (or bimodule), 104 ∗ Maximal C ∗ -algebra, Cmax (A), 69 Maximal operator space, Max(E), 10 Maximal tensor product, A ⊗max B, 233 Minimal operator space, Min(E), 10 Minimal tensor product, X ⊗min Y , 27, 57 Module complementation property, 275 Module Haagerup tensor product, X ⊗hA Y , 119, 315 Module operator space projective tensor

product, X ⊗ A Y , 119 Module tensor product, X ⊗A Y , 119 Morita equivalence, 297, 318, 319 Multiple of a Hilbert space, 3, 110 Multiple of an operator or representation, 3, 110 Multiplier algebra, M (A), 84 Multiplier matrix norms, 170 Noncommutative H ∞ , 159 Noncommutative Shilov boundary, 157, 164, 326 Noncommuting variables, 200, 206 Nondegenerate module, 368 Nondegenerate representation, 51

Normal delta norm, σδ, 251 Normal delta tensor product, X⊗ σδ M , 251 Normal dual bimodule, 137 Normal Haagerup tensor product, X ⊗σh Y , 41 Normal Hilbert module, N HM OD, 138 Normal minimal tensor product, X⊗Y , 39 Normal spatial tensor product, M ⊗ N , 90 Normal tensor product, A ⊗nor M , 233 Normal virtual h-diagonal, 287 Nuclear C ∗ -algebra, 238, 259 Nuclearity, 259, 269, 349 One-sided M -ideal, 184 One-sided M -projection, 174 One-sided M -summand, 184 Operator algebra, 49, 62 Operator module (or bimodule), OM OD, 102, 124 Operator space, 5 Operator space projective tensor product,

X ⊗ Y , 35 Operator space structure, 5 Operator system, 17 Oplication, 175 Opposite operator space, X op , 12, 60, 108 Orthogonally complemented C ∗ -module, 300, 305, 337 Orthonormal basis, 340 p-summable sequences, p , Op , 209, 211 Paulsen system, S(X), 21 Prolongation, 107 Q-algebra, 215 Q-space, 219 Quasi-equivalent, 110 Quasibasis, 311 Quasimultipliers, 192 Quotient map, 2 Quotient module, 106, 109, 117, 139 Quotient operator algebra, 66 Quotient operator space, 8 Reducing property, 275 Reducing submodule, 109 Reﬂexive subspace, 360 Rigged module, 313, 345 Right multiplier algebra, RM (A), 84 Right multipliers of an operator space, Mr (X), 172 Rigid, 153 Ruan’s axioms, 7

Index Schatten spaces, S p , 3, 225, 359 Schur product, 214 Selfadjoint algebra, 269 Selfdual C ∗ -module, 332 Semi-invariant subspace, 109 Semidiscrete W ∗ -algebra, 238, 259 Semidiscreteness, 260, 271, 348 Semigroup operator algebra, O(G), 72 Shilov boundary, ∂E, 149 Shilov inner product, 165 Similarity, 195 Spatially equivalent, 110 Stabilization, 312, 342 Standard form, 138 State, 57, 363, 365 Subtriple, 161, 323 Ternary ring of operators (TRO), 103, 161, 336 Trace class, 23, 359 Triangular algebra, T n , T ∞ , 242 Triple envelope, T (X), 164, 326 Triple extension, 326 Triple ideal, 324 Triple morphism, 161, 322 Triple product, 161, 164 Triple system, 322 Ultraproduct, 16, 62 Ultraweak direct sum, ⊕wc i∈I Yi , 341 Uniform algebra, 59, 147 Unital function space, 147 Unital operator space, 16 Unital-subalgebra, 4 Unital-subspace, 4 Unitary C ∗ -module map, 300 Unitization, A1 , 52, 55, 364 Universal Hilbert module, 111 Universal representation, πu , 70, 365 Virtual h-diagonal, 285 Virtual diagonal, 283 W*-algebra, 89, 365 W*-full, 332 W*-module, 332 Weak Morita equivalence, 336 Weak expectation property (WEP), 270, 348 Weak* Haagerup tensor product, X ∗ ⊗w∗ h Y ∗ , 42 Weak* module Haagerup tensor product, X ⊗w∗ hM Y , 141, 347 X-projection, 152

387