Introduction to Subfactors V. Jones & V. S. Sunder
CAMBRIDGE UNIVERSITY PRESS
London Mathematical Society Lecture Note Series. 234
Introduction to Subfactors V. Jones University of California, Berkeley V.S. Sunder
Institute of Mathematical Sciences, Madras
CAMBRIDGE UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
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© V. Jones and V.S. Sunder 1997 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published 1997 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this book is available from the British Library
ISBN 0 521 58420 5 paperback
Dedicated to
Our wives
Contents Preface Acknowledgement 1
ix xii
Factors
1
. von Neumann algebras and factors 1 . 4 The standard form Discrete crossed products . . . 7 . . 1.4 Examples of factors . . 10 1.4.1 Group von Neumann algebras . . 10 1.4.2 Crossed products of commutative von Neumann algebras 12 . . 17 1.4.3 Infinite tensor products
1.1 1.2 1.3
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Subfactors and index 2.1 2.2 2.3
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The classification of modules dimMN . . . . Subfactors and index . .
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3 Some basic facts 3.1 3.2 3.3
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The basic construction . Finite-dimensional inclusions The projections e, and the tower .
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4 The principal and dual graphs .
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33 36 41
47
. More on bimodules The principal graphs . . . Bases . . 4.4 Relative commutants vs intertwiners
4.1 4.2 4.3
19 23 28
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5 Commuting squares
47 50 54 59
65
The Pimsner-Popa inequality . Examples of commuting squares 5.2.1 The braid group example . . 5.2.2 Spin models . . . 5.2.3 Vertex models . 5.3 Basic construction in finite dimensions 5.1 5.2
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65 70 70 71 71 72
CONTENTS
viii 5.4 5.5 5.6 5.7
. . Path algebras The biunitarity condition . Canonical commuting squares Ocneanu compactness . .
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6 Vertex and spin models .
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80 89 92
101
Computing higher relative commutants . Some examples On permutation vertex models . . 6.4 A diagrammatic formulation 6.1 6.2 6.3
77
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101 111 115
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122
133
Appendix A.1 Concrete and abstract von Neumann . . . algebras A.2 Separable pre-duals, Tomita-Takesaki . . . . . theorem A.3 Simplicity of factors . .
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A.4 Subgroups and subfactors A.5 From subfactors to knots
Bibliography Bibliographical Remarks Index
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133
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134 135 136
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141
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151 156 160
ix
Preface It must be stated at the outset that this little monograph has no pretensions to being a general all-purpose text in operator algebras. On the contrary, it is an attempt to introduce the potentially interested reader - be it a graduate student or a working mathematician who is not necessarily an operator algebraist - to a selection of topics in the theory of subfactors, this selection being influenced by the authors' tastes and personal viewpoints. For instance, we restrict ourselves to the theory of (usually hyperfinite) III factors and their subfactors (almost always of finite index); thus, factors of type III do
not make an appearance beyond the first (introductory) chapter, and the Tomita-Takesaki theorem makes only a cameo appearance in the appendix. It is hoped that such `simplifications' will help to make the material more accessible to the uninitiated reader. The aim of this book is to give an introduction to some of the beautiful ideas and results which have been developed, since the inception of the theory of subfactors, by such mathematicians as Adrian Ocneanu and Sorin Popa; an attempt has been made to keep the material as self-contained as possible; in fact, we feel it should be possible to use this monograph as the basis of a twosemester course to second year graduate students with a minimal background in Hilbert space theory. A remark is in order, as far as the references are concerned; when we state certain standard facts without proof, the reader is often referred to a text in operator algebras; if, in this process, it seems that the text by the second author is cited more often than other texts, that is simply because, a reference for that exact fact already being known to exist at that particular place, it was possible to avoid a search for that fact in other texts. We now give a brief outline of the contents of this volume for the sake of the possibly interested specialist. The first chapter begins with a quick introduction to some preliminary facts about von Neumann algebras (Murray-von Neumann classification of
factors and introduction to traces); this first section contains no proofs, but most of the sequel - with the notable exception of Popa's theorem on amenable subfactors - is self-contained modulo the unproved facts here. The
next section starts with the GNS construction and goes on to discuss the standard form of a finite von Neumann algebra and, in particular, identifies the commutant of the left-regular representation with the range of the rightregular representation. The chapter continues with a discussion of crossed products by countable groups and concludes with some examples - the left von Neumann algebra of an ICC group, factors of the three types coming from crossed products of commutative von Neumann algebras with countable groups acting ergodically and freely, a model of the hyperfinite factor which demonstrates that its fundamental group is the entire positive
x
line, and finally infinite tensor products and the definition of the hyperfinite factor. The second chapter starts with the classification of (separable) modules over a von Neumann algebra (with separable pre-dual), continues with the definition and some elementary properties of the M-dimension of a (separable) module over a III factor M, and concludes with the definition and some elementary properties of the index of a subfactor of a III factor, the statement of the result on restrictions on index values and a proof of the fact that all index values in the interval [4, oo) are possible. The third chapter begins with a section on the fundamental notion of the basic construction; the next section gathers together the basic facts about inclusions of finite-dimensional C*-algebras (including the fact about the basic construction and `reflection of Bratteli diagrams' as well as the notion of a Markov trace); the final section introduces the all-important sequence {e,,,}, derives the basic properties of this sequence, and indicates the relation between the theorem on restrictions of index values and the classification of non-negative integral matrices of norm less than 2, as well as an outline of the procedure originally adopted to prove the existence of hyperfinite subfactors
of index 4 cost n The fourth chapter is devoted to the principal (or standard) graph invariant of a subfactor. It starts with a discussion of ('bifinite') bimodules over a pair of III factors (contragredients and tensor products); the second section gives the two descriptions (in terms of the sequence of higher relative commutants as well as in terms of the bimodules that occur in the tower of the basic construction) of the principal graph of a subfactor, reduces the result on restriction of index values to the classification of non-negative integral matrices of small norm, and concludes with some examples of principal graphs; the chapter concludes with a discussion of `Pimsner-Popa bases' and a proof of why the higher relative commutants have an interpretation as intertwiners of bimodules.
The fifth chapter starts with Pimsner and Popa's minimax characterisation of the index of a subfactor, the consequent estimation of the index of a hyperfinite subfactor in terms of an approximating `ladder' of finitedimensional C*-algebras, and introduces the important notion of a commuting square; the second section discusses examples of commuting squares (vertex and spin models, and the braid-group example); the next section is devoted to the relation between commuting squares and the basic construction, and the consequent importance of symmetric or non-degenerate commuting squares (with respect to the Markov trace); the next two sections are devoted to the path-algebra model for a tower of finite-dimensional C*-algebras, and the reformulation of the commuting square condition in terms of `biunitarity', respectively; the sixth section is a discussion (without proofs) of the canonical commuting square associated to a subfactor and Popa's theorem on the completeness of this invariant; the final section of this chapter centres around
xi
Ocneanu's compactness theorem and the prescription it provides for comput-
ing the higher relative commutants of a subfactor built from an arbitrary initial commuting square.
The sixth and final chapter is devoted to the rich class of examples of subfactors provided by the so-called vertex and spin models. This chapter systematically develops a diagrammatic formulation to discuss the higher relative commutants in these examples, and also shows how to push this diagrammatic formulation through for general commuting squares. The book concludes with an appendix, which contains some facts used in the text - such as the non-existence of two-sided (algebraic) ideals in finite factors - as well as a computation of the principal and dual graphs of the `subgroup-subfactor' and the original derivation of the one-variable polynomial invariant of knots. Finally, the book comes equipped with such customary trappings as a bibliography, some remarks of a bibliographic nature, and one index containing both terms and symbols used.
xii
Acknowledgement These notes have a two-part genesis. To start with, the first author gave a series of about seven lectures at the centres of the Indian Statistical Institute at New Delhi and Bangalore during the winter of '92-'93; these lectures were `panoramic' in nature and did not contain too many proofs (because of the obvious time constraints). Later, the second author visited the Research Institute of Mathematical Sciences at Kyoto for nine months during '94-'95, during which period he gave a course of about 25 lectures - which was an expanded version of the earlier abridged version, and which contained proofs of most assertions and was addressed to the graduate students in the audience. These notes were the by-product of those two courses of lectures. The first (resp., second) author would like to express his gratitude to Professors Kalyan Sinha and V. Radhakrishnan, the Indian Statistical Institute, the Raman Research Institute and the National Board for Higher Mathematics in India (resp., Professor Huzihiro Araki and the Research Institute of Mathematical Sciences, Kyoto University) for the hospitality and the roles they played in making these visits possible.
Chapter 1 Factors 1.1
von Neumann algebras and factors
The fundamental notion is that of a von Neumann algebra. This is, typically, what may be called the `symmetries of a group'. The precise way of saying
this is that a (concrete) von Neumann algebra is nothing but a set of the form M = ir(G)' - where 7 is a unitary representation of a group G on a Hilbert space 7-L, and S' denotes, for S a subset of L(7-l) (the algebra of all bounded operators on 7-l), the commutant of S defined by S' = {x' E L(R) : x'x = xx' for all x in S}. In other words M is the set of intertwiners of the representation ir: thus, x E M 4 xir(g) = ir(g)x for all g in G. The usual definition of a von Neumann algebra is: `a self-adjoint subalgebra of L(7-l) satisfying M = M"' (where we write M" for (M')'). This is equivalent to the definition we have chosen to give. Reason: if M = 7r(G)',
then M is a self-adjoint subalgebra and M = M", since S' = S"' for all S C L(7-l); conversely, if M = M" is a self-adjoint subalgebra, we may set G equal to the unitary group of M' and appeal to the almost obvious fact (cf. [Sunl], Lemma 0.4.7) that G linearly spans M' (so that G' = (M')'). The canonical commutative examples of abstract von Neumann algebras turn out to be L°°(X, µ), while the basic non-commutative example is L(7-l). The fundamental `double commutant theorem' of von Neumann states that a self-adjoint unital subalgebra M of L(7-l) is weakly closed - i.e. (x, rl) y E 7-l, x" E M Vn x c M) (if and only if M is a-weakly closed - see §A.1) if and only if M = M". The importance of the notion of a von Neumann algebra was recognised in 1936 by Murray and von Neumann (although of course, they called them `rings of operators') - see [MvNI] - who also quickly realised that the `building blocks' in the theory of von Neumann algebras were (what they, and people after them, called) factors. If M = 7r(G)', then M is a factor precisely when the representation is `isotypical'. (If G is compact and it is a strongly continous representation, this says that it is a multiple of an irreducible representation;
CHAPTER 1. FACTORS
2
for general G, this says that any two non-zero subrepresentations of 7r admit non-zero subrepresentations which are equivalent.) More precisely, a von Neumann algebra M is called a factor if it has trivial
centre - i.e., Z(M) = M fl M' = Cl; and von Neumann proved - see [vN3] - that any von Neumann algebra is `a direct integral of factors'. Projections play a major role in the theory. It is true that any von Neumann algebra is the norm-closed linear span of its projections - i.e., elements p satisfying p = p* = p2. (Reason: if M = L°°(X, µ), .this is because simple functions - i.e., finite linear combinations of characteristic functions of sets - are dense in L°°(X, µ). The preceding statement and the spectral theorem show that any bounded normal operator is norm-approximable by finite linear combinations of its spectral projections. This proves the assertion about general von Neumann algebras.)
Two projections e and f in a von Neumann algebra M are said to be Murray-von Neumann equivalent - written e - f (or e - f (rel M)) - if there exists (a partial isometry) u in M such that u*u = e and uu* = f. It is not too hard to show that if (and only if) M is a factor, any two projections in M are comparable in the sense that one is Murray-von Neumann equivalent to a sub-projection of the other.
Factors were initially classified into three broad types by Murray and von Neumann, on the basis of the structure of the lattice P(M) of projections in M. The key notion they use is that of a finite projection; say that a projection e E P(M) is finite if e is not equivalent to any proper subprojection of e. It should be obvious that a minimal projection of M, should one exist, is necessarily finite. It should also be equally clear that any factor is one and exactly one of the types I-III as defined below. DEFINITION 1.1.1 (a) A factor M is said to be of type:
(i) I, if there exists a non-zero minimal projection in M;
(ii) II, if M contains non-zero finite projections and if M is not of type I; and
(iii) III, if no non-zero projection in M is finite.
(b) A factor M is said to be finite if 1 (the multiplicative identity of M, which always exists - cf. the definition of a concrete von Neumann algebra) is a finite projection in M; equivalently M is finite if M does not contain any non-unitary isometry.
One of the basics facts about finite factors is contained in the following result; for a proof, see, for instance [Takl], Theorem V.2.6. PROPOSITION 1.1.2 If M is a finite factor, then there always exists a unique faithful normal tracial state (henceforth abbreviated simply to `trace) on M;
1.1. VON NEUMANN ALGEBRAS AND FACTORS
3
i.e., there exists a unique linear functional T on M such that: (i) (trace) T(xy) = T(yx); (ii) (state) T(x*x) > 0 and T(1) = 1; (iii) (faithful) T(x*x) 0 0 if x zA 0; and (iv) (normal) T is (a-)weakly continuous. Further, if e, f E P(M), then e - f G T(e) = T(f).
(1.1.1)
We conclude this section with the rudiments of this Murray-von Neumann classification of factors. Suppose then that M is a factor with separable predual. Then consider the above-mentioned three possibilities for the type of M:
(I) If M is a factor of type I, it turns out (and it is not hard to prove cf. [Sunl], Exercise 4.3.1, for instance) that M = .C(7) for some separable Hilbert space 7-l and that M is finite precisely when 7-l is finite-dimensional. In particular, any finite factor of type I is necessarily finite-dimensional. We say that the type I factor M is of type I,,, if 7-l is an n-dimensional Hilbert space, for n = 1, 2, , oo. If M is of type I,,,, 1 < n < oo, then M ^ M,,, (C) and the unique trace tr of Proposition 1.1.2 is just the usual matrix-trace after suitable normalisation: thus, tr((xzj)) = _I E 1 xii (II) If M is a factor of type II, there are two possibilities, according to whether or not M is a finite factor in the sense of Definition 1.1.1(b). We say
that a type II factor M is of type Ill or of type III according to whether or not M is a finite factor. Recall, from Proposition 1.1.2 that every II1 factor is equipped with a faithful normal tracial state T. It is true - for instance, see [Sunl], Propo-
sition 1.3.14 - that if M is a Ill factor, then {T(p) : p E P(M)} = [0, 1]. Thus the trace T induces a bijection between the collection of Murray-von Neumann equivalence classes of projections in a II1 factor and the continuum [0,1]. What attracted von Neumann to II1 factors was the possibility of `continuously varying dimensions'. On the other hand, suppose M is a III factor. Fix an arbitrary finite proj ection p1 E M. It is true, then, that there exists - see, for instance, [Sunl] - a sequence {p,,, 1 < n < oo} of mutually orthogonal projections in M such that (i) pn - p1(rel M)Vn, and (ii) En pn = 1. Pick a partial isometry un E M such that u*nun = pn, unun = p1. (We briefly digress to remark that it is true - see [Taki], for instance - that if M is a von Neumann algebra, and if p E P(M), :
then the so-called `corner' of M defined by Mp = pMp = {pxp : x E M} is again a von Neumann algebra.) It then follows easily that Mp, is a II factor, and that the mapping x ((u?xuj )) establishes an isomorphism of M onto Mp, ®C(B2). Now consider the map Tr : M+(= {x E M : x > 0}) -> [0, co] defined by Tr x = En trM(unxu*n). It is true of this map - see [vNI] - that
(1) {Trp: pEP(M)}=[0,00];
CHAPTER 1. FACTORS
4
(ii) if p, q E P(M), then p - q(rel M) t-. Tr p = Tr q; and
(iii) if p, q E P(M), and pq = 0, then Tr (p + q) = Tr p + Tr q. Notice that this choice of Tr is so `normalised' that it takes the value 1 at pi; it is true that but for this possible choice in scaling, the function Tr is uniquely determined by the above properties. This function Tr is referred to as the faithful normal semifinite trace on the III factor. (III) The factor M is of type III precisely when every non-zero projection is infinite; under our standing assumption on the separability of the pre-dual
of M, it turns out - cf. [Sunl], Corollary 1.2.4(b) - that any two infinite projections are equivalent. Thus M is a type III factor precisely when any non-zero projection is Murray-von Neumann equivalent to the identity 1.
1.2
The standard form
Suppose cp is a normal state on a von Neumann algebra M - i.e., cp is a linear functional on M which is:
(i) positive - i.e., cp(x*x) > 0;
(ii) a state - cp(1) = 1; (iii) normal - i.e., cp is
continuous.
Consider the sesquilinear form on M defined by (x, y) -4 (x, y) = cp(y*x). This satisfies all the requirements of an inner product except positive definiteness - i.e., the set N,o = {x E M : cp(x*x) = 0} may be non-trivial. It is, in any case, a consequence of the Cauchy-Schwarz inequality that N,p is a descends to a genuine inner product on left-ideal of M. Hence the form the quotient space MIN,,; further, the equation 7r (x) (y + N,p) = xy + N,p defines, not just a well-defined, but even a bounded - with respect to the norm Jjy+NwjI2 = cp(y*y)1/2 - linear operator 7r(x) on MIN,,, which hence extends to the completion 7-l.. It is painless to verify that if 7r,(x) denotes the extension to R. of 7r(x), then 7r, defines a normal representation of M on 7-1,x. Further, if ,p = 1+No, then ,o is a cyclic vector for 7ro, i.e., ii
(where [S] denotes the closed subspace spanned by a subset S of Hilbert space), and the given state cp is recovered from the triple (7-l,w, 7r., .) by (p (x) =
x E M.
We summarise the foregoing construction - called the GNS construction after Gelfand, Naimark and Segal - in the following: PROPOSITION 1.2.1 Let cp be a normal state on a von Neumann algebra M. Then there exists a triple (7-l, 7r, ) consisting of a Hilbert space 7L carrying a normal representation it of M, and a distinguished cyclic vector of the representation satisfying V(x) = for all x in M.
1.2. THE STANDARD FORM
5
Notice, incidentally, that if the state co is faithful (meaning y(x*x)
0 if x
0), so is the representation 7r.
EXAMPLE 1.2.2 If M = L°°(X, µ), a typical normal state on M is of the form f f dv, f E L°°(X, p) - where v is a probability measure on X which is absolutely continuous with respect to p. One GNS triple associated to is given by 7-1, = L2(X, v), ir,(f ) = f l;, and is the constant function 1.
It must be clear that a necessary and sufficient condition for a normal representation it of M to occur as a GNS representation (i.e., to be unitarily equivalent to one) is that the representation 7r be a cyclic representation. An easy application of Zorn's lemma shows now that every (separable) normal representation of a von Neumann algebra is a (countable) direct sum of GNS representations.
In the rest of this section, we consider the special case when M admits a faithful normal tracial state, and analyse the GNS construction. Assume thus that there exists a faithful normal tracial state r on M. We write L2(M, 'r) for the Hilbert space underlying the GNS representation
associated with T. This representation is faithful, since ir(x) = 0 = 0 = r(x*x) = x = 0. Hence we may - and do - identify x with ir,(x) and we assume that M C_ G(L2(M, r)) so that there exists a cyclic vector SZ (which is the notation we shall employ for what we earlier called c,-) such
that T(x) = (xe, 0) for all x E M. Besides being a cyclic vector for M, the vector S2 is also separating for M in the sense that x1 = 0(= T(x*x) = IIXQ112 = 0) = x = 0. Hence the Hilbert space L2 (M, T) contains a vector SZ which is simultaneously cyclic and separating for M. We need the fact - which is ensured by the next lemma - that S2 is also a cyclic and separating vector for M'.
LEMMA 1.2.3 If M C G(7i) is a von Neumann algebra, a vector cyclic for M if and only if is separating for M'. Proof : Let
in 7i is
E H. Let p' be the projection onto [Mf], the closed M-cyclic
subspace spanned by . Note that p' E M' and that (1 - p')l; = 0. So, if is separating for M', then p' = 1 so is cyclic for M. Conversely if l; is cyclic for M, then x' E M' and x'l; = 0 = x'rl = 0 for all rl in [Me], since x in M. Thus L2 (M, T) has the vector S2 which is cyclic and separating for M as well as for M'. For any vector in L2, consider the two operators defined by:
fre( )(x'SZ) = x'
x
Vx' E M', b'x E M.
(1.2.1) (1.2.2)
CHAPTER 1. FACTORS
6
M'12 (and dom Me); these operators are Thus, dom densely and unambiguously defined since 12 is a cyclic and separating vector
for M and for M'. Call a vector left-bounded (respectively right-bounded) if the operator ire() (resp., extends to a (necessarily unique) bounded operator on all of 7-l (or equivalently, is bounded on its dense domain of definition). If is left- (resp. right-) bounded, we shall continue to write 7re(6 (resp. 7rr(6)) for the unique continuous extension to all of L2(M,T). In order to state the fundamental proposition concerning L2 (M, T) - or the standard form of the finite von Neumann algebra M, as it is referred to - we need one last bit of notation: the map x12 H x*S2 is a conjugate-linear isometry from M12 C L2 (M, T) onto itself. Denote its extension to L2 (M, T)
by J. It must be clear that J is an anti-unitary involution - i.e., J is a conjugate linear isometry of L2 (M, T) onto itself whose square is the identity;
in particular J = J* = J-1. This operator J is referred to as the modular conjugation operator for M, and sometimes denoted by JM.
THEOREM 1.2.4 (1) JMJ = M. (2) The following conditions on a vector E L2(M, T) are equivalent: (i) 6 is left-bounded; (i)' = x12 for some (uniquely determined) element x of M; (ii) 6 is right-bounded; (ii)' = x'12 for some (uniquely determined) element x' of M'.
Proof: (1) The definition of J shows that if x, y E M, then (Jx*J)(y12) _ yx12; since `left multiplications commute with right multiplications', we find
that
JMJ C M'.
(1.2.3)
On the other hand, if x E M, x' E M', then by the `self-adjointness' of the anti-unitary operator J, we have (Jx 12, x12) = (Jx12, x'12)
= (x*Q, x'Q) = (12, xx'1l) = (12, x'x1l) = (x'*12, x12). Since x was arbitrary, this implies that Jx'12 = x'*12, Vx E M'.
(1.2.4)
The above equation, together with the same reasoning that led to (1.2.3), now shows that
JM'J C M, and the equality in (1) follows from equations (1.2.3) and (1.2.5).
(1.2.5)
1.3. DISCRETE CROSSED PRODUCTS
7
(2) Denote the set of left- (resp., right-) bounded vectors in L2 (M, T) by Lie
(resp., Ur); it is clear from the definitions that E Lie $ ire(e) E (M')' = M and = conversely, if x e M, then xSl E Lie and x = irl(xS2). Thus the map H ireO establishes a bijective linear map of Lie onto M, with inverse being given by x -> xQ. In an identical manner, we also have Ur = M'Q. Thus (i) (i)' and (ii) (ii)'. To complete the proof, we only need to prove that MSl = M'Q. If x E M, then XQ = Jx* JSl E OR hence showing that MSZ C M'fl. An identical reasoning, with the roles of M and M' reversed, proves the reverse inclusion, and hence the theorem.
1.3
Discrete crossed products
The two constructions used initially by Murray and von Neumann - see [vNl] and [vN2] - to construct examples of factors of all possible types were (i) the crossed product construction, and (ii) the infinite tensor product construction. This section is devoted to a discussion of the former, while the latter will be discussed at the end of this chapter. The starting data is a (discrete) group G acting on a von Neumann algebra M - i.e., suppose we are given a group homomorphism t i--> at from G to the
group Aut(M) of (normal) *-automorphisms of M. (It is a fact (cf. [Sunl], Exercise 2.3.4(a)) that all *-algebra automorphisms of a von Neumann algebra
are automatically normal.) The crossed product is a sort of maximal von Neumann algebra containing copies of M and G with `commutation relations governed by the given action of G on M'; a little more precisely, the crossed product of M by the action a of G is a specific von Neumann algebra of the form M = (ir(M) U A(G))" where it : M -> M (resp., .\ : G - U(M), the unitary group of M) is an injective normal *-homomorphism (resp., injective unitary representation) of M (resp., of G), such that the copies -7r(M) and )(G) of M and G satisfy the commutation relations A(t)-7r(x)A(t)* = 7r(at(x)), for all x E M,t E G.
(1.3.1)
The construction of the crossed product (which is usually denoted by M xa G, or simply M x G ) goes as follows: Suppose M C L(7-i). The Hilbert space 7-l on which M will be represented has three (unitarily equivalent) descriptions:
(i) tEG[
[
(ii) 7i = ®x = {((S(t)))tEG : E IIS(t)II2 < Do} tEG
tEG
CHAPTER 1. FACTORS
8
where we think of a typical element of 7i as a column-vector with normsquare-summable entries from R; although this seems an artificial variant of (i), it is this description, in view of the availability of the convenience of matrix manipulations, which will prove most useful for dealing with M x c, G; (iii) 7-l = 7-l ® £2(G), the Hilbert space tensor product of 7i and the Hilbert
space £2(G) of square-summable complex functions on G (or £2(G; C) in
the notation of (i)). Before proceeding further, let us remark that £2(G) carries two natural - the so-called left-regular and right-regular - unitary representations .\, p : G --> 6(u-lt) and (pu6)(t) = 6(tu). AlternaU(G(f2(G))), defined by tively, if {6t t c G} denotes the canonical (or distinguished) orthonormal basis of £2(G) - thus t is the characteristic function of the singleton set {t} - then ),u(6t) = Sut and pu(t) = Stu-1. It is a basic fact - see §1.4 - that :
A(G)' = (p(G))", p(G)' =
A(G)".
To return to the definition of the crossed product, define 7r £ (7-l)
and
(1.3.2)
:M-
A :G-L(R) by:
(iv(x))(t) = at-1WO),
(1 3 3) .
(A(u))(t) =
(u
t).
.
J
It is trivial to verify that -7r : M ---> L (7-l) (resp. A : G --> £(77f)) is a faithful
normal *-homomorphism (resp., faithful unitary representation) and that it and A satisfy equation (1.2.1). Now define M = (ir(M) U ),(G))", so that M is the smallest von Neumann subalgebra of L(R) containing 7r(M) and A(G). This M is, by definition, the crossed product M x a G. We used some realisation of M as a concrete von Neumann algebra to define the crossed product M xc, G. It is true, however - cf., for instance, [Sunl], Proposition 4.4.4 - that the isomorphism class of the von Neumann algebra M x aG so constructed does not depend upon which faithful realisation
M on Hilbert space one started with. We shall now pass to a closer analysis of M, by using the second picture of 7-l as ®tEG 7-1. In this form, it is clear that any bounded operator x E L (?-1) t))),S,tEG where fi(s, t) E G(H) for all is represented by a matrix x =
s, t E G, and
(s) =
x(s,
the sum on the right being intepreted
tEG
as the norm limit of the net of finite sums. In this language, it is clear that (ir(x))(s,t) = 8stat-1(x)
and
(\(u))(s,t) = 6s,ut
for x in M, u, s, t in G.
1.3. DISCRETE CROSSED PRODUCTS
9
The point is that while elements of MatG(C) have two degrees of freedom, the above generators of the crossed product have only one degree of freedom. More precisely, we have the following matricial description of the crossed product (where we identify an element x E C(111) with its matrix ((x(s, t))) (with respect to the orthonormal basis {fit : t e G})).
LEMMA 1.3.1 With the preceding notation, we have t) = at-i (x(st-1)) Vs, t E G}. M = {x E G(?{) : 3x : G -F M s.t.
Proof: On the one hand, the right side is seen, quite easily, to define a (weakly closed self-adjoint algebra of operators and hence a) von Neumann subalgebra of G(?-l), while on the other, the algebra generated by ir(M) U .(G) is seen to be the dense subalgebra, corresponding to finitely supported functions, of the one given by the right side. It follows from Lemma 1.3.1 that the crossed product Mx,G is identifiable
- via the association x H x(s) = x(s, 1) - with a space of functions from G to M - viz. M = {x : G - MI3JE E G(?-l) s.t. x(s, t) = at-,(x(st-1)) Vs, t E G}. (1.3.4)
It is a matter of easy verification to check that the algebra structure inherited from (7r(M) U A(G))" by the set M defined by equation (1.3.4) is:
(x * y)(s) = E at-1(x(st 1))y(t), tEG
(1.3.5)
x*(s) = as-i(x(s-1)*). (The series above is interpreted as the limit, in the weak topology, of the net of finite sums; this converges by the nature of matrix multiplication.) In the new notation, note that 7r (x) (s) = 6s1 x, x E M, and
A(u)(s)=6su 1,uEG. We conclude this section by determining when the crossed product is a finite von Neumann algebra - i.e., admits a faithful normal tracial state. PROPOSITION 1.3.2 Let M = M xa G be as above (cf. the discussion preceding equation (1.3.5)). Then M admits a faithful normal tracial state 'r if and only if M admits a faithful normal tracial G-invariant state T (where G-invariance means T o at = T Vt E G).
CHAPTER 1. FACTORS
10
Proof: If T is a faithful normal tracial state on M, then T(x) = T(ir(x)) defines a faithful normal tracial state on M which is G-invariant since T(at(x)) _ T(7r(at(x))) = T(A(t)7(x)A(t)-i) = T(1r(x)) = T(x). Conversely, if T is a G-invariant faithful normal tracial state, define "F(x)
T(x(1)) (= r(fl(I,1))). Then T is clearly a normal positive linear function; further, T is faithful (since x E M+ and T(T (1, 1)) = 0 implies T(1,1) = 0, whence x(t, t) = at-, (x(1,1)) = 0 for all tin G, whence T, 0 (since a positive operator with zero diagonal is zero)). Finally, T is a trace, since T(x * y) = T((x * y)(1)) T (F_
at-1 (x(t i))y(t))
tEG
_ E T(at-1(x(t 1))y(t)) tEG
(since T is normal) _
E T(x(t-i)at(y(t))) tEG
(since T is G-invariant)
ET(at(y(t))x(t i)) tEG
(since T is a trace)
_
T(as-1(y(s i))x(s)) sEG
= T(y * x).
Before discussing when the crossed product is a factor, we digress for some examples, one of which will motivate the definitions of the necessary concepts.
1.4 1.4.1
Examples of factors Group von Neumann algebras
In the notation of §1.2, the (left) group von Neumann algebra LG of the discrete group G is defined thus:
LG = C x G = A(G)" c £(Q2(G)),
where the crossed product is with respect to the trivial action - at(z) _ z, Vt E G, z E C - of G on C, and, of course, A denotes the left-regular representation of G on £2(G). The analysis of §1.2 translates, in this most trivial case of a crossed prod-
uct, as follows: the elements of LG are those T E G(t2(G)) whose matrix, with respect to the standard orthonormal basis {fit t E G} of £2(G), has :
1.4. EXAMPLES OF FACTORS
11
the form 2(s, t) = x(st-1) V's, t E G for some function x : G -> C. Note that E x(t, so that the function x : G ---> C is nothing but Thus there exists a unique function g : LG -> £2(G) - defined by
such
that , (s,t) = 77(x)(st-1) Vs,t in G. Some of the main features of the von Neumann algebra LG are contained in the next proposition. PROPOSITION 1.4.1 (1) LG is a finite von Neumann algebra; more precisely, defines a faithful normal tracial state on LG; the equation T(x) = (2) (p2(G), idLG, 1) is a GNS triple for the trace T of (1) above, where idLG is the identity representation of LG in f2 (G))- hence LG, realised as operators on £2(G), is in standard form in the sense of §1.2; in particular, A(G)' = p(G)", p(G)' = A(G)".
(1.4.1)
(3) The following conditions on G are equivalent:
(i) LG is a factor;
(ii) G is an ICC group' - i.e., every non-trivial conjugacy class C(t) _ {sts-1 : s E G}, for t 0 1, is infinite. In particular, if G is an ICC group, then LG is a II, factor. Proof : (1) The equation T(A) = A, A E C, is clearly a trace on (M =) C
which is invariant under the trivial action of G and C. Hence (the proof of) Proposition 1.3.2 implies that the equation T(om) _ (1 1) = rl(x)(1) _ (ij,, 1) defines a faithful normal tracial state on LG. t E G}] = £2(G) and so 1 is a cyclic vector (2) D [{fit = for LG such that T(om) _ (xl, 1) Vi E LG. This proves the first assertion in (2).
As for the second, note that the canonical conjugation J is the unique anti-unitary operator in f2(G) such that Jet = t-1 for all t E G, and that
JA Jet = JAs t-i = StS-1
pst Deduce now from Theorem 1.2.4(2) that
A(G)' = A(G)"' = (LG)' = J(LG)J = JA(G)"J = (JA(G)J)" = p(G)". (1.4.2)
Thus A(G)" and p(G)" are commutants of one another, thus proving (2). x 1. An easy computation (3) As before, write g : LG -* £2(G) for shows that Ic E Z(M) if and only if r7(x) is constant on conjugacy classes.
CHAPTER 1. FACTORS
12
Suppose now that G is an ICC group. Then the conditions that constant on conjugacy classes and that 77 (x) E £2(G) force
is
0 for t
i.e., x = (97((1)) idp2(G). Thus Z(LG) = C and LG is a factor. Conversely suppose G is not an ICC group so that there exists a finite conjugacy class 10 C C G; then if 9 = 1c denotes the characteristic function of the finite set C, it follows that n Q2 (G) and that the matrix defined by x(s, t) = lc(st 1) defines a bounded operator on £2(G), so that x E LG and 1
97(x) = 77. The earlier discussion shows that E Z(LG) (since 77 is constant on conjugacy classes) while x V C, since 97 V C 1i thus LG has non-trivial
centre and is not a factor. Finally, if G is an ICC group, then LG is a finite factor by (1) and (2)
of this proposition; the fact that G is an infinite group shows that LG is an infinite-dimensional vector space over C and hence not a type I factor (since finite type I factors are (isomorphic to some M,,,(c) and hence) of finite dimension over C).
Examples of ICC groups are given by: 00
(i) So,, = U S, = the group of those permutations of 1,2,3,... which fix all n=1
but finitely many integers;
(ii) F,, (n > 1), the free group on n generators. Hence LSD and LF., n > 1, are (our first examples of) IIl factors.
1.4.2
Crossed products of commutative von Neumann algebras
Let M = L°°(Sl, .F, µ), where (S2, Jc', µ) is a separable probability space. (The assumption of separability is equivalent to the requirement that L2 (S2, .F, µ) is separable. Also, it is true - see [Takl], for instance - that such an M is the most general example of a commutative von Neumann algebra with separable pre-dual.) By an automorphism of the measure space (S2, .F, µ), we shall mean a bimeasurable bijection T of S2 such that ,u o T-1 and p have the same null sets. (Thus, T : S2 -> S2 is a bijection satisfying: (i) if E is a subset of S2, then
E E F t* T-1(E) E .F; and (ii) if E E F, then µ(E) = 0
µ(T-1(E)) = 0.
Such a map T is also called a non-singular transformation in the literature.) The reason for the above terminology is that the most general automorphism of M is of the form 9(W) = co o T, Vco E L°°(S2, J , /11), for some automorphism
T of (c,.F,µ). If M, 0, T are as above, it is not hard to prove - cf. [Sunl], Exercise 4.1.12 - that the following conditions are equivalent: (i) T acts `freely' on S2 - i.e., µ({w E S2 : Tw = w}) = 0;
(ii)zEM,xy=0(y)xdyEM=x=0.
1.4. EXAMPLES OF FACTORS
13
This is the justification for the following general definition.
DEFINITION 1.4.2 (1) An automorphism 0 of an abstract von Neumann algebra M is said to be free if x E M, xy = 0(y)x Vy E M = x = 0.
(2) An action a of a group G on M - i.e., a group homormorphism a G -> Aut(M) - is said to be free if the automorphism at is free (in the sense of (1) above) for every t 0 1 in G. We need one more definition before we proceed further:
DEFINITION 1.4.3 (1) If a : G -> Aut(M) is an action of a group on any von Neumann algebra, write M' = {x E M : at(x) = x Vt E G} for the fixed-point subalgebra of the action.
(2) The action a : G -+ Aut(M) is said to be ergodic if M' With the notation of Definition 1.4.3, it must be clear that M' is always a von Neumann subalgebra of M. Since any von Neumann algebra is generated, as a von Neumann algebra, by its lattice of projections, it must be clear that the action a is ergodic if and only if P(Ma) = 10, 11.
In the special case when M = L°° (1, .F, µ), it must be clear from the foregoing discussion that any action a : G -+ Aut(M) must be of the form at(co) = cp o T,7', where t -# Tt is a group homomorphism from G into the group of automorphisms of the probability space (S2, F, /L); and that the action
a is ergodic if and only if
E E F, µ(EOTt 1(E)) = 0 Vt E G
µ(E) = 0 or µ(E°) = 0
(1.4.3)
where A denotes symmetric difference (so that AAB = (A - B) U (B - A)) and E° denotes the complement of the set E. (This is the classical notion of ergodicity and the reason for Definition 1.4.3(2).) It would have been more natural to include the next proposition - which concerns general von Neumann algebras - in the last section, but it has been included here since the notions of freeness and ergodicity of an action are most natural in the context of the abelian examples considered in this section.
I
PROPOSITION 1.4.4 Let a : G -> Aut(M) be an action of a discrete group G on any von Neumann algebra M. Let M = M xa G. Then: (i) ir(M)' n M = ir(Z(M)) 4* the action a is free; (ii) Assume the action a is free; then the crossed product k is a factor if and only if the restricted action alz(M) is ergodic. (Here, we use the notation aIM0 to mean the restricted action - (alM0)t(xo) = at(xo) for all xo E Mo of G on any von Neumann subalgebra Mo of M which is invariant under the action a.) In particular, if a is a free action of a discrete group G on L°°(S2, then the crossed product is a factor if and only if the action a is ergodic.
CHAPTER 1. FACTORS
14
Proof: As before, if i E M, we write so that x(s, t) =at-, (x(st-1)). In view of equation (1.3.5), it is easily seen that the condition E 'ir(M)' n M translates into the condition that
x(t)y = at-1 (y)x(t) Vy E M, t E G.
(1.4.4)
Hence, if the action is assumed to be free, then x'7' E7r(M)'nM
x(t)=Obtl ir(x(1)) E 7r(M).
Since the map yr : M --- M is injective, we see that indeed i E ir(M)' n M = 7r(z) for some z(= x(1)) E Z(M).
Conversely, if the action is not free, there exist t 1 in G and xo E M 0 and x0y = at(y)x0 for all y c M. If we now define the such that x0 function x : G - M by x(s) = 6st-1 xo, it is clear that the equation ; (s, u) = a.,n-i (x(su-1)) defines a bounded operator 1 on ?-l; further, it must be equally
clear that Jr E Rr(M)' n M and that 1 V 7r(M) (since ((x(s, u))) is not a diagonal matrix). (ii) Assume the action a is free. It then follows from (i) above that
Z(M) = ir(Z(M)) n )(G)' = {7r(z) : z E Z(M) and z = at(z) dt E G}, whence the desired conclusion.
Thus if a countable group G acts freely and ergodically on M = L°°(S2,
.F, µ), then M = M x G is a factor. The complete details of the Murrayvon Neumann type of the factor thus obtained are contained in the following beautiful theorem due to von Neumann ([vNl]).
THEOREM 1.4.5 Let M = M xa G be as above. (Thus, M = L°°(S2, .F, lc), where li) is a separable probability space, and G is assumed to act freely and ergodically on M via the equations at(p) = cp o Tt 1, where {Tt : t E G} is a group of automorphisms of the measure space (1, ,F, p).) (1) M is of type I if and only if p) contains atoms - i.e., ]E E .F
such that µ(E) > 0 and whenever Eo E F, E0 C_ E, either p(Eo) = 0 or µ(E - Eo) = 0. In this case, there exists a countable partition S2 = 11iEA E2, where each E2 is an atom; further, M is of type In,,1 < n < oo, if and only if IAI = n.
(2) M is of type II if and only if there exists a a-finite measure v on (Q, .F) such that (i) v is non-atomic - i.e v has no atoms, (ii) v and ,i have
the same null sets, and (iii) v is G-invariant - i.e., v o Tt1 = v Vt E G. Further, M is of type IIl or III according as v(S2) < 00 or v(S2) = oo. (3) M is of type III if and only if there exists no or-finite G-invariant measure v which is mutually absolutely continuous with M.
1.4. EXAMPLES OF FACTORS
15
We shall not prove the theorem here. (The interested reader may find such a proof, for instance, in [Sunl], Theorem 4.3.13.) We shall, instead, content ourselves with a discussion of the Ill case, which is the case of interest in this book. We know from Proposition 1.3.2 that M is a finite factor if and only if there
exists a faithful normal tracial state T on M = L°° (S2, F, p) (equivalently, if there exists a probability measure v which has the same null sets as p) which is G-invariant. If v is non-atomic, then M - and hence M - is infinitedimensional, so that M cannot be a factor of type I,,,, n < oo. Since M is a finite factor, the non-atomicity of v implies that M is a factor of type Ill. On the other hand, if v is atomic, the finiteness of v and the ergodicity of the action imply that S2 is the union of a finite number, say n, of atoms. This means that M = C. The assumed freeness (and ergodicity) of the action now forces G to have exactly n elements, and consequently M is a finite factor with n2 elements - i.e., it is a I,,, factor. EXAMPLE 1.4.6 Suppose G is a countably infinite dense subgroup of a compact group. (Two examples to bear in mind are:
(a) K = T = {eZ° : 0 E IR}, G = {ez2n,,I.: n E z}, where Bo is irrational; and
(b) K = {0,1}N = set of all (0, 1) sequences, with group operation being coordinate-wise addition mod 2, and G = the subgroup of those sequences which contain at most finitely many 1's.)
Set S2 = K, F = the a -algebra of Borel sets in K, and p = Haar measure. (In example (a), p is normalised arc-length, so that µ(I) = 6/27r if I is an arc subtending an angle 0 at the origin, and in example (b), µ is the countable product of the measure on {0, 1} which assigns 1/2 to {0} and to {1}.) Fort E G and k E K, define Tt(k) = tk. It should be clear that t --> Tt is a homomorphism from G into the group of automorphisms of the measure space (K, .F, µ). Further, if we define at(c) = ¢ o Tt-,, t E G, 0 E L°° (K,.F, µ), it
follows that a is an action of G on L°°(K,.F, µ) which preserves µ and is ergodic (since G is dense in K). Hence L°°(K) x,, G is a 11, factor, whenever G is an infinite countable dense subgroup of a compact group.
EXAMPLE 1.4.7 A slight variation of the construction in the preceding example yields factors of type III - the so-called Powers factors - see [Pow]. Suppose S2 = {O,1}', equipped with the Borel a-algebra. Now take p to be the countable product of a measure µo on {O, 1} which assigns unequal masses to {0} and {1} thus: µo{0} = 1/(1+.1) and µo{1} = .X/(1+A), where A # 1. If Tt, t E G, is defined as before, it is true that each Tt is still an automorphism
CHAPTER 1. FACTORS
16
of (52, .P, µ), although µ o T ' t for t 1. It is still the case that a, as defined before, is an ergodic action of G on L°°(SZ, .F, A). The crucial difference is that, now, the action does not admit any a-finite invariant measure which is mutually absolutely continuous with t, and hence RA = L°°(S2,.F, t) x,,, G is a factor of type III. In fact, it turns out that RA and RA' are non-isomorphic for 0 < A A' < 1. We shall not prove these facts; the interested reader may consult [Suns], for instance.
EXAMPLE 1.4.8 The group G = SL(2, 7L) admits a natural (linear) action on (1R2, B, µ) - where µ is Lebesgue measure defined on the a-algebra 13 of Borel sets in 'R2. It is true - cf. [Zim], Example 2.2.9 - that this action is ergodic. Since µ is an infinite measure, it follows from Theorem 1.4.5 that M = L°°(1R2,13, p) x SL(2, z) is a factor of type III. ( Note that Lebesgue measure is preserved by linear transformations of determinant 1.)
In the next proposition, we single out a certain feature of the preceding example that we shall need later.
PROPOSITION 1.4.9 Let M = M x,,, G, where M = L°°(1R2,,a) and G = SL(2, 7[), are as in Example 1.4.8. Let pl = 1® denote the characteristic function of the unit disc D {(x, y) E IR2 : x2 + y2 < 1} in IR2 (so that pl is a projection in M). Then (a) R = ir(p1)Mir(p1) is a IIl factor, where it : M -> M is the canonical embedding; and
(b) R = pRp for any non-zero projection p in R.
Proof : We use the notation of §1.3, so that a typical element of M is represented by a matrix of the form ((x(s, t)))8,tEG - which induces a bounded operator on ®7-lt, 7-lt = L2 (J 2, ,a) Vt E G - such that : (s, t) = at-1(x(st-1)), tEG
where {x(s) : s c GI 9 L- (R2, /Z).
Since M is a factor and ir(p1) E P(M), it follows that R ir(p1)M7r(p1) is a factor. Further, the non-atomicity of t shows that MP,, and
hence R, is infinite-dimensional over C. To show that R is a III factor, it suffies, therefore, to exhibit a faithful normal tracial state on R. More generally, let pr = 1IDr, where IDr = {(x, y) E IR2 : x2 + y2 < r2}, for 0 < r < oo. It is easy to see that if x E M, then f E M1f(pr) if and only if
x(s,t) = a,-1(pr)x(s,t)at-1(pr);
(1.4.5)
in other words, I E MM(Pr) if and only if I(s, t) (which is an L°°-function on IR2) is supported in Tt 1(Dr) fl T,-' (B,). In particular, if I E M,r(pr), then
1(1, 1) E L- (B,, µ). It is not hard, using the fact that t o Ti- = µ Vt E G, to show that the equation Tr(x)
f 1(1,1)dµ
(1.4.6)
1.4. EXAMPLES OF FACTORS
17
defines a faithful normal tracial state on Mn(Pr). Hence M, (Pr) is a II1 factor for all r > 0, thus proving (a).
Next, begin by noting that if A > 0, then the equation (BA(f)(x) f (.fix), x e R2, f E L°°(1R2, µ), defines, for each A > 0, an automorphism OA of L°°(R, µ) = M. Since linear maps commute with scalar multiplication, it must be clear that Ba oat = at o Ba VA > 0, t E SL(2, Z). This implies that there exists, for each A > 0, a unique automorphism fix of M such that Ba((x(S1 t))) = ((0A(x(S, t))))
Since 9(i) = ljj
for r, A > 0, it follows from equation (1.4.5) that
M, (PA,); in particular, setting r = 1 and 0 < A < 1 we find (since (Mn(p))_(pa) = R-(pa)) that R = pRp if p = v(PA), 0 <
A<1. Note, finally, that, if 0 < A < 1, then T1(7r(pA)) _ p(IDa) A2, where vrl is as defined in equation (1.4.6), with r = 1). Hence if 0 p E P(R) and A= then T1(p) = T1(7r(pA)); so there exists (by equation (1.1.1)) a unitary u E R such that upu* = 7r(p),), and consequently Rp = R,(p.) (via the isomorphism RP E) x - uxu*), thus completing the proof of the proposition. T1(P)1/2
1.4.3
Infinite tensor products
Fix an integer N, and let
A. = ®" MN
(C)
= MN_ (c).
Regard A. as a subalgebra of A,,,+1 via the embedding x
0
0
0 x 0
X
...
0
0
0
...
0
Xy
0
0 0
0
= x®1N.
X
Then it is clear that A., = U An, is a *-algebra, and that there exists a unique tracial state on A., which restricts on the IN.. factor A. to the unique tracial state trA,,. (In fact, it is clear that this is the only tracial state on A., since finite factors admit unique tracial states.) Let (7-l, 7r, Q) denote the associated GNS triple. We define R(N) = ®N MN(C) = 7r(A°o)".
(1.4.7)
The faithfulness of trA. implies that it embeds A,, as a weakly dense *subalgebra of R(N). Since the equation trR(N) (z) = (x12,12)
18
CHAPTER 1. FACTORS
clearly defines a tracial state on R(N), it follows that the von Neumann algebra R(N) has the property of admitting a unique normal tracial state. This implies
that R(N) is a III factor. (Reason: if h e Z(R(N))+ is arbitrary, then also trR(N) (h.) is a trace; further, in view of the infinite-dimensionality of Ate, the finite factor R(N) cannot be of type I.) (Here is an alternative way of seeing that R(N) is a III factor: consider the compact group K = zN obtained as the direct product of countably infinitely many copies of the discrete group ZN; let G be the dense subgroup consisting of those sequences from ZN which differ from the identity element in at most finitely many places; then R(N) is isomorphic to the III factor constructed as in Example 1.4.6.) Notice that R(N) is clearly approximately finite-dimensional in the sense of the following definition. Not quite so clear, however, is the striking statement of the subsequent theorem due to Murray and von Neumann (which we shall not prove here). (See [MvN3] and also [Conl].)
DEFINITION 1.4.10 A von Neumann algebra M is said to be approximately finite dimensional, or simply AFD, if there exists an increasing sequence
AICA2C...CA, CA,,+IC... of finite-dimensional *-subalgebras, whose union in weakly dense in M.
THEOREM 1.4.11 Every AFD III factor is isomorphic to R(2).
Thus we find that among the class of all III factors, there is one, the socalled hyperfinite III factor - which we shall always denote by the symbol R - which is uniquely determined up to isomorphism by the property that it is approximately finite-dimensional. Thus, for instance, if S" = u' I S. is the group of those permutations of N which move at most finitely many integers, then the group von Neumann algebra LS,, is one model of the hyperfinite
III factor. Less obvious, but also true, is the fact that the algebra denoted by R in Proposition 1.4.9 is another model for the hyperfinite III factor.
Chapter 2 Subfactors and index 2.1
The classification of modules
If M is any von Neumann algebra, we shall, by an M-module, mean a Hilbert space 7-1 equipped with an action of M, i.e., a unital normal homomorphism from M into L(7-I). We shall generally suppress specific mention of the underlying representation it : M - L(7-I) and just write x instead of ir(x) when
If 7-l and K are M-modules, we shall say that a bounded linear operator T : 7-l -* K is M-linear if T Vx E M, E R, and we shall denote the collection of all such M-linear operators from 7-l to K by ML(7-l, K).
When 7-l = K, we shall simply write ML(h) for ML(H,H). (If it denotes the underlying representation of M on 7-1, note that ML(H) _ ir(M)' and consequently ML(H) is a von Neumann algebra. In general, it is true similarly that ML(H, K) is a weakly closed subspace of L(7-1, K) which is well-behaved
under polar decomposition in the sense that if T E C(7-I,IC) has polar decomposition T = UITI, then T is M-linear if and only if U and ITI are.) Two M-modules 7-l and K are said to be isomorphic if there exists an M-linear unitary operator of 7-l onto K. (In view of the parenthetical general remark of the last paragraph, this is equivalent to the existence of an invertible M-linear map of 7{ onto K.)
EXAMPLE 2.1.1 (i) If 0 is a normal state on any von Neumann algebra M, we shall use - here and elsewhere in the sequel - the symbol L2(M, 0) to denote the Hilbert space underlying the GNS representation of M associated with 0, and we shall denote the cyclic vector by Q0. Then, by definition, the Hilbert space L2 (M, 0) is an M-module. (Further, as has already been remarked, every cyclic M-module is isomorphic to such a module, for some normal state 0.) (ii) If {7-i i E I} is any family of M-modules, then their direct sum ®ic, Hi is again an M-module. In particular, if I = N and 7-12 = 7{ Vi, :
this direct sum will be denoted by 7{ ® £2 (for the obvious reason that the
CHAPTER 2. SUBFACTORS AND INDEX
20
underlying Hilbert spaces are isomorphic and the action, in the tensor-product formulation, can be identified thus: x( ®rl) = 7r(x) 0 77). The following result is crucial in the classification of modules over factors.
PROPOSITION 2.1.2 Let M be any von Neumann algebra and let K be a faithful M-module - i.e., a module such that the underlying representation of M is faithful. Then any separable M-module 7-1 is isomorphic to a submodule of K®P2. Proof: First consider the case when 7-l is a cyclic M-module and consequently isomorphic to L2 (M, /) for some normal state 0 on M. The faithfulness assumption says that M is embedded as a von Neumann subalgebra of L(K) and consequently, we may (extend 0 to a normal state on all of L(K) - thanks to the Hahn-Banach extension theorem - and consequently) 1 and O(x) _ find a sequence of vectors in K such that En Vx E M. Notice now that, for each fixed n, we have, for all x E M,
IIx&II2 = (x*g.'' .) < b(x*x) = IIzQPII2;
(2.1.1)
K such that hence there exists a unique bounded operator R,,, : L2(M, x& Vx E M. It follows at once from the definitions that each R is actually an M-linear map. Now consider the (clearly M-linear) operator U : L2(M, 0) -> K ® £2
defined by U _ >n
where {E,,,}°°_1 denotes the standard orthonormal basis for f2. It must be clear that U is an isometric operator, whence L2 (M, b) - and consequently 7i - is isomorphic, as an M-module, to a submodule (viz., the range of the operator U) of K ® P2. Thus we have proved the proposition for cyclic modules. The general case
follows from the facts that (i) every separable module is isomorphic to a countable direct sum of cyclic modules, and (ii) as an M-module, a countable direct sum of copies of the module K ® f2 is isomorphic to K 0 f2 itself.
REMARK 2.1.3 One consequence of the preceding result is a fact which is sometimes termed the `structure of normal isomorphisms'. Suppose then that Mi C L(7-1z), i = 1, 2, are von Neumann algebras and suppose that there exists a normal isomorphism 0 of Ml onto M2. Then put M = M1 and regard 0 as a faithful representation of M, and deduce from Proposition 2.1.2 that 7-11 is isomorphic, as an M-module, to a submodule of 7-12 0 £2. Similarly, 7-12 is isomorphic, as an M-module, to 7i ® e2. Also, it follows that each of the M-modules, 7-li 0 £2, i = 1, 2, is isomor-
phic to a submodule of the other. This implies - by the same reasoning as is employed to prove the classical Schroeder-Bernstein theorem - that these two modules are actually isomorphic. We thus have the so-called `structure
2.1. THE CLASSIFICATION OF MODULES
21
of normal isomorphisms', viz.: the isomorphism B is a composition of three maps, (i) a dilation or 'ampliation' (id 0 le2), (ii) a spatial isomorphism - i.e., one implemented by a unitary isomorphism of the underlying Hilbert spaces (the one establishing the isomorphism of the M-modules ®B2), and (iii) a `reduction' - i.e., the restriction to a submodule.
For us, the importance of Proposition 2.1.2 lies in the classification of modules. For one thing, it says that it is enough to classify, up to isomorphism,
the various submodules of one module - which we may choose as H,,, _ 7Ci ® Q2, where 7-1k = L2(M, 0) for some arbitrary faithful normal state on M. Before proceeding further, we single out two simple facts as lemmas, for ease of reference. LEMMA 2.1.4 Let M be a von Neumann algebra and let K be an M-module.
Then the map p H ran p sets up a bijection between the set P(ML(K)) of M-linear projection operators in K and the set of M-submodules of K. Further, two projections p, q E P(ML(JC)) are Murray-von Neumann equivalent (relative to the von Neumann algebra ML(K)) if and only if their ranges are isomorphic as M-modules. Proof: Easy.
As in our treatment of crossed products, we adopt the convention that if K is a Hilbert space, then K ® £2 is identified with the Hilbert space direct sum of countably infinitely many copies of K; also, we shall identify an operator T E C(K 0 £2) with an infinite matrix ((TTY)) with entries from C(K) (in such
a way that if x E L(K), then the operator x ® ide2 gets identified with the matrix with x on each diagonal entry and zeros elsewhere). Recall that if M C L(K), N C L(f2) are von Neumann algebras, then M ON denotes the von Neumann subalgebra of K ®f2 generated by operators
of the form x ® y, x E M, y E N. Under the identification discussed in the previous paragraph, it should be fairly clear that M ® L(B2) corresponds to the set of those operators T on K®f2 for which all the entries of the associated matrix ((TTY)) come from the von Neumann algebra M. LEMMA 2.1.5 If M C L(K) is a von Neumann algebra, then (M ® 1)' = M' ® L(B2).
(2.1.2)
Proof: This is a routine computation. Now we are ready for the classification of separable modules over factors with separable pre-duals.
CHAPTER 2. SUBFACTORS AND INDEX
22
THEOREM 2.1.6 Let M be a factor with separable pre-dual and let N be a separable M-module. , oo}} (i) If M is of type I, there exists a sequence {N,,, : n E M = {1, 2, of pairwise non-isomorphic M-modules, and there exists (a necessarily unique)
_
n E N such that H = Hn. (ii) If M is of type II, there exists a family {Nd : d E JR = [0, oo]} of pairwise non-isomorphic M-modules, and there exists (a necessarily unique) d E IR such that H = Hd. (iii) If M is of type III, there exists a separable non-zero M-module which is unique up to isomorphism. Proof: We consider the several possible cases.
Case(i): M is of type I As mentioned in §1.1, there exists a separable Hilbert space K such that M = L(K). Deduce from Lemma 2.1.5 that k = mL(K ® £2) (= 1 (9 L (f2)) is a factor of type Imo; it follows that for any projection p E M, it is the case that Mp = pMp is again a factor of type I. On the other hand, it is a consequence of Proposition 2.1.2 that if H is an arbitrary M-module, then H is isomorphic, as an M-module, to p(K®.E2) for some p E M and consequently, that M1(H) is a factor of type I and that in fact, N is isomorphic, as an M-module, to the direct sum of n copies of K, if p = 1 ®po, po E L(.&2) and n = dim ranpo.
Case(ii)1: M is of type II1 In this case, set K = L2(M, tr), where, of course, the symbol tr denotes the unique normal tracial state on M. It follows from Theorem 1.2.4 and Lemma 2.1.5 that ML(K) = JMJ, and consequently, M,C(K ® £2) = JMJ ®,C(E2) is a factor of type III. On the other hand, it follows from Proposition 2.1.2 and Lemma 2.1.4 that the set of isomorphism classes of M-modules is in bijective correspondence with the set of Murray-von Neumann equivalence classes of projections in this III factor; the latter set is, according to the discussion in §1.1, in bijection with [0, oo].
It follows that if 71 is any separable M-module, then there exists p E P(mL(K ® £2)) such that M is isomorphic, as an M-module, to ran p and that ML (H) is a factor of type II; further, the isomorphism class of 7-l depends only upon Trp.
Case(ii)c,,,: M is of type II., In this case - as already mentioned in §1.1 there exists a finite projection p1 E M such that Mo = p1Mp1 is a factor of type 111, and such that M = Mo®c(e2). Set Ko = L2(Mo, tr) and K = K®J2. It is seen then that ML(K) = JMOMoJMO 0 1 - where, of course, the symbol JMo denotes the `modular conjugation' operator on Ko - which is a IIl factor. It follows, as in the last case, that if 7-l is any separable M-module, then there exists a projection p E P(MG(K 0 £2)) such that M is isomorphic, as an Mmodule, to ran p and that ML (H) is a factor of type II .
2.2. DIMMf
23
Case (iii): M is of type III Notice from the cases discussed so far, that if P C G(M), M an arbitrary Hilbert space, and if P is a factor of type I (resp., II), then also P is a factor of type I (resp., II). Hence, if P is a factor of type III, it follows by exclusion that so also is P'. Suppose now that K is an arbitrary faithful M-module. It follows from the previous paragraph that ML (K (& Q2) is a factor of type III. Since any two non-zero projections in such a factor are Murray-von Neumann equivalent, the proof of this case follows at once from Proposition 2.1.2 and Lemma 2.1.4.
0
2.2
dimM?
In order to really be able to use Theorem 2.1.6, we shall find it convenient to work with bimodules. For this reason, we shall make a slight change in terminology: thus, if 7-l is what we have so far been calling an M-module, we shall henceforth refer to 7-l as a left M-module. On the other hand, a Hilbert space 71 is called a right M-module - where M is an arbitrary von Neumann algebra - if there exists a o--weakly continuous linear map lrr : M -+ £(N) which preserves adjoints and reverses products (i.e., 7rr(x*) = (7rr(x))* and iIr(xy) = 7rr(y)lrr(x) for all x, y E M). As with left modules, we shall often omit referring to the map lrr, and simply write x instead of 7rr (x) whenever x E M, E H. (The assumed product-reversal is consistent with writing the operator on the right, as above, in the sense that (xy) = 0) Y dx, y E M, E 7-l.) Every statement about left modules has a corresponding statement about right modules, via the following observation. If M is a von Neumann algebra, recall that there is an opposite von Neumann algebra MP such that there exists a linear isometry x0 f--> x from MOP onto M which preserves adjoints and reverses products. (Note that the conditions of the previous sentence
ensure that (MOP)* = M* and determine the von Neumann algebra MP uniquely up to isomorphism.) Thus, we may - and shall - think of a right M-module as a left MP-module, i.e., as a Hilbert space 7-l equipped with a unital normal homomorphism 2r°P : MOP -> £(7I) - so that 7r°P(x°) _ DEFINITION 2.2.1 (i) If M, N are von Neumann algebras, a Hilbert space 71 is said to be an M-N-bimodule if:
(a) 71 is a left M-module;
(b) 71 is a right N-module; and
(c) the actions of M and N commute; i.e., (mi)n = m(in) Vm E M,n E N, l= E R.
CHAPTER 2. SUBFACTORS AND INDEX
24
Thus, in order for 7-1 to be an M-N-bimodule, there must exist unital normal homomorphisms 71 : M -> L(7-1) and 70 : N°P -> L (H) (i.e., a unital normal anti-homomorphism 7rr : N -> L(7I)) such that 7r°(N°P)
irr(N)) C (ii) If 7-1, IC are M-N-bimodules, an operator T E L(7-i,1C) will be called
M-linear (resp., N-linear) if T(x6) = xT6 (resp., T(6y) = (T6)y) for all x E M, y E N, 6 E H. The collection of such M-linear (resp., N-linear) operators will be denoted by MG(', IC) (resp., LN(',1C)), and an operator is said to be M-N-linear if it is both M-linear and N-linear; we denote the collection of all such operators by ML N(', IC).
We shall be concerned primarily with III factors in this book, and it will serve us well to spell out exactly what Theorem 2.1.6 says in this special case.
If M denotes a III factor with separable pre-dual, the symbol trM will always denote the unique normal tracial state on M and we shall simply write L2(M) for L2(M,trM). We shall assume throughout this section that M denotes a III factor with separable pre-dual, and we shall write 7-11 = L2 (M) and 7i = H1® £2. Consequently, we shall also use the symbols 7r1i ire to denote the representations underlying the above M-modules. (Thus, for instance, ir,,, (x) = 7r1(x) (9 idez.) Also we shall write MO(M) = M 0 G((2) and think of elements of MO(M) as infinite matrices ((xij)) with entries from M .
Notice now that '1 is actually an M-M-bimodule, with the right action of M being given by 7rr(y) = y = Jy*J6; in fact, Theorem 1.2.4 says that in this case, we have an equality ir1(M)' = 7r, (M). In order to deal with 7-1,,,, we shall find it convenient to think of 7-i as Mat1xo,('I) - by which we mean the Hilbert space of norm-square-summable
sequences with entries from '1. A moment's thought should convince the reader that 7-i is actually an M-MO(M)-bimodule with respect to matrix multiplication (explicitly, if = (1i 6, . .), x E M, y = ((yij)) E M. M, and that in fact, we have 7r (M)' _ and r1 = then 71j = >i 9rr(M.(M)). Notice now that MO(M) is a III factor, and the `faithful normal semifinite trace' Tr on it - as discussed at the end of the last section - is given by the obvious formula Tr((pij)) = E-1 trM(pii). (This is the `natural' normalisation to choose, in the sense that Tr(1M 0 q) = 1 whenever q is a projection in £(f2) of rank 1. Throughout the sequel, the symbol Tr - if it is used in the context of MO(M) - will always mean the one defined in this paragraph.) In the above terminology, the content of Theorem 2.1.6, at least as far as III factors are concerned, may be reformulated thus: THEOREM 2.2.2 If'H is any separable M-module, then there exists a projec-
tion p E MO(M) such that ' = 'gyp, and such a projection p is determined uniquely up to Murray-von Neumann equivalence.
2.2. DIMM7-l
25
We now come to a fundamental definition.
DEFINITION 2.2.3 Let 7-l denote an arbitrary separable module over a 11, factor M with separable pre-dual. Then define dimM h = Tr p where p E as an M-module.
is any projection such that 7-(0p is isomorphic to 7-C
We shall soon derive some of the basic properties of the assignment 7-l dimM 7-l, which should convince the reader that there is every reason to call this quantity the M-dimension of the module. Before that, however, we pause to (a) mention an example that might make the reader more favourably disposed to this definition, and (b) make a remark of a historical nature.
EXAMPLE 2.2.4 Suppose r is a discrete subgroup of a semisimple Lie group G such that covol(r) < oo. Assume further that r is an ICC group in the sense discussed in §1.4. Suppose now that it is a 'discrete-series representation' of G, meaning that it is a subrepresentation of the left-regular representation of G, or equivalently, that every (equivalently, that some non-zero) matrix77) where , 77 belong coefficient of 7r (i. e., a function on G of the form to the Hilbert space 7-l, underlying the unitary representation 7r) is squareintegrable with respect to Haar measure. It is then the case that 7rlr extends to an isomorphism of Lr onto 7r(r)"; consequently, the algebra ir(r)" is also a 11, factor; and it can be shown - see
[GHJ] - that dim,(r),, (7-l,) = covol(r) x d,r,
where d.7, denotes the so-called `formal dimension' of the discrete series representation 7r. REMARK 2.2.5 We should mention here that what we have termed dimM 7-l occurred first in the work of Murray and von Neumann ([MvN1]) as the socalled `coupling constant' of the module. Their definition is different from
the one presented here (and depends upon a result we do not prove here, since that is not really essential for our purposes). They define this coupling constant as infinity if ML (7-l) is an infinite factor, and if this is a factor of type III, they show the following is true: pick any 6 54 0 in 7-C and let p E M (resp., p' E mL(R)) be the projection operator whose range is (resp., [M6]); then the quotient trM(p) tr(Mc(x)) (p')
turns out to be a positive finite constant which is independent of the initial choice of vector t;; this is the number that they call the `coupling constant' of the module, and this number agrees with what we have defined as dimM7-l.
CHAPTER 2. SUBFACTORS AND INDEX
26
(In order to minimise notation, we assume in the next proposition that M C L(N) rather than that 7-L is some abstract M-module; there is no real distinction here since every unital normal homomorphism of a IIi factor is an isomorphism onto its image - see the appendix.) PROPOSITION 2.2.6 Let h be a separable Hilbert space, and suppose M C .C(H) is a IIi factor. (i) For each d E [0, oo], there exists an M-module Hd .iuch that dimMHd = d.
(ii) There exists a unique d E [0, oo] such that 7i = qtd; in particular, two M-modules are isomorphic if and only if they have the same M-dimension.
M' is a Ill factor. (iii) dimM 7-l < oo (iv) dimM L2(M) = 1. (v) If {K,,,}n is any countable collection of separable M-modules, then dimM(®n Kl) = E dimM /Cf. n
(vi) If dimM h < oo so that M' is a IIi factor (see (iii) above), and if p' E 'P(M), then dimM(p'9-l) = trM,(p') dimM H. (vii) p E P(M) dimMM(p11) = (trM(p))-' dimM7i. (viii) If dimM h < oo - see (iii) above - then dimM 7i = (dimM ?-()-i.
Proof: The first two assertions follow immediately from Theorem 2.2.2, and from the facts concerning the semifinite trace Tr that were listed out in §1.1.
We shall find the following description of hd, d < oo, convenient.
For d = n E N, H is the direct sum of n copies of ?-ll. As in the case of d = oo, we think of Hn as Matlxn,(7ii); observe, as before, that ?-t,, is naturally an M-M.,,(M)-bimodule (with respect to matrix multiplication) in such a way that MG(q-ln) = irr(Mn(M)).
If d E [0, oo), pick an integer n which is at least as large as d, pick a and set projection q in the Ill factor M,,(M) such that xd = H.q. (iii) Notice that if 7-l = ?-tip for some p E P(MM(M)), then M' _ ,
M,),) (M)P, and that Trp < oo 4= p is a finite projection.
(iv) If p = 1M ® ell, where ell denotes the matrix-unit (with 1 in the (1,1)-place and zeros elsewhere), then 74,0p = L2(M). (v) Let dimM K,, = d.. We can find mutually orthogonal projections pn in M,),) (M) such that Tr p.,. = dn for each n. If P = En p,., the desired assertion
follows from the fact that Tr is `countably additive' on P(M,),)(M)), since
®n, In = H-p. (vi) We may as well assume that 7-t = q'ad, where xd has been constructed
as in the discussion preceding the proof of (iii) of this proposition. Let n, q have the same meanings as above. In this case, since M1(Hd) = Mn(M)q,
27
2.2. DIMM7-l
it follows that there exists a projection p E M,,,(M) such that p < q and p' = 7r, (p). Then, p'R = Hp and consequently, we see that dimM(px) = ntrM (M)(p). On the other hand, it follows from the uniqueness of the trace in a 111 factor
that trM, (p) = trm. (M), (p) trM,(M) (p) trMM.(M) (q)
1-1 dimM(p'iI)
1 dimM H
(vii) Notice that the conclusion is `additive in 7-l', in the sense that if I-l = (D H is a decomposition of 7-l into countably many M-submodules, and if the desired assertion is valid for each 9-1n, then the desired assertion is valid for 7-l as well. Hence there is no loss of generality in assuming that dimM l-l < trMp(< 1). Hence - see the discussion just prior to the proof of (iii) above - we may
assume that R = (i11)q , where q E P(M) and q < p. It is then seen that dimMP(p1) = dimMp(p(iu1)q) = dimMM(p(xl)p - q) = dimMp(L2(MP)q) (by (vi) and (iv)) = trMp(q) . 1 trMq trMp dimM 9-l
trMp
(viii) We may assume that R = 7-ld is constructed as described in the comments preceding the proof of (iii) above. First, if d = 1, this is a consequence of Theorem 1.2.4 (1). Next, if d = n, note that ?-knell = R1 and that 7rr(ell) is a projection in 7r,(Mn(M)) = 7r1(M)' with trace -11; it follows from (vii) above that 1
= dim(M,c(n1)) 7i: dimM c(nn ), , (,11)) (Hnell ) i dim(MG(nn.))(xn); n
(?-1,,,) = 11, thus proving the assertion when d = n. A similar hence dimM reasoning, applied to the fact that 7 d is obtained by `cutting down' / I , proves the general case.
CHAPTER 2. SUBFACTORS AND INDEX
28
2.3
Subfactors and index
We begin with a look at some examples of bimodules (or correspondences, as introduced and popularised by Connes - see [Con2]).
Suppose, to be specific, that M is a III factor and that 7-i = L2(M). Then, as has been already observed, Ill is an M-M-bimodule in a natural fashion; but there are several other bimodule structures on L2 (M) as follows.
Fix an automorphism 8 of M, and define the bimodule 7-le to be L2(M) y = xl`8(y), as a Hilbert space, but where the actions are given by: x where the actions in the right side are as in Rl (and the actions on the left refer to the actions in 718). It is easy to see that 7-1e, so defined, is indeed an M-M-bimodule, which is irreducible in the sense that He has no proper sub-bimodules (or equivalently, the von Neumann algebra ML M(HO) reduces
to the scalar multiples of the identity operator). Thus, there are plenty of irreducible M-M-bimodules, although there are no irreducible M-modules.
Further, since the only left-M-linear maps on L2 (M) are of the form irr (x), x E M, it follows that if Oi E Ant M, i = 1, 2, then the bimodules 7-le, are isomorphic if and only if the automorphisms are -outer equivalent - meaning that there exists a unitary element u E M such that 81(x) = 92(uxu*) Hx E M. Thus, the set of equivalence classes of irreducible M-Mbimodules is at least as rich as the group of outer automorphisms (viz., the quotient of Ant M by the normal subgroup of inner automorphisms). In fact, it is much richer, as can be seen, for instance, from the fact that we could have defined the bimodule 7-i by only requiring B to be a unital endomorphism.
(It is a fact that if M is a factor of type III, then every M-M-bimodule is isomorphic to 7.18, for some endomorphism 0 - see the third paragraph of §4.1.)
It is clear that the assignment 7-l H (dimM_(R),dim_M(71))
(2.3.1)
defines an isomorphism-invariant of the M-M-bimodule 7-l. The bimodules 11e show that the above is not a complete invariant of the bimodule. Motivated by the success of Theorem 2.1.6, we are naturally led to the following question.
QUESTION 2.3.1 What can be said about the set {(dimM_(7-i),dim_M(7I)) :7-l is a separable M-M-bimodul}?
Before we get to this question, pause to notice that if M is a factor and if R is an M-M-bimodule, then, by definition of a bimodule, we see that irj(M) is a subfactor of 7rr(M)'; further, we see that, at least in this example, the subfactor has `trivial relative commutant' in the ambient factor precisely when the initial bimodule is irreducible. We formalise all of this in the next definition.
2.3. SUBFACTORS AND INDEX
29
DEFINITION 2.3.2 A subfactor of a factor M is a subalgebra N C M such that N is also a factor and N contains the identity element of M. The subfactor N is said to be irreducible if it has `trivial relative commutant' - i.e.,
ifN'nM=c1.
EXAMPLE 2.3.3 (a) Suppose ro is a subgroup of r, and suppose that both r and r0 are ICC groups. Notice then that Lro sits naturally as a-subfactor of the IIl factor Lr. Notice, further, that 22 (r) = L2(Lr) and that if r = Li r0si is the partition of r into disjoint cosets of ro, then 22(11) = ®(f2(ro))si is an orthogonal decomposition of f2(r) into (left) Lro- submodules (each of which is isomorphic to 22(ro) as an Lro-module, and consequently, we have dimLro(12(r)) = [r : ro].
(b) Suppose G is a finite group acting on a h i-factor P. It is true - see the first paragraph of §A.4 - that an automorphism 0 of a IIl factor is `free' in the sense of Definition 1.4.2 if and only if it is not an inner automorphism (i.e., not of the form Adu = u in the algebra). It follows that if P, G are as above, then the crossed product algebra P x G is a IIl factor if and only if the action is outer - meaning that no non-identity element of G acts as an inner automorphism. Suppose then that G acts as outer automorphisms of P, so that M = P x G is a IIl factor. It should be clear now that every subgroup H of G would yield a
subfactor N = P x H of M. Notice now that there is a natural identification L2(M) = L2(P) ® £2(G) and that if G = 11Hsi is the partition of G into distinct cosets of H, then (i) each of the subspaces 'Hi = L2(P) ® hE H}] is an N-submodule of L2(M), which is isomorphic, as an N-module, to L2(N), and (ii) L2(M) = ®7I,. It follows that dimN(L2(M)) = [G : H]. Motivated by the preceding examples, we make the following definition.
DEFINITION 2.3.4 If N is a subfactor of a IIl factor M, define the index of N in M by the expression [M : N] = dimN(L2(M)).
The next proposition shows that the index [M : N] can be read off from any M-module of finite M-dimension. PROPOSITION 2.3.5 Let N C M be an inclusion of IIl factors. Let 7I be any separable M-module such that dimM7-l < oo. Then dimN 7-l < oo -# [M : N] < oo;
in fact, we have the identity dimN 7-1 = [M : N] dimM R.
(2.3.2)
CHAPTER 2. SUBFACTORS AND INDEX
30
Proof: Suppose, to start with, that Ki, i = 1, 2, are any two M-modules with dimM Ki < oc. It follows then that there exists n E N and a projection q' E MG(K2 0 c") such that ICI = q'(/C2 (9 c"); thus we may deduce that dimN)C1 < dimN(K2 (9 E") = n x dime, K2. Since the roles of ICI and K2 are interchangeable, we see that dimN KI < oo ( dimN K2 < oo. In particular, we see that dimN h < 00 [M: NJ < oo. To prove the asserted equality, we may assume that, as an M-module, " copies 7l = 7l"q, where 7l" = L2(M)® ®L2(M),q E dimM7-l. Then dimN?-l" < oo so N'(= NL(7-l")) is a III factor, and hence, it follows that dimN 7-l = trN' 7rr (q) dimN 7-l"
= trM'lrT (q)n dimN hI = dimM?-l[M : N].
11
Before proceeding further, we record an immediate consequence of this fact and Proposition 2.2.6(viii).
COROLLARY 2.3.6 (a) If N C M is an inclusion of III factors, and if M C L ('H) and dimM 7-l < oo, then
[N':M']=[M:N]. (b) If N C M C P is a tower of III factors, then
[P: N]=[P:M][M:N]. In particular, if N is an M-M-bimodule which is bifinite in the sense that both dimM_ 7-l < oo and dim-MN < oc, it follows then that dimM_7-l dim_M7-l = [irT(M)' : irj(M)].
Thus, we find that the answer to Question 2.3.1 is tied up with the following related question.
QUESTION 2.3.7 What are the possible values of [M : N], where N C M is an inclusion of III factors ?
The following result - see [Jonl] - completely answers Question 2.3.7, and the answer is quite unexpected in the light of our experience with the classification of modules.
THEOREM 2.3.8 If N C M is any inclusion of III factors, then
[M: N] E {4cos2n : n = 3,4,
}U[4,oc].
n = 3, 4, } U [4, oo]), then there exists a subfactor Further, if .A E ({4 cos' RA of the hyperfinite III factor such that [R : RA] = A.
2.3. SUBFACTORS AND INDEX
31
In this section, we shall content ourselves with describing a construction of subfactors of all possible index values > 4. LEMMA 2.3.9 Let N C_ M be an inclusion of IIl factors, and suppose M C_ L (H) where dimM fl < oo. If p E P(N' n M), then
[pMp : Np] = trN'(p) trM(p) - [M: N]. Proof: It follows from Proposition 2.3.5 that
[pMp : Np] -
dimNp(pl) dimmMp(pH)
= (trN'(p)dimM7-l) dimM,,(p7-l)
= (trN'(p) dimN7i) (trM(p) dimM l), as desired.
PROPOSITION 2.3.10 Suppose M is a IIl factor such that Mp = Ml_p for some projection p E M. Let 0 : Mp -; Ml_p be such an isomorphism, and define N = {x + 0(x) : x E M}. Then N is a subfactor of M with [M : N] _ 1
l
trMp + trM(1-p)
Proof: Note that p E N'nM and that pMp = Np, so that [pMp : Np] = 1. Hence, by Lemma 2.3.9, 1 = trN (p) . trM(p) . [M : N],
and so trM(p)
= trN' (p) - [M : N]
.
Similarly, we also have 1
trM(1 - p)
= trN , (1 -p) - IM:
N]
Add the above equations to obtain the desired conclusion.
13
The existence of subfactors of the hyperfinite II, factor R with index at least 4 is an immediate consequence of proposition 1.4.9 and the preceding proposition (and the trivial fact that the mapping t H (t + l l t) maps (0,1) onto 0; oo) ). The preceding construction used the existence of non-trivial projections in the relative commutant. The following question, which naturally arises, still remains unanswered. QUESTION 2.3.11 Describe the set of index values [R subfactors Ro of the hyperfinite factor R.
:
Ro] of irreducible
CHAPTER 2. SUBFACTORS AND INDEX
32
We conclude this section with some useful consequences of Lemma 2.3.9.
PROPOSITION 2.3.12 Suppose N C M is an inclusion of IIl factors such that [M : N] < oo. If there exist pairwise orthogonal non-zero projections pl, , pn E N' n M, then [M: N] >n 2. In particular, (i) [M : N] < oo = N' n M is finite-dimensional;
(ii)[M N] <4
N'nM=E1.
Proof: Deduce from Lemma 2.3.9 that n
[M : N] > EtrM(pi)[M : N] i_1
n
>
1
E i1 trVi(pi)
> n2
(since rl, , r n E (0, 1), Ei ri = 1 = L i r > m2 The above inequality clearly implies (i) and (ii).
Chapter 3 Some basic facts 3.1
The basic construction
Suppose N C M is an inclusion of finite von Neumann algebras. Fix some faithful normal tracial state tr on M, and consider the M-M-bimodule 7i = L2 (M, tr), with its distinguished cyclic trace vector Q. Since 7-l is the completion of MS2, it follows that the subspace 7-1 = [NS2] of 7-l can be naturally identified with L2(N, tr). Let eN denote the orthogonal projection of 7-l onto the subspace 7-l1. It is a consequence of Theorem 1.2.4(1) that eN(MS2) C NS2, and hence the projection eN induces, by restriction, a map E : M -* N. The
map E is called the (tr-preserving) conditional expectation of M onto N, and is easily seen to satisfy the following properties: (i)
eNxeN = E(x)eN Vx E M,
(3.1.1)
and consequently E defines a (Banach space) projection of M onto N.
(ii) E is N-N-bilinear, meaning that E(nimn2) = n1E(m)n2.
(iii) tr o E = tr. It is clear from equation (3.1.1) that the conditional expectation is a preserving map, or, in other words, that JCN = eNJ,
(3.1.2)
where J denotes the modular conjugation operator on H. DEFINITION 3.1.1 The passage from the initial inclusion N C M to the von Neumann algebra (M, eN) = (M U {en})", and consequently to the tower N C M C (M, eN), is called the basic construction. We list some simple relations between the various objects involved in the basic construction. Also, we adopt the convention that M (and hence, also N) is identified, via 7r1, with a subalgebra of G(7-l); thus, for instance, the right action of M on 7i is given by ir,.(x) = Jx*J.
CHAPTER 3. SOME BASIC FACTS
34
PROPOSITION 3.1.2 With the foregoing notation, we have:
(i)eNEN'. (ii)N=MfI{eN}'. (iii) (M, eN) = JN'J. (iv) Assume both M and N are II, factors. Then: N] < oo; in this case, we
(a) (M, eN) is a II, factor i f and only i f [M have [(M, eN) : M] _ [M : N]. (b) tr(M,eN)(eN) _ [M: N]-1.
(c) (Markov property) EM(eN) = tr(M,eN)(eN) - where EM denotes the conditional expectation of the II, factor (M, eN) onto the subfactor M.
Proof: The first assertion is obvious; as for the second, the fact that the xeN is injective, trace vector SZ is separating for M implies that the map x and consequently, if x E M, it follows from equation (3.1.1) that x commutes with eN if and only if x = Ex. (iii) It follows from (ii) that
JN'J = J(M', {eN}")J = (JM'J, JeNJ) = (M, eN). (iv) (a) The first assertion is a consequence of (iii) above and Proposition 2.2.6(iii). As for the second, note that
[(M, eN) : M] = (dim(M,gN) = (dimJN'J 7-l)-1
= (dimN,
L2(M))-1
7-L)-1
= dimN 7-l
= [M:N]. (b) On the one hand, tr(M,eN)eN = trN,eN, while on the other,
1 = dimN L2 (N) = dimN(eN7I)
= (c) We need to verify that tr(M,eN)(xeN) = TtrM(x) Vx E M,
where T = tr(M,CN)(eN) = [M : N]-1. We first verify this identity whenever
x E N; for this, note that, in view of (i), the map x
tr(xeN) is a trace
3.1. THE BASIC CONSTRUCTION
35
on the 11, factor N which takes the value T at the identity; appeal to the uniqueness of the trace in a finite factor to conclude the proof in the special
case. The case of a general x E M reduces to the above case because of equation (3.1.1), thus: tr(M,eN) (xeN) = tr(M,eN) (xeN)
= tr(M,eN) (eNxeN) tr(M,eN) (EN(x)eN).
11
REMARK 3.1.3 In the sequel, if N C M is an inclusion of finite von Neumann algebras, if tr is a fixed tracial state on M, if (M, eN) is as above, and if 0 denotes an extension of tr to (M, eN), we shall say that eN satisfies the Markov property with respect to the algebra M (and 0) if it is the case that EM(eN) = q(eN), where EM denotes the unique q-preserving conditional expectation of (M, eN) onto M. Before proceeding further, we record two simple consequences of the last proposition.
COROLLARY 3.1.4 Suppose N C M is an inclusion of IIl factors. Then
[M : N] =1
N = M.
Proof: If eN and (M, eN) are as above, then the assumption [M : N] = 1 implies that tr(M,eN)eN = 1, and the faithfulness of the trace now implies that eN = 1; thus, in the notation of the first paragraph of this section, we have 71i = H. An appeal to Proposition 3.1.2 (i) completes the proof. COROLLARY 3.1.5 If N C M is an inclusion of IIl factors, then [M : N] V (1, 2).
Proof- On the one hand, it follows from (iv)(a),(b) of Proposition 3.1.2 that [(M, eN) : N] = [M : N]2. On the other hand, since eN E N' fl (M, 6N), the non-triviality of the relative commutant implies, via Proposition 2.3.12(ii), that we must have [M: N] > 2.
We conclude this section with one way of constructing subfactors of the hyperfinite IIl factor. EXAMPLE 3.1.6 Let us take the model R = R(2) (in the notation of §1.4) of the hyperfinite Ih factor. (Everything we say can be said just as well with any N in place of 2.) Let An have the same meaning as in the above-mentioned construction of R(N). Notice, to start with, that, for any n, there is an obvious identification A® ® A2 = An+2; for any x E A2, let us denote the image of
CHAPTER 3. SOME BASIC FACTS
36
1®x under this identification by xn+i,n+2. (Thus, xn+l,n+2 is just the element
x `sitting in slots n + 1, n + 2'.) Now fix a unitary element u E A2, and notice that the equation O.(x) =
(3.1.3)
(x)
defines a unital normal endomorphism 9,l of R = R(2). Consider the subfactor Ru = 9,,(R) of R. Thus, we get a family of subfactors of R parametrised by the unitary group U(4, C). It must be clear that the subfactor R1, corresponding
to u = 1, is the trivial subfactor Ri = R. On the other hand, if we take u to be the unitary operator implementing the `flip' on C2 0 C2 - or, in other words,
u=
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
- then it is easy to see that Bu defines the `shift', thus: O,(xi ®x2 (9 ...) = 10 X1 ®x2
®...
E M2(C). Thus, we find that in this case, R = M2(C)®R,,, and consequently, [R: R,l] = 4. Since the unitary group U(4, c) is connected, we find that `the index is a discontinuous function of the subfactor'! whenever x1, x2i
3.2
Finite-dimensional inclusions
In this section, we recall various elementary facts concerning finite-dimensional von Neumann algebras. To start with, any finite-dimensional C*-algebra A is semi-simple and hence, by the Wedderburn-Artin theorem, is isomorphic to the direct sum of finitely many matrix algebras over C. To be specific, if A is a finite-dimensional C*-algebra, then A = Mnl (C) ® Mnz (c) ® ... ® Mn, (C)
(3.2.1)
for a uniquely determined integer k(= dim Z(A)) and a uniquely determined nk} of positive integers. (The fact is that A admits exactly k set {n1, equivalence classes of (pairwise inequivalent) irreducible representations, and the dimensions of these representations are precisely the integers n1, , nk.) nk) If equation (3.2.1) is satisfied, we shall say that A is of type (ni, nk) as the dimension vector of A. and we shall refer to the vector n = (ni, Further, since a matrix algebra is a factor, it follows that if A is as above, then there is a bijective correspondence between the set of faithful tracial ,
,
,
3.2. FINITE-DIMENSIONAL INCLUSIONS
37
states on A and the open simplex Ak = {t = (tl, , tk)' E IRk : t, > 0 Vj and YET = 1} given by T " t , where tj = T(pj) is the trace of a minimal (= rank one) projection in the j-th summand of A. (The reason for the superscript i (which denotes `transpose') is that we shall find it convenient to adopt the convention that dimension-vectors are row-vectors while tracevectors are column-vectors.) That's about all there is to say about finite-dimensional C*- (and hence von Neumann) algebras. Suppose now that we are given an inclusion A C B of finite-dimensional C*-algebras. Then, besides the dimension-vectors, say n = (nl, nk) and ,
m = (ml,
ml), of A and B, respectively, there is more data needed to describe how the smaller algebra is included in the larger. The more that is needed is the k x l inclusion matrix A = AA described thus: Azj is the number of times that the i-th irreducible representation of A features, in the restriction, to A, of the j-th irreducible representation of B. Less formally, but more transparently, this is the number of times the i-th summand of A gets repeated in the j-th summand of B. This data can clearly be encoded equally efficiently in a bipartite (multi-) graph with k even and 1 odd vertices, where the i-th even vertex is joined to the j-th odd vertex by A2, bonds. (This graph is sometimes referred to as the Bratteli diagram of the inclusion A C_ B.) It is a fact that the isomorphism-class of the inclusion is uniquely determined by the inclusion matrix and the dimension vectors of A and B. EXAMPLE 3.2.1 The inclusion
A = {([0 0] '[O y],[Y1):XEM2(C),YEC} n
B = M3 (C) ®M2(E) ED Mi (IC) is described by the data
A
n=
n
AA
B
m=
Bratteli diagram
(2,1) 1
[
1
0 2
0 1
(3,2,1)
It must be clear from the definitions that if the inclusion is unital - i.e., if the subalgebra contains the identity of the big algebra, and this is the only kind of inclusion that we shall ever consider here - then the dimension-vectors are related to the inclusion matrix by the equation m = nA.
(3.2.2)
CHAPTER 3. SOME BASIC FACTS
38
Similarly, if a tracial state T on A (resp., of on B) corresponds to the 'trace-vector' t (resp., s), then (3.2.3)
It should be clear that if we had a whole tower
A1CA2C...CA,CA,+1C ... of finite-dimensional C*-algebras, then we could stack up the Bratteli diagrams corresponding to the several inclusions and obtain the Bratteli diagram of the tower. (Of course, for this to be possible, one must consistently label the direct summands of any algebra in the tower.) Or, in terms of the inclusion matrices, one should observe that - with the summands of the Ak's AAn+2 = consistently labelled . AA.+2 AAn+I
For instance, we might have the following tower:
A1C A2C A3C A4C A5C . .
A good feature about a tower {A,,,}' 1, as above, is that it has `finite width' - meaning that sup, dim Z(A,,,) < oo - and the Bratteli diagram is connected and is periodic (of order two). The reason this is `good' is that for such a tower, the algebra U An admits a unique tracial state and yields the hyperfinite IIl factor as its completion in the GNS representation associated with the trace - as in our discussion of R(N) in §1.4. (This is a consequence of the general fact that if {A}1 is an arbitrary tower of C*-algebras and if T is a tracial state on A0,, = U A,,,, and if ir, denotes the GNS representation associated with T, then 7r, (A,,,)" is a factor if and only if T is an extreme point in the set of tracial states of Ate.)
We now consider the basic construction, when applied to an inclusion A C B of finite-dimensional C*-algebras. Notice that (B, eA) is, by definition, an algebra of operators on the finite-dimensional space L2(B, a), and is hence also finite-dimensional.
LEMMA 3.2.2 (a) Let A C B be a unital inclusion of finite-dimensional C*-
algebras. Fix a faithful tracial state ¢ on B, and let J denote the modular conjugation operator on L2(B, 0). Let e denote the orthogonal projection of L2(B, 0) onto the subspace L2(A, OJA); let EA denote the 0-preserving condi-
tional expectation of B onto A; and let B1 = (B, e) denote the result of the
3.2. FINITE-DIMENSIONAL INCLUSIONS
39
basic construction. Then:
(i) ze = Jz*Je, Vz E Z(A); (ii) if b E B, then b c Z(B)
b = Jb*J;
(iii) z H Jz*J is a *-isomorphism of Z(A) onto Z(Bi); (iv) if pl,
, p,,,,,
(resp., q1,
,
q,) is an ordering of the set of minimal cen-
tral projections of A (resp., B), and if we set pi = JpiJ, 1 < i < m, then pl, , p,,,, is an ordering of the minimal central projections of B1; if A is the inclusion matrix for A C B, computed with respect to the pi's and qt's, then the inclusion matrix for B C B1, computed with respect to the qt's and the pi's, is the transpose matrix A. (b) Furthermore,
(i) the map a H ae is a *-isomorphism of A onto eBie;
(ii) if p is a minimal central projection in A, and if po is a minimal projection of A such that po < p, then poe is a minimal projection in B1 which is majorised by the minimal central projection JpJ of B1. Proof: (a)(i) For all z E Z(A), b E B, we have zebOB = zEA(b)OB = EA(b)zQB = Jz*JEA(b)QB = Jz*JebOB. (ii) This is obvious.
(iii) The map a - Ja*J is an anti-isomorphism of A into L(LZ(B)), and hence its restriction to the abelian subalgebra Z(A) is an isomorphism onto J(A n A')J = B1 n B1. (iv) Only the assertion about the inclusion matrices needs proof. If A denotes the inclusion matrix for B C B1, the definitions imply that Aij = [dimc(pig7A'pigj npiggBpiqa)]2,
while Aji = [dimc(gjpiB'g1pi n q.jpiBlg1pi)[ 2.
Notice, by (ii) and (iii) above, that gjpi = piqj = JpiqjJ, so that gjpiB'gjpi n gjpiB1g7pi = JpigjJB'JpigjJn JpigjJB1JpigjJ = J(piqjBpigq n pigjA'pig9)J,
thereby showing that indeed, A,i = Aid. (b) (i) Clearly the assignment a F--> ae defines an injective *-homomorphism
of A into eBie. To see that it is surjective, notice first that - thanks to the fundamental equation ebe = E(b)e Vb E B, where E denotes the 0-preserving
40
CHAPTER 3. SOME BASIC FACTS
conditional expectation of B onto A - the set {bo + E2 1 bieb' : n > 0, bi, bi E B} is a *-subalgebra of B, which contains B U {e} and is consequently equal to all of B1; since ebe = E(b)e, e(beb')e = E(b)E(b')e, it follows that indeed eBle = Ae. (ii) This follows at once from (b)(i), (a)(iii) and (i). We thus find that the isomorphism-type of (B, eA) is independent of the initial state o-. It turns out, however, that there is one `good choice' of trace on the big algebra - which is unique, under mild `irreducibility' assumptions - as shown by the next result. A word about notation before we state the next result (which is necessary because traces on finite-dimensional algebras are far from unique): if A C_ B are as above, and if o- is a tracial state on B, we shall write o- = tr7 if o- and s are related as in equation (3.2.3); further, in this case, we shall write EAT for the unique trs preserving conditional expectation of B onto A. PROPOSITION 3.2.3 Let A = AA denote the inclusion matrix, where A C B is an inclusion of finite-dimensional C*-algebras as above; and let r = trt be a tracial state on B, as described above. The following conditions on the trace r are equivalent:
(i) r extends to a tracial state r1 = trig on (B, CA) such that EB (eA) _ for some scalar A;
1
(ii) A'At = A-lt. Further, when these conditions are satisfied, the scalar A must be the reciprocal of the Perron-Frobenius eigenvalue of the (positive semi-definite and entrywise non-negative) matrix AA, and hence, 1 = IIA'AII = IIAII2.
Proof: Let pi, qj, pi be as in Lemma 3.2.2(a)(iv); we assume that it is these ordered sets of minimal projections in the three algebras A, B and (B, eA) with respect to which inclusion matrices and trace-vectors are described. Thus, the inclusion matrix for B C (B, eA) is just A' (by Lemma 3.2.2(a)(iv)).
Let r1 = trig be any tracial state on (B, eA) which extends r (or equivalently, ti is a trace vector for (B, CA) such that A71 = t). For each i, pick a minimal projection p° in A such that p° < pi, and set p° = p°e. It follows from Lemma 3.2.2(b)(ii) that (At)i = T(p°)
and (t1)i = 'r, (AD) Vi.
(3.2.4)
On the other hand, it follows, by reasoning exactly as in the proof of Proposition 3.1.2(iv)(c), that Ea (eA) = Al
EA (eA) = Al.
(3.2.5)
3.3. THE PROJECTIONS EN AND THE TOWER
41
Hence condition (i) of the proposition is seen to be exactly equivalent to the requirement (i)' There exists a trace-vector tl for (B, eA) such that A'11 = t and.-'tl = At.
If (i)' is satisfied, then A'At = .-'A't, = .\-'I and (ii) is satisfied. Conversely, if (ii) is satisfied, and if we define tl = )Ai, we see that (ii)' is satisfied.
DEFINITION 3.2.4 A trace T which satisfies the equivalent conditions of Proposition 3.2.3 is said to be a Markov trace for the inclusion A C B. COROLLARY 3.2.5 Let A C B be an inclusion of finite-dimensional C*algebras.
(a) If T is a Markov trace for the inclusion A C B, then it extends uniquely
-
to a state trig on (B, eA) which is a Markov trace for the inclusion B C (B, eA) (b) The following conditions are equivalent:
(i) there exists a unique Markov trace for the inclusion A C B; (ii) the Bratteli diagram for the inclusion A C B is connected. Proof: (a) This is clear from (the proof of) Proposition 3.2.3. (b) The content of Proposition 3.2.3 is that trt is a Markov trace for the inclusion A C B precisely when t is a Perron-Frobenius eigenvector of the (entry-wise non-negative) matrix AA'. The point is that the general theory
says - see [Gant], for instance - when such an eigenvector is unique, and that answer, when translated into our context, is that (ii) is precisely what is needed to ensure that uniqueness.
3.3
The projections en and the tower
In this section, we shall be interested in an (initial) inclusion M_1 C_ Mo of finite von Neumann algebras, which falls into one of the following cases:
Case (i) Mo and M_1 are III factors, and [M0 : M-1] < oo; Case (ii) Mo and M-1 are finite-dimensional; in this case, we shall always assume that the inclusion is `connected' (meaning that the Bratteli diagram is connected), and we shall reserve the symbol trM0 for the Markov trace for the inclusion, which is unique by Corollary 3.2.5(b). In either case, write M1 = (Mo, el), where we write el for the projection we denoted earlier by eM_1. The reason for the changed notation is that then the inclusion Mo C M1 falls into the same `case' above, as did the initial
CHAPTER 3. SOME BASIC FACTS
42
inclusion M_1 C_ Mo; hence, we can iterate the above procedure to obtain a tower
M19M29 9 M.9Mn+1C...
(3.3.1)
where Mn+1 = (Mn, en+I) is the result of applying the basic construction to the inclusion .a_1 C_ Mn, and e,,,+I denotes the projection implementing the (trM, -preserving) conditional expectation of Mn onto Mn_1. It must be clear that in case (i), the tower (3.3.1) is a tower of III factors such that [Mn+l : Mn] is independent of n; and that in case (ii), it is a tower of finite-dimensional C*-algebras. In either case, we see that U Mn comes
equipped with a tracial state tr (whose restriction to Mn is trMJ; in fact this is the unique tracial state on U Mn, and consequently we see that M,,, _ nrt,(U Mn)" is a III factor (which is hyperfinite in case (ii), and also in case (i) provided that each of Mo and M_1 is). EXAMPLE 3.3.1 (i) Suppose M_1 = Cl C MN(C) = Mo; then the inclusion
matrix Am', is the 1 x 1 matrix [N]. It follows that Mn_1 - MN. (C) on MN (C) do > 0, and we find that M,),, = R(N). If we identify Mo with the subalgebra MN(C) 0 1 of MN(C) 0 MN(C) = M1, then the projection el is given, in terms of the usual system {eij} j=1 of matrix units of MN(C), by the formula el = N N_I eij ®eij. (ii) (In a sense, this is a `square root' of the last example.) 1] and it follows Suppose M_1 = C1 C_ CN = Mo. Then AM 1 = [1 1 MN-() do > 1. If we think of Mo as the diagonal subalgebra that M2n_I of MN(C) = M1, then the projection el is the N x N matrix with all entries equal to N The sequence {en}°n°_1 of projections plays a central role in the theory of
subfactors. We list below some properties of this sequence.
PROPOSITION 3.3.2 Let {en}' 1 be the sequence of projections in the IIl factor Mme, constructed, as above, from an initial inclusion M_1 C_ Mo which
falls into either case(i) or case (ii). Write tr for the unique tracial state on
M. Then: (i) the number T = tr en is independent of n; further, 1_ T
[Mo : M_1] IAMo I
1
I I2
in case (i), in case (ii);
(ii) tr(xen) = T tr x Vx E Mn_1, n > 1; (iii) en E M;,_2 fl Mn do > 1, and in particular, ene,n = emen
(iv) en+lenen+l = Ten+I do > 1; (v) even+1en = Ten do > 1.
if Im - nj > 1;
3.3. THE PROJECTIONS EN AND THE TOWER
43
Proof: The Markov property and the fact that e,,, has trace equal to T are proved in Proposition 3.1.2 (for case (i)) and Proposition 3.2.3 (for case (ii)); this takes care of (ii) and (i). Assertion (iii) is a consequence of Proposition 3.1.2 (i). (iv) Since tr restricts, on M,,,, to trMM, it follows from equation (3.1.1) that en+Ixen+i = EMn_1(x)e,,,+i Vx E M,,,; in particular, if we choose x = e,,, and use the already proved (ii), we get (iv). (v) It follows from (iv) that u = T-2e,,,en+1 is a partial isometry in Mme, with initial projection given by u*u = en+1 The final projection of u clearly satisfies uu* < e,. On the other hand, we must have
T = tr en > tr(uu*) = tr(u*u) = tr en+i = T; thus we must have tr uu* = tr en; since tr is a faithful trace, this implies that uu* = en, as desired.
Notice the two expressions for T in Proposition 3.3.2(i); this is just an indication of the deeper relationship between index of subfactors and squares of norms of non-negative integer matrices. Parallel to Theorem 2.3.8, there is an old result which is essentially due to Kronecker - see [GHJ] - which says that if A is a finite matrix with integral entries, then 11A11 E {2 cos
n n = 2,3,4,
} U [2, oo].
To better understand this statement, recall that - exactly as the inclusion matrix AA is related to the Bratteli diagram for the inclusion A C_ B of finite-dimensional C*-algebras - there is a bijective correspondence between finite matrices with non-negative integral entries and finite bipartite (multi-) graphs. The matrix G which corresponds to a bipartite graph G is referred to as the `adjacency matrix of 9' and JI( Cl will also be called the norm of the graph 9. Closely tied up with Kronecker's theorem (and the well-known classification of Coxeter graphs or Dynkin diagrams) is the fact that the only (bipartite) graphs with norm less than two are the ones listed below, where the subscript refers to the number of vertices in the graph. An Dn
-0 ... 0- ... a I
I
44
CHAPTER 3. SOME BASIC FACTS
Now return to the case of a general initial inclusion M_1 C_ Mo (which falls into case (i) or case (ii)). Let e,,, ,r be as in Proposition 3.3.2(i). (Thus, according to the foregoing discussion and Theorem 2.3.8, either 7-1 _> 4 or 71 = 4 cost , for some integer m > 3.) Consider the algebras defined by Q /,r_11 n
_
{ C
ifn>0, "
if n E{-1, 0}.
(This notation turns out to be justified, as the following discussion shows.) It is an easy consequence of the properties in Proposition 3.3.2(iii), (iv) and (v) that each Qn(T-1) is finite-dimensional. It turns out - see [Jonl] for details - that for any T < 1, the Bratteli diagram for the tower {Q-(T-1)}° _1 is given by a 'half-Pascal-triangle', and that the Bratteli diagram for the tower {Q,(4 cost M)}- -1 is obtained by starting with the half-Pascal-triangle and slicing it off in a vertical line between the vertices labelled (m-3) and (m-2). (The cases m = 4, 5 have been illustrated below.)
T-1 > 4
T 1 = 4 COS2
4
T 1 = 4 cost
The analysis in [Jonl] goes on to show that if r = 4sec2, where m > 3, the algebra UQn(T-1) admits a unique tracial state, and hence has a copy Ro of the hyperfinite h i factor as a von Neumann algebra completion (meaning weak closure of its image under the associated GNS representation); and if R_1 denotes the von Neumann subalgebra of Ro generated by {en : n > 2}, then R_1 is a subfactor of Ro such that [Ro : R_1] = T-1. Most of what was said in the last paragraph is also true when T < 2 (and
is proved in [Jonl]); the only statement that needs to be modified is that when T < 2, it is no longer true that UQ,, (T-1) admits a unique trace, but it is true nevertheless that the completion of UQn(T-1) with respect to the GNS representation associated with the Markov trace is the hyperfinite factor R.
45
3.3. THE PROJECTIONS EN AND THE TOWER
Hence, in order to exhibit a subfactor of R with index 4 cos' , it suffices to find an inclusion M_1 C Mo whose Bratteli diagram has norm 2 cos and pick an inclusion with this and for this, we may choose the graph then R_1 C Ro meets our requirements. as Bratteli diagram - and Before concluding this section, we wish to point out that if'r, R_1i Ro are as above, we have an instance of simultaneous approximation of the subfactorfactor inclusion R_1 C Ro by finite-dimensional inclusions in the following sense : (el)
C
U C
(e1, e2)
C
...
C
U C
(e2)
(e1,e2i...
en)
C
U
U C
...
C
(e2i ... efl)
Ro
C
R-1
Chapter 4 The principal and dual graphs 4.1
More on bimodules
Suppose M, P are arbitrary von Neumann algebras with separable pre-duals, and suppose 7-l is a separable M-P-bimodule. Pick some faithful normal state co and set 7-ll = L2(M, cp) and 7-l = 7-l® ®Q2. It follows from Theorem 2.2.2 that 7-l may be identified, as a left M-module, with 7-l,,.q for some projection
q E M(M) (which is uniquely determined up to Murray-von Neumann equivalence in MO(M)); further, we have MG(R) = 7rT(Moo(M)q). Since 7i
is an M-P-bimodule, it follows from our identification that there exists a normal unital homomorphism B : P -> such that the right action of P is given by y = 60(y). Conversely, given a normal homomorphism B : P -> M(M), let 7-I, denote the M-P-bimodule with underlying Hilbert space 7-I 9(1), and with
the actions given, via matrix multiplication, by m 6 p = m,B(p). The content of the preceding paragraph is that every separable M-P-bimodule is isomorphic to 7-t for suitable B.
If M is a factor of type III, then so is MO(M), and hence every non-zero projection in MO(M) is Murray-von Neumann equivalent to 1M 0 ell. Consequently, every M-M-bimodule is isomorphic to 7-i for some endomorphism
9:M-*M.
Suppose M and P are II, factors and suppose 7-t is as above. (In this case, we naturally take cp = trM.) Notice then that dimM-f is finite precisely when Tr 0(1) < oo, while dim_p 7-l is finite precisely when the index [M, (M), : B(P)] is finite. This suggests that we define a co-finite morphism of P into M as a normal homomorphism B : P -* M(M) such that (i) 0(1) is a finite projection, say q, in MO(M), and (ii) B(P) has finite index in M,,(M)q. The point is that if B is a co-finite morphism of P into M, then 7-t = 7-I 9(1) is a bifinite M-P-bimodule, and every bifinite M-P-bimodule arises in this fashion. Motivated by the case of automorphisms, we shall say that two co-finite
morphisms Bi, i = 1, 2, of P into M are outer equivalent if there exists a
48
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
partial isometry u c MO(M) such that u*u = 01(1), uu* = 02(1) and 02(x) = u01(x)u* Vx E P; this is because, with this definition, two co-finite morphisms 0; from P into M are outer equivalent if and only if the corresponding bifinite M-P-bimodules 7-la, are isomorphic. We wish, in the rest of this section, to discuss two operations - one unary (contragredient) and one binary (tensor products) - that can be performed with bimodules. (In the latter case, we shall restrict ourselves to the case when both the algebras in question are II1 factors.) Contragredients: Suppose 7-l is an M-P-bimodule. By the contragredient of 7-l, we shall mean any P-M-bimodule 71 for which there exists an anti- unitary p) = p* J m*. It is clear that such operator J : 7-1 -> 77 such that J(m a contragredient exists and is unique up to isomorphism.
Note that if 7-l = 7-lai and if 7-1 = 7-te, then 0 : P -* MO(M) while M -> MA(P), and the relationship between the morphisms 0 and B is somewhat mysterious. However, in the simple case when M = P and 0 is an automorphism of M, it is easy to verify that 8 (which is, after all, only determined up to outer equivalence) may be taken as 0-1. (The reader should have no difficulty in constructing a J which establishes that 7L -i is a 0
contragedient of R O.)
Tensor products: We shall discuss tensor products of bifinite bimodules over II1 factors. First, however, we want to single out a distinguished dense subspace of such a bimodule, namely the one consisting of the so-called bounded vectors.
Suppose, to be specific, that M, P are II1 factors and that 7-l is a bifinite M-P-bimodule. Say that a vector E 7-l is left-bounded (resp., right-bounded) Ktrp(p*p) Vp E P if there exists a constant K > 0 such that KtrM(m*m) Vm E M), or equivalently, if there exists a (resp., bounded operator Le L2(P) -* 7-l (resp., Re : L2(M) -* 7-l) such that :
Vp E P (resp., R (m12) = m Vm E M). It is true that a vector is left-bounded if and only if it is right-bounded. (Reason: assume 7-l = 7-te, for a co-finite morphism 0 : P -> M,,(M); thus a vector in 7-l is E (Mat1,,,,(L2(M))0(1); it follows easily from of the form = Theorem 1.2.4(1) - and the fairly easily proved fact that if Po is a subfactor of P of finite index, then a vector is left-bounded for the right action of P if and only if it is left-bounded for the right action of PO - that the above vector satisfies either of the boundedness conditions above precisely when each co-ordinate has the form i = xiSl for some xi E M.) Thus we may talk simply of bounded vectors. We shall denote the collection of bounded vectors in the bifinite bimodule 7-t by the symbol 7-10. We list some properties of the assignment 7-l F-+ 7-lo in the following prop o-
sition, which is easily proved by considering the case of the model 7-1g. (In any case, complete proofs of all the assertions in this section may be found,
4.1. MORE ON BIMOD ULES
49
for instance, in [Sun3]; actually, only the case M = P is treated there, but the proofs carry over verbatim to the `more general' case.) PROPOSITION 4.1.1 (a) Let M, P be IIl factors and let 7-l, K denote arbitrary bifinite M-P-bimodules. Then Ho is an M-P-linear dense subspace of R, and the assignment T F--/ T0 defines a bijective correspondence between the Banach space MIP(R, K) and the vector space MLP(R0i Ko) of M-Plinear transformations between the vector spaces Ho and 1C0. (b) There exists a unique mapping (i;, 71) F-4 rl)M from 7-lo x 7-lo into M, referred to as the M-valued inner product on the bimodule R, which satisfies the following properties, for all , rl, C E 7-lo, m E M, p E P:
(i)
rttl) =
(ii)
M
(iii)
rl)M) 0.
rl)M = (07, 0 M)*.
(iv) (m . + (, rl)M = (v)
77) M) + (c, 77) M.
-p*)M.
As has been already remarked, if R = R8, then we may identify 7-lo with Matlxn(M)9(1), and in this case, ((xi, ... , xn), (yi, ... , yn))M = E xiy2 i
Further, if K = R0, then the typical element of MLp(R, K) is of the form
is a matrix satisfying 9(p)T = T¢(p) Vp E P. One reason for introducing the M-valued inner product is to facilitate the formulation of the universal property possessed by the tensor product.
" fT (matrix multiplication), where T E
PROPOSITION 4.1.2 Suppose M, P and Q are IIl factors, and suppose 7i (resp., K) is a bifinite M-P-bimodule (resp., P-Q-bimodule). Then there exists a bifinite M-Q-bimodule, denoted by 7-l®pK, which is determined uniquely up to isomorphism, by the following property: There exists a surjective linear map from the algebraic tensor product7L00 KO onto (7-l ®p K)0, the image of ® rl being denoted by ®p rl, satisfying:
(a) 'p®prl (b)
(c) (S ®p 77, S' ®p 71,)M = (S - 0, Op, 0M.
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
50
The useful way to think of these tensor products is in the reformulation in terms of co-finite morphisms. Suppose, to be specific, that 7L = 7-Ho,1C = 7-14,
for some co-finite morphisms 0 : P -* M,, (M), 0 : Q -4 M,, (P). It is not hard to see that the equation (0 ® I)ij,kl(q) = Oik(Ij1(q))
(4.1.1)
defines a normal homomorphism 0 ®0 : Q - M,,,,,(M); it is further true (although co-finiteness requires some work) that 0 ®0 is a co-finite morphism
from Q into M and that 7-le®o = 7-i ®p 'Ho. In particular, in the special case when M = P = Q and 0, 0 E Aut(M), we find the reassuring fact that
B®Qi=9o0. Another immediate consequence of the morphism description of the tensor product is the multiplicativity of dimension under tensor products, meaning that if 7i (resp. K) is an M-N- (resp., N-P-) bimodule, then dimM_ (7-l ON K) = dimM_ 7-l dimM_ K
(4.1.2)
dim_p(7-l ON)C) = dim-NH dim_p)C.
(4.1.3)
and
4.2
The principal graphs
We assume throughout this section that N C_ M is an inclusion of 11, factors
such that [M : N] < oo, and that (4.2.1)
for n > 0. It follows from Proposition 2.3.12 that {Mi n Mj : -1 < i < j} is a grid of finite-dimensional C*-algebras, which is canonically associated with the inclusion N C M, and is consequently an `invariant' of the initial inclusion. It turns out see Proposition 4.3.7 - that there is a periodicity of order two and hence we need to consider only i = -1 and i = 0. and belongs to N' n Let us consider = -1 first. Notice that is the tower of the basic construction, with M,,,+1 = (Mn,
implements the conditional expectation of N' n Mn onto N' n M,,,_1. It follows contains a copy of the basic construc- from Lemma 5.3.1(b) - that C_ (N' n M,,,), and consequently the Bratteli tion for the inclusion (N' n C (N' n M,,+1) contains a `reflection' of diagram for the inclusion (N' n C (N'nM,,). The graph obthe Bratteli diagram for the inclusion
tained by starting with the Bratteli diagram for the tower {N'nM : n > -1} of relative commutants, and removing all those parts which are obtained by reflecting the previous stage, is called the principal graph invariant of the inclusion N C M. A similar reasoning also applies for i = 0, and the resulting graph is called the dual graph invariant of the inclusion N C M.
4.2. THE PRINCIPAL GRAPHS
51
We shall now give an alternative description of these two graphs, using the language of bimodules. (The equivalence of these two descriptions is established in §4.4.)
The principal graph is the bipartite multi-graph 9 defined as follows: the set gi°) of even vertices is indexed by isomorphism classes of irreducible N-N-bimodules which occur as submodules of NL2(M.M)N for some n >
1; the set g(1) of odd vertices is indexed by isomorphism classes of irreducible N-M-bimodules which occur as submodules of NL2(M.M)M for some
n > 1; the even vertex labelled by the irreducible N-N-bimodule X is connected to the odd vertex labelled by the irreducible N-M-bimodule Y
by k bonds, if k is the multiplicity with which X occurs in the NN-bimodule Y. The dual graph is the bipartite multi-graph it defined as follows: the set 'H(o) of even vertices is indexed by isomorphism classes of irreducible M-Mbimodules which occur as submodules of ML2 (M,)M for some n > 1; the set i-l(l) of odd vertices is indexed by isomorphism classes of irreducible M-Nbimodules which occur as submodules of ML2(M.M)N for some n > 1; the even vertex labelled by the irreducible M-M-bimodule X is connected to the odd vertex labelled by the irreducible M-N-bimodule Y by k bonds, if k is the multiplicity with which Y occurs in the M-N-bimodule X.
It is a consequence of Proposition 4.3.7 that the principal (resp., dual) graph for the inclusion M C M1 may be identified with the dual (resp., principal) graph for the inclusion N C M. Hence any result concerning the principal graph has a corresponding statement about the dual graph. So we shall restrict ourselves, in this section, to making some comments concerning the principal graph. The principal graph has a distinguished vertex *G which corresponds to the even vertex indexed by the isomorphism class of the standard bimodule NL2(N)N. Notice that the odd vertices at distance 1 from *G are indexed by isomorphism classes of irreducible N-M-submodules of NL2 (M)M; hence the case of irreducible subfactors corresponds to the case where *G has a unique neighbour. Further, since the tensor product of bifinite bimodules is a direct sum of only finitely many irreducible submodules, it follows from the description of the principal graph that each vertex in 9 has only finitely many edges incident on it; thus the principal graph is a locally finite connected pointed bipartite graph.
Suppose 9 is the principal graph of the inclusion N C M, as above. Let G denote the (non-negative integer-valued) 9i°> x O' matrix with gxy equal to the number of bonds joining the even vertex X to the odd vertex Y in the graph 9. We shall also sometimes use the suggestive notation gxy = (X, Y). Even if the graph 9 is infinite, the local finiteness of g translates into the fact that the matrix G has only finitely many non-zero entries on any row or column.
52
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
We are now ready to prove an important relation between the index of the subfactor and the norm of the matrix G, which will, in particular, reduce the statement in Theorem 2.3.8 concerning the restriction on index values of subfactors to the classification of graphs of norm less than 2.
PROPOSITION 4.2.1 Let N C M be an inclusion of III factors such that [M : N] < oo, and let G, 9 be as above. (i) G defines, via matrix multiplication, a bounded operator from £2(g(')) into £2(g(0)), with IGII2 < [M : N]. (ii) If [M : N] < 4, then G is necessarily finite and is one of the Coxeter diagrams A,, D,, E6i E7 or E8. Proof: (i) In order to prove (i), we shall prove the clearly equivalent statement that GG' defines a bounded self-adjoint operator on 22(c(°)) with norm
at most [M: NJ. It is a consequence of the `Frobenius reciprocity' statement contained in Proposition 4.4.1 that (NXN) ON (NL2(M)M)
® (X, Y) - NYM, VX E G(°), YE9(1)
where we write m 7-1 to denote the direct sum of m copies of the module H. Equating left and right dimensions, we get
[M: N] dimN_ X = E gxY dimN_ Y, VX E g(°),
(4.2.2)
YE9(1)
and
dimN X = E gxY dim_M Y, dX E g(°).
(4.2.3)
YE9(1)
Similarly, we find that
(X, Y) NXN VY E
(NYM) ®M (ML2(M)N) XE9(0)
Equating dimensions, we get
dimN_ Y = i gxY dimN_ X, VY E 9(l),
(4.2.4)
XEc(°)
and
gxY dimN X, VY E
[M : N] dim_M Y =
(4.2.5)
XE9(0)
It is an immediate consequence of equations (4.2.2) and (4.2.4) that if we define vX = dimN X, VX E g(°), then v is a (column-) vector indexed by 90) such that (GG')v = [M: N]v. On the other hand, it is a consequence of the Perron-Frobenius theorem - see [Gant], for instance - that if the adjacency matrix of a connected graph
4.2. THE PRINCIPAL GRAPHS
53
has a positive eigenvector, then that adjacency matrix defines a bounded operator (on the Q2 space with basis indexed by the vertices of the graph) with norm bounded by the eigenvalue afforded by the positive vector. This completes the proof of (i).
(ii) If [M : N] < 4, it follows from (i) that G < 2, and we may appeal to the classification of matrices of norm less than 2 - see the discussion in §3.3.
We now discuss some examples of principal graphs. EXAMPLE 4.2.2 Let N = (1, e2, e3, ) C M = (1, ei, e2, ), as in §3.3. It is a consequence of a result called Skau's lemma - see [GHJ] for details - that = A,,_1 if [M : N] = 4 cost ; thus, in these cases, we find that the Bratteli diagram for the tower {N' fl M,,, : n > -1} is precisely the truncated Pascal triangle of §3.3. (Since N' n M,,, (1, e1, , en), this means, in view of the description of the Bratteli diagram for the tower {Q.(T 1)} discussed in §3.3, that the inclusion above is actually an equality.) When [M : N] > 4, however, things are quite different - see [GHJ] - and the principal graph turns out to be A_........ which is the infinite path extending
to infinity in both directions, so that we have a strict inclusion N' fl Mn D (1, el, , en) in this case, and the Bratteli diagram for the tower {N' fl Mn n > -1} turns out to be the full Pascal triangle. EXAMPLE 4.2.3 Suppose G is a finite group acting as outer automorphisms
of a III factor P. (i) Let N = P C P x G = M. In this case, the principal graph is an n-star (where n = IGI) with all arms of length one; thus, for instance, if IGI = 3, the principal graph is just the Coxeter graph D4.
(ii) Let N = PG C P = M be the subfactor of fixed points under the action. Then the principal graph 9 has one odd vertex and the even vertices are indexed by the inequivalent irreducible representations of G, with the even vertex indexed by it connected to the unique odd vertex by d, bonds, where d denotes the degree of the representation ir.
It is a fact - see §A..4 - that if Ml = (M, eN) is the result of the basic construction, then the inclusion M C Ml is isomorphic to the inclusion M C_
M x G, and hence the discussion in the previous paragraph amounts to a description of the dual graph of the case considered in (i).
(iii) More generally than in (i) and (ii), if H is a subgroup of G, we could consider the case N = P x H C P x G = M. This case is treated in §A.4, where the principal and dual graphs are explicitly computed, using the bimodule approach. EXAMPLE 4.2.4 Suppose {O
, , G } is a set of automorphisms of R, which is closed under the formation of inverses. It has then been shown by Popa -
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
54
see [Pop6] - that if
N=
x
0
..
0
01(x)
...
0
0
0 0
:xER CM,,+1(R)=M
0,,(x) j
denotes the so-called `diagonal subfactor' (given by the O 's), then the principal graph is described by what might be called the Cayley graph of the subgroup of Aut(R)/Int(R) generated by the O 's - where Int(R) denotes the group of inner automorphisms of R.
4.3
Bases
We assume in the rest of this chapter that N C M is a finite-index inclusion of III factors, and that
N=M_1CM=Mo9M1C...CM.CM.+IC...
(4.3.1)
is the tower of the basic construction, with M,, = (M,i_1i en,) for n > 1. Also
we use the notation r = [M : N]-I. Further, we shall identify M with the dense subspace M1 of L2(M). We begin with a very useful fact.
LEMMA 4.3.1 (i) If x1 E MI, there exists a unique element xo E M such that x1e1 = xoeli this element is given by xo = T-IEM(xlel). (ii) The action of M1 on L2(M) is given by xl(yoPM) = T-'EM(x1yoe1)1M
Proof: (i) Since EM(el) = T, the second assertion and the uniqueness in the first assertion follow. As for existence, we need therefore to show that T-IEM(xlel)el = x1e1 for all x1 E MI. Since the two sides vary (strongly) continuously with x1, it suffices to establish this equality for a dense set of x1's. Note that the set D = {ao + Ti 1 bielci ao, bi, ci E M, n E N} is a self-adjoint subalgebra of M1 which contains M U fell and is consequently strongly dense in MI. Finally it is easily verified that if x1 E D, then the desired equality is indeed valid. (ii) Again, we may assume that x1 E D, as above, and the desired equality :
is easily verified.
REMARK 4.3.2 (a) The proof shows that the preceding lemma is also valid when N C_ M is an inclusion of finite-dimensional C*-algebras, provided the trace we work with is a Markov trace.
4.3. BASES
55
(b) It follows from Lemma 4.3.1 (i) that the set of finite sums of elements
of the form aoelbo, where ao, bo E M, is an ideal in M1 and must consequently be equal to all of M1 by Proposition A.3.1. This shows that there is an isomorphism 0: M ON M , M1 such that q(ao (&N bo) = aoelbo PROPOSITION 4.3.3 Fix n > [M : N]. (a) Suppose q E Mn(N) is a projection such that there exist Al, , ) E M such that
qij = EN(A)) Vi, j.
'v
. Then (4.3.2)
A collection {a1i , An} C M will be called a basis for M/N if the matrix q = ((qij)) defined by equation 4.3.2 above is a projection in Mn(N) such that tr q = M:N n (b) Let {A1i , An} C M be basis for M/N. Let x E M be arbitrary. Then,
(i) x = j 1 EN(x)j))j; (xi,
,
(ii) the row-vector (EN(xAf), belongs to Mlx,,,(N)q; and if , x..) E Mi (N)q satisfies x = E 1 xiAi, then xj = Vj; (iii) further, J:' l A eAi = 1.
Proof:
(a) Let q = ((qij)) E Mn(N) be a projection such that trq = M:N n
Consider the projection E = ((Eij)) E MM(M1) defined by Eij = bije; notice that q and E are commuting projections, and so p = qE is also a projection in the II1 factor Mn(M1). Observe next that
trp =
1
n
tr (giie) = Ttr q =
n a=1
1
n
= tr ell,
where ell denotes the projection in M.M(M1) with (1,1) entry equal to the identity 1, and other entries equal to 0. Hence the projections p and ell are Murray-von Neumann equivalent in Mn(M1); thus, there exists a partial isometry v E Mn(M1) such that v*v = ell and vv* = p; the condition v*v = ell clearly implies that v has the form
v=
v1
0
...
0
v2
0
...
0 (4.3.3)
vn
0
.
0
0
..
0
for uniquely determined v1, , vn E M1. By definition of the vi's, we have
v vi = 1 i=1
(4.3.4)
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
56
and viva = gij e Vi, j
(4.3.5)
In particular, note that vivz = qiie < e, and hence, we must have vi = evi. Then, by the preceding Lemma 4.3.1, there exists a unique )i E M such that vi = eAi Vi. Deduce that gije = eAiA*e = EN(AiA )e, thereby establishing (a).
Before proceeding to (b), we wish to point out that every `basis for M/N' arises in the manner indicated in the proof above. (Reason: Suppose 1/\1, - ', fin} C M is basis for M/N. Define vi = eAi, and define v E by equation 4.3.3; it is then seen that vv* = qE, so that v is a partial isometry; also, it follows that the matrix v*v has 0 entries except at the (1,1) place, and that the (1,1) entry must be f, for some projection f E Ml, which satisfies trf = 1; in other words, v*v = ell, as desired.) (b) To start with, note that (iii) is an immediate consequence of equation 4.3.4 (and the fact that vi = eAi). As for (i), if x E M is arbitrary, then
ex =
n
exvi vi i=1 n
ex,\ eAi i=1
n
e(Y EN(x1\i)Ai), i=1
and deduce from Lemma 4.3.1(i) that x = 1 EN(x,2 )A . Note next that if x E M and if 1 < j < n, then, n
e(> EN(x)\i )qij) i=1
ex,\ egij i=1 71
exA eEN(AiA*) i=1 exA e\iA*e i=1
= ex.\ e eEN(xAj),
and it follows (again from Lemma 4.3.1(i)) that (YEN (xAi)gij) = EN(xAj) i=1
4.3. BASES
57
in other words, the row vector (EN(x)i), Mix.(N)q. Conversely, if (x1,
,
that
, EN(x)*)) indeed belongs to
x,,,) E Ml,,,,, (N) q, and if x = E l xiAi, then observe n
EN(xAj) = i=1 n
EN(xiAiA-) xiEN (Ai )*)
i=1
xj,
thereby establishing (ii) and hence the proposition. A collection {A1, , .Xn} C M which satisfies the condition (i) of the above proposition is, strictly speaking, a `basis' for M, viewed as a left Nmodule, and should probably be called a 'left-basis' (as against a right-basis, which is what the set of adjoints of a 'left-basis' would be); but we shall stick to this terminology. The above proposition is a mild extension of the original construction of a basis (in [PP]). (They only discuss the case when the projection q of the proposition has zero entries on the off-diagonal entries.) The reason for our mild extension lies in Lemma 4.3.4(i) and the consequent Remark 4.3.5. We omit the proof of the lemma, which is a routine verification.
LEMMA 4.3.4 (i) If N C M C P is a tower of Ih factors, with [P : N] < oo, and if {)i : 1 < i < n} (resp., {rlj : 1 < j < m}) is a basis for M/N (resp.,
P/M), then {)irlj 1 < i < n, l < j < m} is a basis for P/N. (ii) If {al, , Xn} is a basis for M/N, then {T 2 e) j 1 < j < n} is a basis for M1/M; hence {T-2Aie,Aj : 1 < i, j < n} is a basis for M1/N. :
REMARK 4.3.5 Notice, from Lemma 4.3..4(ii), that any element of M1 is expressible in the form E j=1 aijaieaj, with the aij's coming from N, which is a slightly stronger statement than the fact - see Remark 4.3.2(b) - that elements of the form aelb, with a, b E M, linearly span M1. Also, an easy induction argument shows that if {Ai : i E If is a basis for M/N, and if we define, for i = (i1, , ik) E Ik k > 1,
A1 =T
a
AilelAi2e2elai3.Aik
lek_l...elAik
then {X(k) : i E Ik} is a basis for Mk_1/N.
,.2k) for i, j E Ik, Notice next, that if we write i V j = (i 1, , ik, jl, it follows from the commutation relations satisfied by the en's - see Proposi-
tion 3.3.2 - that AiVj = T 2 .\j (ek ... el)(ek+l ... e2) ... 2
(e2k-1 ... ek)AJk)
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
58
This equation, together with the second statement in Lemma 4.3.4, should suggest that we might expect the validity of the next result. PROPOSITION 4.3.6 If Mi, ei are as above, then, for each m > 0, k > -1, the algebra Mk+2m is isomorphic to the result of the basic construction applied to the inclusion Mk C Mk+m,, with a choice of the projection which implements the conditional expectation of Mk+m, onto Mk being-given by e [k,k+m]
__(M-1) T
2
(ek+m+lek+m
ek+2) (ek+m+2 ... ek+3) ... (ek+2m
ek+m+1)
Proof: This is true basically because of the relations satisfied by the projections e,,,. As for the proof, there is no loss of generality in assuming that
k = -1. Rather than going through the proof of the proposition in its full generality, we shall just present the proof when k = -1, m = 2; all the ingredients of the general proof are already present here, and the reader should not have too much trouble writing out the proof in its full generality. In any case, the proof may be found in [PP2].
Thus we have to show that N C Ml C M3 is an instance of the basic construction, with a choice of the projection implementing the conditional expectation of Ml onto N being given by f = T-1 e2ele3e2First, the fact that el commutes with e3 implies that f * = f ; and f2 = T-2e2e1e3e2e3e1e2 = T-1e2e1e3e1e2 = f,
so f is indeed a projection. Next, for any xl E M1, note that Jx1f
T- 2e2e1e3e2x1e2e3e1e2
T_2e2e1e3EM(x1)e2e3e1e2
T-1e2e1EM(x1)e3e1e2
T_ Ie2elEM(xl)ele3e2 T- Ie2EN(x1)e1e3e2,
and hence, indeed, we have
fx1f = EN(xl)J dx1 E Ml. It follows that there exists a (unique) normal homomorphism it from P = (Ml, eN1) (the result of the basic construction for the inclusion N C_ Ml) onto
Pl = (M1 U If })", such that ir(eN') = f and lrlM, = idM,. (The normality, as well as the fact that image is a von Neumann algebra, may be deduced from the second statement in Lemma 4.3.4(ii).) Since P is a III factor, it follows that it is an isomorphism and that Pl is a III subfactor of M3 such
4.4. RELATIVE COMMUTANTS VS INTERTWINERS
59
that [P1 : M1] = [M1 : N] = [M3 : Ml]. This means that [M3 : P1] = 1, and the proof is complete.
We conclude this section with a proposition, due to Pimsner and Popa, which shows that the basic construction lends itself to a nice `duality' result, which, among other things, yields the `periodicity of order two' in the tower of the basic construction, which was referred to in §4.2. PROPOSITION 4.3.7 ([PP1]) Let d = [M : N]. Then, for each n > -1, there exists an isomorphism of towers:
(Md(N)9Md(M)c...cMd(M,M))-(M1cM29...cMn+2). Proof: We prove the case n = 0; the general case is proved similarly, by using Remark 4.3.5. Fix a basis {A1, , A,,j for M/N. Let q = 9(1), where Bij(x) = EN(Ais) ) Vx E M, and consider the model gM,,,(N)q of Md(N). Define c : Md(N) _ Ml by
= 0((a'ij))
n
E A aijeAj. i,j=1
An easy computation shows that 0 is a unital normal homomorphism, which is necessarily injective. As for surjectivity, if x1 E M1, deduce from Proposition 4.3.3(b)(iii) that
Si = (EAZel.i)xl(EA e,Aj) =
j
i
E Ai (e,AixA*el)Aj; i,j
but by Lemma 4.3.1(i), there exists mij E M such that mije1 = Aix1.\j*el. It is then clear that x1 = 0((EN(mij))), thereby establishing the surjectivity of 0. Now apply the conclusion above, to the basis {-r-2el.i} for M1/M, to find an isomorphism 01 : Md(M) -+ M2 defined by
gi((mij)) =T 1 E Aielmije2ejAj, i,j=1
which restricts on Md(N) to the map 0 defined above, thus proving the proposition.
4.4
Relative commutants vs intertwiners
This section is devoted to establishing the equivalence of the two descriptions of the principal graphs given in §4.2. We shall consistently use the symbols x,,,, y,,, etc., to denote elements of M. M.
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
60
PROPOSITION 4.4.1 (i) For each n > 0, there exists an isomorphism of squares of finite-dimensional C*-algebras, as follows:
N' n M2n c N' n M2n+1
NLM(L2(Mn)) c NLN(L2(Mn))
=
U
U
M' n M2n c M' n M2n+1
U
U
MGM(L2(Mn)) c MGN(L2(M.))
(ii) For each n > 0, there exists an Mn-M-linear unitary operator un :
ON (NL2(M)M)
(M,L2(Mn+1)M)
(iii) The composite maps
N' n M2n+1 c N' n M2n+2
NLM(L2(Mn+1)),
MLN(L2(Mn)) = Mn M2n+1 C M' n M2n+2
=+ MLM(L2(Mn+1))
NLN(L2(Mn))
are given, in either case, by the formula T F--> un(T ON idL2(M))u*n.
Proof: (i) Appeal first to Proposition 4.3.6 to note that N C_ Mn C M2n+1
(resp., M C Mn c M2n) is an instance of the basic construction, and hence M2n+1 = JM N'JMn, so that (N' n M2n+1) = (N' n JMN'JMj = NLN(L2(Mn)). Similarly we can argue that each of the four corners in each square of algebras, displayed in (i) above, is isomorphic to the corresponding corner in the other square. Completing the proof is just a matter of ensuring that the various isomorphisms are compatible. For this, use Lemma 4.3.1(ii) and Proposition 4.3.6 to find that if we define 0, : M2n+1 ---> G(L2(Mn)) by (0n(x2n+1))(xnQMn) = T-n-'EMn(x2n+lxne[-1,n])IMn,
then Wn (M2n+1) = JM N' JMn and cn Mn = idMM. A pleasant exercise involving the commutation relations satisfied by the en's, coupled with a judicious use of a combination of Lemma 4.3.1(i) and Proposition 4.3.6, as well as such identities as e[k,k+m] =
T-(m-1)e[k+l,k+m]ek+2ek+m+3
' '
' ek+m(ek+2m ' ' ' ek+m+1)i
leads to the fact that if x2n E M2n, then (cbn(x2n))(xnQMn) =T-nEMn(x2nxne[0,n])QMn,
(4.4.1)
and consequently, as before, ¢n(M2n) = JMM'JMM, and this ¢n does all that it is supposed to. (ii) Recall that if P is any h i factor, and if P0, Qo are any two subfactors of finite index in P, and if N = L2(P), regarded as a Po-Qo-bimodule, then
4.4. RELATIVE COMMUTANTS VS INTERTWINERS
61
71o = P and the Po-valued inner product on 7-L is given by (z, y)P0 = Epo (xy*). Define un : Mn ® M --+ M,,+, by un(X® ®Xo) = T
xnen+l...e1x0. 2
It is a consequence of Remark 4.3.5 that un is surjective. An easy computation shows that (un(Zn ® xo),un(yn ® YO)) M.
T (n+1) EMn(Xnen+l ...elxoyoel...en+lyn)
= xnEN(xoyo)yn
= (Xn
(zo, YO) N, yn)n n;
it follows from Proposition 4.1.2 that un extends to an Mn-M-linear unitary operator of (MnL2(Mn)N) ON (nnL2(M)M) onto MnL2(Mn+1)M, as desired. (iii) Note, in general, that if 7-L (resp., 1C) is an N-P- (resp., P-Q-) bimod-
ule, where N, P, Q are III factors, and if T E NLP(7-L), S E PGQ(K), then there is a unique element T®pS of NLQ(7-L(&p1C) satisfying
V E7-Lo,yE/Co. Hence, in order to establish the assertion for either of the composite maps, we find, when all the definitions have been unravelled, that it suffices to verify that for arbitrary xj e M;, j E {0, n, 2n + 1}, we have (4.4.2)
0n+1(x2n+1)T2un(xn ON x0) =T 2 Un(4'n(x2n+1)zn ON zo).
The definitions imply that the right side of equation (4.4.2) is equal to T
-n-1
e1X0;
on the other hand, it follows from equation (4.4.1) that the left side of equation (4.4.2) is equal to T-(n+1) EM
+,(x2n+lxnen+l ... elXoe[o n+1])
T -(n+1) EMn+1(Xzn+lXnen+1 ... e1e[0,n+1])X0
Hence it suffices to prove that, for arbitrary yen+1 E Men+1, we have EMn+1(y2n+1en+1...ele[0n+1]) = EMn(y2n+le[-1,n])en+l
el
An appeal to Proposition 4.3.6 and Lemma 4.3.1 shows that this amounts to showing that EMn+1 (Y2n+1en+1 ... eie[o,n+1])e[0,n+1] = EMn (y2n+1e[-1,n])en+1
e1e[0,n+1],
i.e., that T
_n_1
y2n+len+1 .. ele[O,n+1] = EMn (y2n+1e[-1,n])en+1
e1e[o,n+1]
CHAPTER 4. THE PRINCIPAL AND DUAL GRAPHS
62
On the other hand, it follows immediately from the formula given in Proposition 4.3.6 that en+1 ... ele[o,n+1] = e[-1,n]e2n+2 ... en+2
Hence it follows, again from Lemma 4.3.1, that EMjy2n+le[-1,n])en+1
.
ele[0 n+1] =
T-n-1y2n+1e[-l,n]e2n+2
en+2,
and a final appeal to Lemma 4.3.1 completes the proof.
Before proceeding further, notice that if 7-l is a bifinite P-Q-bimodule, where P, Q are arbitrary III factors, then pLQ(7-l) is a finite-dimensional C*algebra. (Reason: Assume 7-l = 7-lh, for some co-finite morphism B : Q ->
Md(P), where d = dimp_71; then pLQ(7-le) = (Md(P) n 0(Q)', which is finite-dimensional in view of the co-finiteness of B.) Suppose, for notational convenience, that A = pLQ(7-l). It must be clear
that there is a bijective correspondence between P(A) and the collection of P-Q-sub-bimodules of 7-l (given by p H ran p). It follows that, under the above correspondence, the (canonical) decomposition of the identity as a sum of minimal central projections in A corresponds to the (canonical) decomposition of 7-l into its isotypical components; and that, the further (non-canonical) decomposition of a minimal central projection as a sum of minimal projections of A corresponds to the (non-canonical) decomposition
of the corresponding isotypical submodule of 7-l as a direct sum of mutually equivalent irreducible submodules. It follows that if 7-l is a bifinite P-Q-bimodule, then 7-l is expressible as the direct sum of finitely many irreducible P-Q-bimodules, and that if IC is an irreducible bifinite P-Qbimodule, then there is a well-defined multiplicity with which `IC occurs in 7-l'.
If we apply the foregoing remarks to the case when 7-l = L2(Mn), P = Q = N, we find from Proposition 4.4.1 that the minimal central projections of (N' n Men+1) are in bijective correspondence with isomorphism classes of irreducible N-N-sub-bimodules of L2(Mn). Similarly, the minimal central projections of (N'nM2n) are in bijective correspondence with isomorphism classes of irreducible N-M-sub-bimodules of L2(Mn). Further, Proposition 4.4.1(i) implies that in the Bratteli diagram for the inclusion (N'nM2n) C (N'nM2n+1), the vertex indexed by an irreducible N-M-sub-bimodule X of L2(Mn), is connected to the irreducible N-N-sub-bimodule Y of L2(Mn), by k bonds, where k is the multiplicity with which X, when viewed as an N-N-bimodule, occurs in Y. Similarly, the assertion (iii) of Proposition 4.4.1 can be re-interpreted as a version of the Frobenius reciprocity theorem, thus: the inclusion matrix for the inclusion (N' n Men+1) c (N' n Men+2) corresponds to `induction', in the same way that the inclusion matrix for the inclusion (N'nM2n) C (N'nM2n+1) corresponds to `restriction'.
4.4. RELATIVE COMMUTANTS VS INTERTWINERS
63
Similar remarks apply also for the relative commutants of M in the members of the tower of the basic construction, and we have completed the justification for the alternative description of the principal and dual graphs, in terms of bimodules, which is given in §4.2.
Chapter 5 Commuting squares 5.1
The Pimsner-Popa inequality
The `inequality' referred to in the title of this section is actually not an inequality, but a `minimax' characterisation of the index of a subfactor which can be taken as an alternative definition of the index. (This definition has the advantage of making sense for any inclusion of von Neumann algebras for which there is a conditional expectation of the bigger algebra onto the smaller.) Before getting to the inequality, we pause to record a lemma - see [Joni] - which shows that the basic construction is `generic'. LEMMA 5.1.1 If Mo C Ml is an inclusion of II, factors with [M1 : M0] < no, then there exists a subfactor M_1 such that M1 = (Mo, eM_1).
Proof: The first step in the proof is to realise L2(Mo) as an M1-module in such a way that the action of MO agrees with the standard action. To do this, start with a projection pl E Ml satisfying trp1 = [Ml Mo]-1, observe that L2(Ml)p1 is an Mo-module with dimMo(L2(M1)p1) = 1, and use an Mo-linear unitary operator u from LZ(Mo) onto L2(Ml)pl (noting that LZ(Ml)pl is actually an Ml-submodule of L2(M1)) to transfer the action of :
M1 to L2(Mo), as desired. So we may assume that M1 C L(L2(Mo)). Now define M_1 = JMIJ, where of course J denotes the modular conjugation operator on L2(Mo). Note then
that
WO :M_1]=[M'1:Mo]=IM 1J:JMMJ]=[M1:M0]. If el denotes the orthogonal projection of L2(Mo) onto L2(M_1), then el E M' 1, and so el = Je1J E M1. It follows that if P = (Mo, el), then P C_ M1 and [M1 : P] = 1; an appeal to Corollary 3.1.4 completes the proof.
We first state a weaker version of the inequality (which we prove), and later state the stronger version (which we do not prove).
CHAPTER 5. COMMUTING SQUARES
66
PROPOSITION 5.1.2 (The Pimsner-Popa inequality) If N C M is an inclusion of III factors with finite index, if EN denotes the unique trace-preserving conditional expectation onto N, and if we write T = [M : N]-', then T = sup{A > 0 : EN(x) > Ax Vx E M+}.
(5.1.1)
Proof: In the course of the proof, we shall need the following simple fact, which is an immediate consequence of equation (3.1.1), for instance: The trace-preserving conditional expectation of a finite von Neumann algebra onto a von Neumann subalgebra is completely positive - meaning that
if P C N is an inclusion of finite von Neumann algebras, and if ((aij)) (M.(N))+, n E N, then ((EP(aij))) E (MM(P))+. To prove the proposition, begin by appealing to Lemma 5.1.1, and assume that M = (N, ep), for some subfactor P of N. Let x E M+; write x = zz* for some z in M. By Remark 4.3.2(b), we may assume that z = E', xiepyi for some xi, yj E N, n E N. An easy computation now shows that n
zz* _ E xiEP(yiyj*)epxt, i,j=1 n
EN (zz*)
=
rE
i,j=1
It follows from the complete positivity of Ep (referred to at the start of this
proof) that the matrix y' = ((Ep(yiyj))) is positive. Since ep E P' n M, it follows that
zz* _ [x1 ... xn]y 2
[x1 ... xn y
= 7 -1EN(zz*).
Thus we see that ENX > rx Vx E M+. On the other hand, we know that EN(ep) = 7-1. The previous two sentences complete the proof of the proposition.
Cl
Now we state, without proof, the stronger version (see [PP1]), which states that the previous proposition is valid without the assumption of finite index.
THEOREM 5.1.3 If N C M is an inclusion of III factors, and if EN denotes the unique trace-preserving conditional expectation onto N, then
[M : N] = (sup{A > 0: EN(x) > Ax Vx E M+})-1 .
(5.1.2)
5. 1. THE PIMSNER-POPA INEQUALITY
67
We single out the amount by which Theorem 5.1.3 is stronger than Proposition 5.1.2, as a separate corollary.
COROLLARY 5.1.4 If N C M is an inclusion of IIi faictors, then [M : N] < co if and only if there exists a positive constant A such that ENX > Ax Vx E
M. The next proposition is an important first step in the computation, by approximation, of the index of a subfactor.
PROPOSITION 5.1.5 Suppose N C M is an inclusion of IIi factors. Suppose that the subfactor and factor can be simultaneously approximated by a grid of von Neumann subalgebras of M in the following sense (where 'convergence' means that the `limit' algebra is the weak closure of the union of the increasing sequence of subalgebras that form the members of the sequence of `approximants'):
C B. C
Bo C Bi C U
U
Ao C Ai C
-M U
U ...
C An C
->
N.
Define
An = \(Bn, An) = sup{.\ > 0:
E B,,+};
then
[M : N] < (limsupAn)-i. n
Proof: Consider the simple fact that if {14} n 1 is an increasing sequence of closed subspaces of a Hilbert space 7-l, whose union is dense in 7-l, and if p, denotes the orthogonal projection onto 7-h, then p,,, -> V E H. This fact has the following consequences in the context of this proposition: (i) EA,,,z -* ENZ weakly, for all z in M; and
(ii) if x E M and x,,, = EB,x, then x7, -> x strongly. Fix x E M+, and let x,,, be as in (ii) above. Let {r.,,} be a sequence in (0, 1) such that rn, -> 1 as m -> oo. Now fix an integer n, and note that since xn E B,,,,+ Vm > n, the definition of .Am ensures that EA_ (xn) > rm.\mxn Vm > n. Now use (i) and (ii) above, and conclude, by first letting m -> oo and then letting n -> oo, that ENX >_ (limsupn An)x; this completes the proof, in view of Theorem 5.1.3.
CHAPTER 5. COMMUTING SQUARES
68 A
EXAMPLE 5.1.6 Let Bo C Bl be a connected inclusion of finite-dimensional C_ Bn C denote the tower of C*-algebras, and let B0 C B1 C_ B2 C the basic construction with respect to the Markov trace. Then, as has already been observed, there is a unique trace on Un Bn and the von Neumann algebra completion of this union is the hyperfinite III factor R. Set B_1 = C, and pick arbitrary unitary elements un E B'_1 n Bn+1; it follows exactly as in Example 3.1.6 that there exists a unique normal endo(This morphism a : R -> R with the property that aIB. = construction is fairly general; for instance, if each Bn is a factor, then this is the form of the most general endomorphism of Un Bn which maps Bn into Bn+1 for all n.) Set An = a(Bn_1), n > 0, M = R, N = a(R), and note that for n > 1, we have
(An C An+1 C Bn+1) = Adu,u2...un (Bn_1 C Bn C Bn+1),
and consequently, (An C An+1 C Bn+1) is an instance of the basic construction; it follows from this and the proof of Proposition 5.1.2 that, in the notation of Proposition 5.1.5, we have, for all k > 2, Ak =
IIAII-2.
Thus, we find, by Proposition 5.1.5, that no matter what the initial sequence {un} of unitary elements is, we always have [M : N] < IIAII2 < oo. It is known - see [Jon6] and [Ake], for instance - from examples that the exact value of the index could be just about any admissible index value in the range [1, IIAI I2]; it would be very interesting to determine the exact dependence of the index on the initial sequence of unitary elements. The possible usefulness of Proposition 5.1.5 becomes enhanced in the light
of a result, due to Popa, to the effect that if N C M is an inclusion of hyperfinite III factors, then it is always possible to find finite-dimensional C*subalgebras An, Bn of M satisfying the conditions of the above proposition. Further, Proposition 5.1.5 would be even more useful, if we could sharpen
the inequality to an equality. It becomes clear very quickly, however, that it is unreasonable to expect any such equality without further conditions on the approximating grid; for instance if {An} is any increasing sequence of finite-dimensional C*-subalgebras of R whose union is weakly dense in R, and if we set Bn = An+k for some arbitrarily fixed k, then in the notation of Proposition 5.1.5, we would have N = M = R, while the numbers An could all be quite large! This leads us to the very important notion of a `commuting square of finite von Neumann algebras', which, as we shall soon see, very satisfactorily plays the role of the `further conditions' that were mentioned above. DEFINITION 5.1.7 Suppose B1 is a finite von Neumann algebra, and tr is a faithful normal tracial state on B1. Suppose Ao, Al and B0 are von Neumann
5.1. THE PIMSNER-POPA INEQUALITY
69
subalgebras of B1 such that AO C Al fl Bo, and suppose the following clearly equivalent conditions are satisfied: (i)
Bo
B1
EA, I
1 EA,
Al
AO
is a commutative diagram of maps; (i)' EA, (Bo) = Ao; (ii) Al
°->
EAO I
AO
B1 I EBo Bo
is a commutative diagram of maps; (ii)' EBo (A,) = AO; (iii) EA, EB0 = EAo;
(iii)' EB0EA, = EAo. When the preceding conditions are satisfied, we shall (often suppress specific reference to the trace tr, and simply) say that
Bo C B1 U
U
Ao C Al is a commuting square of finite von Neumann algebras. We shall discuss several examples of such commuting squares in the next section. Before that, however, we conclude this section by making a general remark about the notion of a commuting square, and then fulfil the assertion made earlier about this notion by showing that in the presence of the commuting square condition, we can sharpen Proposition 5.1.5 to the desired equality.
REMARK 5.1.8 Let us temporarily adopt the convention that if G is a directed graph without multiple bonds, then by a representation of G we shall mean an
assignment of an algebra A to each vertex v in the graph 9 and inclusion whenever e is an edge in G from v1 to v2. It can then maps ie A,,, -+ :
be shown that any tree admits an essentially unique representation, while if G has cycles, there are some 'parameters' or in other words, there is a 'moduli space'. A commuting square may be considered as a point in this moduli space corresponding to the graph given by a square.
CHAPTER 5. COMMUTING SQUARES
70
PROPOSITION 5.1.9 Suppose M, N, A,,,, B,,,, A,,, are as in Proposition 5.1.5,
and suppose further that C Bn+1
B,,
U
U
An C An,+1 is, for each n, a commuting square. Then
[M:NJ=(li A,,,)-1. Proof: The point to observe is that the result of stacking up (finitely or countably many) commuting squares is again a commuting square. The necessary formal argument is simple (and uses essentially only the simple fact about increasing sequences of projections in Hilbert space which was mentioned at the start of the proof of Proposition 5.1.5), and may be safely omitted.
5.2
Examples of commuting squares
In this section, we shall discuss three families of examples of commuting squares. Each of them is of the special form
uAlu* C A2 U
U
Ao
C Al
(5.2.1)
where (i) A0 C Al is a `connected' inclusion of finite-dimensional C*-algebras, (ii) A2 = (A1, e2) is the result of the basic construction applied to A0 C A1;
and (iii) u is a suitable unitary element in Ao f1 A2 such that (5.2.1) is a commuting square, with respect to the Markov trace on A2.
5.2.1
The braid group example
Consider the square (5.2.1), where u = (q +1)e2 - 1, for some scalar q which must necessarily satisfy Iql = 1 (in order for u to be unitary). It is an easy exercise to check that for such a u, the square (5.2.1) satisfies the commuting square condition if and only if 2+q+
q_1
= (tr e2)-1.
On the other hand, since the condition I qt = 1 clearly implies that 12 + q +
q-' < 4, this construction can work only when T-1 = 4 cost ,, for some m
>3.
The reason for calling this the `braid group example' is clarified in §A.5 of the appendix.
5.2. EXAMPLES OF COMMUTING SQUARES
5.2.2
71
Spin models
The reason that these examples are called spin models stems from statistical mechanical considerations - see, for instance [Jon3]. Consider the square (5.2.1), where we now assume that A0 = C and Al = A is the diagonal subalgebra of A2 = MN(C) - see Example 3.3.1(ii). Another
easy computation shows that in this case, the square (5.2.1) satisfies the commuting square condition if and only if Juij I = --IVi, i. V_N A square matrix u, as above - i.e., a unitary matrix, all of whose entries have the same modulus - is called a complex Hadamard matrix. The simplest exp(2ir example of such a matrix is given, for any N, by uij U)ij
where w denotes a primitive N-th root of unity. (This matrix may be called the finite Fourier transform, as it diagonalises the left regular representation of the cyclic group 7ZN.) To see that there are many complex Hadamard matrices, observe the following one-parameter family of 4 x 4 com-
plex Hadamard matrices, indexed by the parameter t ranging over the unit circle T in C: ut
1
1
t -1
t
1
t
t -1
-1
1 t t t -1 t 1 The reason for the adjective `complex' in the previous paragraph is that there is a fair amount of history related to real orthogonal matrices with constant modulus - these were studied by Hadamard and bear his name. We
2
record here a few facts about such real Hadamard matrices: (i) Call a positive integer N an Hadamard integer if there exists an N x N real Hadamard matrix. It is easy to show that if N is an Hadamard integer and if N > 2, then N - 0(mod4). The open question concerning real Hadamard matrices is whether every multiple of 4 is in fact an Hadamard integer. (This has been verified to be true for some fairly large multiples of 4.)
(ii) It is not hard to show that two squares of the form (5.2.1) that are given by real Hadamard matrices are isomorphic (as squares of algebras) if and only if the corresponding Hadamard matrices are equivalent in the sense that it is possible to obtain one by (pre- and post-) multiplying the other by diagonal orthogonal matrices and permutation matrices. The number h of isomorphism classes of real Hadamard matrices of size N, for small N, is given as follows:
5.2.3
N
2
4
8
12
16
h
1
1
1
1
5
Vertex models
These examples also get their names from statistical mechanical considerations - see [Jon3].
CHAPTER 5. COMMUTING SQUARES
72
Consider the square (5.2.1), where we now assume, as in Example 3.3.1(i), that Ao = C, Al = MN(C) ® 1, A2 = MN(E) ® MN(C). If we make the identification Ai = MN (C), and if u = ((ukl))1
x3
uk1 k T
= bk k dl
x,7
x30'
uk1 kd'
6j,9's&,l' .
i,k
The simplest example of a vertex model is provided by the `flip' - which was briefly encountered in Example 3.1.6 - for which ukl = SilSki.
We shall consider the last two classes of examples in greater detail in Chapter 6 (where, among other things, we shall see the reason for the above `biunitarity' condition).
5.3
Basic construction in finite dimensions
Recall that if e is a projection in a von Neumann algebra, then the projection V {ueu* : u E M, a unitary} is also given by Al f E P(Z(M)) : e < f }; this projection, which we shall denote by zM(e), is called the central support of the projection e.
LEMMA 5.3.1 Let A C B be a unital inclusion of finite-dimensional C*algebras. Fix a faithful tracial state 0 on B, and let J denote the modular conjugation operator on L2(B, 0). Let e denote the orthogonal projection of L2(B, 0) onto the subspace L2(A, 01A); let EA denote the 0-preserving condi-
tional expectation of B onto A; and let B1 = (B, e) denote the result of the basic construction. (a) zB1(e) = 1. (b) Suppose C is a finite-dimensional C*-algebra containing B. (We only consider unital inclusions.) Suppose C contains a projection f satisfying:
(i) C = (B, f ); (ii) f b f = EA(b) f, Vb E B; and
(iii) a H a f is an injective map of A into C.
Then B f B = C f C = Czc(f), and there exists a unique isomorphism V) : B1 -4 C f C such that b(e) = f and 0(b) = bzc(f) Vb E B. Proof- (a) Suppose z E P(Z(B1)) and e < z. Then also e = JeJ < JzJ E ae P(Z(JB1J)) = P(Z(A)); but then (1-JzJ)e = 0, and since the map a is an injective map of A, it follows, as desired, that z = 1.
5.3. BASIC CONSTRUCTION IN FINITE DIMENSIONS
(b) The assumptions clearly imply that C = {bo + > 1 bi f bi
73 :
nE
N, bi, bi E B}. It follows - from this and (ii) - that C f C = B f B. On the other hand, every two-sided ideal in the finite-dimensional C*algebra is of the form Cz, for a uniquely determined central projection z c C. It follows from the definition of the central support that C f C = Czc(f).
In order to complete the proof, we need to show that the assignment E 1 biebi F--> > 1 bi f bi yields a well-defined injective mapping from B1 to C f C. (It will immediately follow that such a map would be the desired isomorphism.) That this assignment is indeed an unambiguously defined bijection to C f C from B1 is an immediate consequence of (two applications, once to C, and once to B1, of) the following:
Assertion: Let n E N, bi, bi c B,1 < i < n. Then Ii
n
EA(bbi)EA(b''b') = 0 Vb, b' c B.
bi f bi = 0 i=1
(5.3.1)
i.1
bi f bi = 0. Pre-multiply by f b and postProof of assertion: Suppose multiply by b'f, and appeal to the hypotheses (ii) and (iii), to conclude that indeed, En 1 EA(bbi)EA(bb') = 0 Vb, b' E B. Conversely, if the latter condition is satisfied, pre-multiply by x f and post-multiply by f x', to deduce that x f b(> 1 bi f bi)b' f x' = 0, for arbitrary x, x' E B. Since the finite-dimensional C*-algebra C f C = B f B has a unit, this implies that indeed >? 1 bi f bi = 0, and the proof of the assertion, and consequently of the lemma, is complete.
COROLLARY 5.3.2 Let A, B, B1, C, e, f be as in Lemma 5.3.1(b). Then the following conditions are equivalent:
(i) there exists an isomorphism 0 : B1 -> C such that 01B = idB and
c(e) = f; (ii) zc (f) = 1. Proof: Obvious.
We come now to a very important relation between commuting squares and the basic construction. LEMMA 5.3.3 Suppose
H
Bo C B1 KU
UL
Ao C Al
(5.3.2)
CHAPTER 5. COMMUTING SQUARES
74
is a commuting square of finite-dimensional C*-algebras with respect to a trace
on B1 which is a Markov trace for the inclusion B° C B1, with inclusion matrices as indicated. Let B2 = (B1, e) denote the result of the basic construction for the inclusion 13° C 131, where e denotes the projection in B2 which implements the conditional expectation of B1 onto B°. Define A2 = (A1, e) C B2. Then the following is also a commuting square (with respect to the unique trace on B2 which extends the given trace on B1 and is a Markov trace for the inclusion B1 C B2), with the inclusion matrices as indicated: H'
Bi C B2 Uxl
LU
(5.3.3)
GI
Al C A2, where G1 (resp., K1) is a matrix with block-form [G' r) (resp.,
1), I
A
where
F (resp., A) is the submatrix of G1 (resp., K1) determined by the columns (resp., rows) indexed by minimal central projections of A2 which are orthogonal to e. Proof: Begin by observing that, in view of the assumed commuting square
condition, the tower A° C Al C A2 = (Al, e) satisfies the conditions of Lemma 5.3.1(b); in particular, any element of A2 is of the form x = a + E 1 bieci, where a, bi, ci E Al; and EBIx = a + E 1 Tbici E A1, where T = IIHII-2, and consequently (5.3.3) is also a commuting square. , q<<)) be an ordering of the minimal central Let pll), pl (resp., qh>, i projections of Al (resp., B1) for 1 = 0, 1, with respect to which the inclusion matrices are G, H, K, L as in (5.3.2). If we define q(2) = JBIgkO)JBI,1 < k < no, it follows from Lemma 3.2.2(a)(iv) that q12), , qno) is an ordering of ,
the minimal central projections of B2, and hence the inclusion matrix for and the gk2)'s is H. B1 C B2, computed with respect to the On the other hand, it follows from Lemma 5.3.1(b) that, if we write z = (A1, eAo) -> A2z such that zA2 (e), then there exists an isomorphism O(eA,) = e and d(al) = a1z dal E A1. Define p12) _ cb(JAIpI°)JAI),1 < i < ,p(22 be a listing of the minimal central projections of m°i and let p(2)+1i A2 which are orthogonal to z (or equivalently, to e). It follows, again from :
is a listing of the minimal central projections , of A2 such that the inclusion matrix for Al C A2, computed with respect to the p2*s and the pk2)'s, is [G' FJ, as asserted. Lemma 3.2.2, that p12),
Finally, fix 1 < i < m°i1 < k < no, and fix a minimal projection f E A°pl°). By definition of the inclusion matrix, there exists a decomposition f q(k0) = EK`i r., of f q(°) as a sum of (mutually orthogonal) minimal projections in B°qk°). Notice that 0(f eAo) = 0(f)o(eAo) = f z e = f e. Hence we may
5.3. BASIC CONSTRUCTION IN FINITE DIMENSIONS
75
deduce from Lemma 3.2.2(b)(ii) that fe is a minimal projection in and further,
f
eg(k2)
= f g(k 2)e f q(k0) e
(by Lemma 3.2.2(a)(i))
Kik rse;
s=1
but, again by Lemma 3.2.2(b)(ii), each rse is a minimal projection in B2q 2). This completes the proof that the inclusion matrix for A2 C B2 is indeed as asserted. COROLLARY 5.3.4 Let H
Bo C Bl KU UL G
(5.3.4)
Ao c Al be as in Lemma 5.3.3.
(a) Then G'K < LH' in the entry-wise sense. (b) Assume that each of the four inclusions in (5.3.4) is connected. Then the following conditions are equivalent:
(i) G'K = LH'; (ii) if A2, B2 are as in Lemma 5.3.3, then zA2 (e) = 1; (iii) V{ueu* : u E A1, u unitary} = 1, where the 'supremum' is computed in A2 or equivalently in C(L2(Bl));
(iv) B1 is linearly spanned by Al Bo = {a,bo : a1 E A1, bo E Bo};
(i)' KG = HL'; (ii)' if C1 = (B1, f) is the result of the basic construction for the inclusion Al C B1 (with respect to the Markov trace for this inclusion), and if Co = (Bo, f), then zco (f) = 1;
(iii)' if Co and f are as in (ii)' above, then V{u f u* : u E Bo, u unitary} = 1, where the `supremum' is computed in Co or equivalently in C(L2(Bl));
(ivy B1 is linearly spanned by BOA1 = {boar : a1 E A1, bo E Bo}. (c) Assume that each of the four inclusions in (5.3.4) is connected, and that the equivalent conditions of (b) are satisfied; then: (i) IIGII = IIHII;
(ii) IIKII = IILII;
A2pZ2),
CHAPTER 5. COMMUTING SQUARES
76
(iii) if tr denotes the trace on B1 which is the Markov trace for B0 C B1, then tr is also the Markov trace for Al C B1, and trIA, (resp., trIB0) is the Markov trace for A0 C Al (resp., Ao C Bo.) Proof: (a) By multiplicativity of inclusion matrices, we find from Lemma 5.3.3 that (KH = GL and) LH' = G1K1 = G'K + FA, thus establishing (a). (b) By the above reasoning, if (ii) is satisfied, it follows from Lemma 5.3.3 that r' and A are absent, whence we have (i). On the other hand, suppose (ii) is violated. In the notation of the proof of Lemma 5.3.3, this means there exists i, mo < i < m2i such that p,2) 0. Since the inclusion Al C A2 is unital, there must exist j < m1 such that Fhi : 0. Also, by the definition of an inclusion, there must exist k,1 < k < n2, such that Aik 0; it follows that (FA)jk 0 0, and hence (G'K),k < (LH');k, thus
establishing that (i) '# (ii).
(In the following, we shall write S2 for the cyclic trace vector in L2(B1).)
Notice that the condition (ii) amounts to the requirement that V{ueu* u E A2, n unitary} = 1, which amounts to requiring (since the range of :
ueu* is just uBoS2) that B1S2 is spanned by U{uB0Q : u E A2, U unitary}, which is equivalent to the requirement that B11 is spanned by {xboQ : x E A2, bo E B0}. Since A2 is spanned by Al together with elements of the form aeal, a, a1 E A1, and since the range of e is just B0S2, it follows that (ii) is equivalent to the requirement that B11 is spanned by A1B0S2 (which, in view of S2 being a separating vector for B1, is the same as (iv)) which is again seen to be equivalent (by reversing the earlier reasoning) to (iii). By taking adjoints, it is clear that the conditions (iv) and (iv)' are equivalent, while the equivalence of (i)'-(iv)' follows from the equivalence of (i)-(iv). (c) The assumption that (b)(ii) is satisfied implies - by Proposition 3.2.3 IIHII-2) and that - that trIA, is the Markov trace for A0 C Al (since EB,e = IIGII = IIHII As for (iii), let us write tB' for the trace vector corresponding to the trace
tr. Let us also write tC for the trace vector corresponding to tric, where
C E {A0, A1, B0}. The already established fact that trIA, is the Markov trace for A0 C Al implies that (5.3.5)
G'tA0 = IIGII2tA'-
Hence
H'HL'tA' = H'K'GtA' = H'K'tA0 = L'GtA0 =
IIGII2L'tA'.
Thus, L'tA' is a, and hence the, Perron-Frobenius eigenvector for H'H; i.e., there exists a positive constant p such that L'tA' = ptB'; but then, also LL'tA' = pLtB' = ptA'; and hence p = IILII2, and indeed tr is the Markov 1 trace for Al C B1. REMARK 5.3.5 The proof shows that the conditions (ii), (iii), (iv), (ii)', (iii)', and (iv)' in Corollary 5.3.4(b) are equivalent even without requiring either
5.4. PATH ALGEBRAS
77
that the trace in question is the Markov trace, or that all the inclusions are connected. In fact, these conditions are equivalent under just the assumption that we have a commuting square of finite von Neumann algebras, provided that the expression `linearly spanned' in conditions (iv) and (iv)' is replaced by `contained in the closed subspace of L2(B1) generated by'. Such a general commuting square of finite von Neumann algebras has also been called a nondegenerate commuting square by Popa. DEFINITION 5.3.6 The commuting square (5.3.4) is called a symmetric commuting square if the equivalent conditions in Corollary 5.3.4(b) are satisfied.
5.4 Suppose
Path algebras A0cA,CA29...9AnC...
(5.4.1)
is a tower of finite-dimensional C*-algebras. We know that the `isomorphism class' of this tower is completely determined by the dimension vector of Ao and the Bratteli diagrams of the successive inclusions. The path-algebra formalism gives a model of such a tower (for any prescribed data as above), which is quite useful in some computations. Before we get to discussing this formalism, we set up some notation.
If A is a finite-dimensional C*-algebra, we shall write ir(A) for the set of minimal central projections of A. Thus, if A _C B is a unital inclusion of finite-dimensional C*-algebras, then the inclusion matrix is the ir(A) x ir(B) matrix, with (p, q)-th entry given by [dim,(pgA'pgfl pgBpq)] 2. We shall think of an edge a in the Bratteli diagram for the inclusion A C B, which joins p E 7r(A) and q E ir(B), say, as being oriented so that it starts from p and finishes at q, and we shall write p = s(a), q = f (a). Suppose now that the Ak's are as in (5.4.1), where we assume that all the inclusions are unital. For convenience, we set A_1 = Cl _C A0; write ir(A_1) = {*}. For k > 0, let 1 k denote the set of edges in the Bratteli diagram for the inclusion Ak_1 C_ Ak, oriented as discussed in the last paragraph.
For -1 <1 < k
For -1 < l < k < oo, let l([l,k] denote a Hilbert space with an orthonormal basis indexed by I [1,k]; we shall identify an operator x on l1 [l,k] with its matrix
((x(a,a))) (with respect to this basis).
If -1 < l < l1 < k1 < k < oo, and if a = (a1+1, a1+2,
-
write s(a) = s(a1+i), f (a) = f (ak) and a[11,k11 = (a11+i, a11+2)
.
,
,
ak) E Q[l,k], ,
ak1)
CHAPTER 5. COMMUTING SQUARES
78
We shall use the following obvious simplifying notational conventions: k] _ [-1,k], 1[l = 1 [l,oo],
_ [ li
?k] = 'H[-1,k]) H[1 =H[1,.], 7-t = x[-1;
ak] = a[-1,k], a[l = a[,,.) Finally, for -1 < n < oo, define B,,, = {x E ,C(7-l) :3a E G(7in]) such that x(a, Q)
= b«ln,P[na(an], l3n]), Va, 0 E 1 }.
PROPOSITION 5.4.1 With the foregoing notation, we have: (i) for -1 < n < oo, Bn is a *-subalgebra of G(7-[) and (5.4.2)
(ii) if-1
Bk n Bn = {x E L(R) : 3a E C(7-l[k,n]) such that x(a, Q) = b«k1,Pklb«ln,P[na(a[k,n], 0[k,n]) Va, Q E Q};
(iii) for fixed n > -1 and p E 7r(An), define p E Bn by p(a, Q) = b«,Pbf(«n]),p
Vc , a E 1;
then 7r(Bn) _ {p : p E ir(An)}; (iv) for n > -1, and A,, E Stn] satisfying f (A) = f (µ), define ea,N, E Bn by
eA,,(a, a) =
Va,'3 E S1;
then
{e),,,, : A, µ E Qn], f (A) = f (µ)}
is a `system of matrix units' for Bn; (v) there exists an isomorphism z/) of the tower (5.4.1) onto the tower (5.4.2) such that //'(p) = p Vp E ir(An), do > -1. Proof: The proof of (i)-(iv) is routine computation. Assertion (v) is easily seen, via induction, to amount to the following statement which is indeed true: If Ao C Al and B0 C_ B1 are inclusions of finite-dimensional C*-algebras, if 8 : A0 -> B0 is an isomorphism, and if there exists a bijection ¢ : 7r(A1) ->
7r(B1) such that the two inclusion matrices AA and AB satisfy AA(p, q) _ AB(O(p), o(q)), then there exists an isomorphism B : Al --> B1 which simultaneously extends both 0 and 0.
5.4. PATH ALGEBRAS
79
REMARK 5.4.2 (1) If x E Bn and a E G(7-ln)) are related as in the definition of Bn, the assignment x a defines a faithful representation on of Bn on 'Hn); we shall consequently simply write x(a, 0) for (pn)(x))(a, 0), whenever x E Bn, a, a E Stn) . Further, dimc ?-ln) = EPE,,(A.) [dimc (BfP)] 2 , and pn contains each distinct irreducible representation of Bn exactly once. (2) Suppose o,,b' : An -> Bn are two isomorphisms which both map Ak to Bk and p E 7r(Ak) to P for each p E 7r(Ak), for 0 < k < n. Then 0 = 0-1 o 0' is an automorphism of An with the following two properties: (a) O(Ak) _ Ak, 0 < k < n; and (b) Ba(sk) = idz(Ak), 0 < k < n. On the other hand, any automorphism of MM(c) is inner; hence an antomorphism of a finite-dimensional C*-algebra is inner if and only if it acts trivially on the centre; this implies, by an easy induction argument, that an automorphisr 0 of An satisfies properties (a) and (b) of the preceding paragraph if and only if it has the form 0 = Aduov,,...u,,, where Uk is a unitary element of A'k_1 fl Ak, for 0 < k < n. Suppose now that tr is a faithful tracial state on An, where we temporarily
fix n > 0. For -1 < k < n, let t(k) : lr(Ak) -> [0, 1] be the trace-vector corresponding to trIAk; thus _t (k) = trpo, where po is any minimal projection in Akp.
In the following proposition, whose proof is another straightforward verification, we identify the Ak's with the corresponding Bk's of Proposition 5.4.1.
PROPOSITION 5.4.3 (i) If -1 < k < n, and if x E Bk, then tr x=>«Ec k, t(())
x(a a) (ii) If -1 < k < n, and if x E Bn, then the tr-preserving conditional expectation EBk of Bn onto Bk is given by t(n)
/
(EBk x)(a
f(B)
6 - Ct[k,Q[k
k)
x(ak)
9E(1 [k n
o B Ok) '
B)
3(B)=f(«k])
for all a, 0 E On], where we have used the symbol o to denote concatenation of paths.
(iii) In particular, if -1 < k < n, and if ea,,, is as in Proposition 5.4.1(iv), then t1(A)
EBk (ea,P) - 6a[k,fi[k t (k) f(ak))
eAk],Fik]
11
We conclude this section by discussing the above path-algebra formulation in the important special case when the tower (5.4.1) (equivalently, (5.4.2)) is
the tower of the basic construction applied to the initial connected inclusion A0 C A1, with respect to the Markov trace. In this case, suppose A (resp., S2o) denotes the inclusion matrix (resp., the set of oriented edges in the Bratteli diagram) for A0 C A1. Let us inductively identify p E 7r(An)
CHAPTER 5. COMMUTING SQUARES
80
with JAn+1pJA,+1 E ir(An+2). Further, if a E S20, let us write & for the same edge with the opposite orientation, and let us write S2o = {& : cx E S201. It then follows from Lemma 5.3.3 that if An denotes the inclusion matrix for An c An+1i then (An, Stn) is (A, S2o) or (A', S2o) according as n is even or odd. Let i(n) ir(An) -> [0, 1] denote the trace-vector corresponding to the restriction to A. of tr, the unique consistently defined trace on A,,, = Uk Ak. Then, with our identification of 7r(A2n) (resp., ir(A2n+1)) with ir(Ao) (resp., ir(Al)), it follows that :
i(2n+1)
= .-ni(l), (2n) = T-nt(0), t(0) = AP),
where r = IIA11-2. Again, in the following result, which is easily verified, we identify An above
with the corresponding Bn of Proposition 5.4.1. PROPOSITION 5.4.4 With the foregoing notation, a candidate for the projection en+1 E A; _1 n An+1 is given by en+i(a, a)
Ptf(
=
n1)tf(n[)
(n-1)
(5.4.3)
tf(a, q)
5.5
The biunitarity condition
In this section, we discuss the important re-formulation, due to Ocneanu, of the commuting square condition, as a biunitarity condition. Before we get to that, it would be good to make a slight digression concerning matrix units. Recall that if e;,j is the n x n matrix with 1 in the (i, j) position and 0's elsewhere, then {e2j 1 < i,j < n} is called a (or, sometimes, the) standard system of matrix units for Mn(E); more generally, if A is a In factor, a set of matrix units for A is any set of the form {0r'(eij) : 1 < i,j < n}, where 0 : A -> Mn(c) is any isomorphism. Slightly more generally, if A is a finitedimensional C*-algebra, we shall refer to a union of sets of matrix units for each central summand of A as a system of matrix units for A. (See Proposition 5.4.1(iv).) Finally if we are given a tower of finite-dimensional C*-algebras, say Ao C Al C ... C An as in the last section, we would like to work with a system of matrix units for An which is compatible with the tower in a natural :
fashion.
For this, notice that, in the notation of Proposition 5.4.1, the set {eA,/, A, p E S2n], f (A) = f (p)} is a system of matrix units for An with the following
5.5. THE BIUNITARITY CONDITION
81
property: whenever -1 < k < 1 < n, A, µ E Q[k,1], s(\) = s(µ), f (A) = f (µ), if we define ea.µ =
eaoAo0,aoµo/3,
F_
-Enk],0Ec[,
f(a)=s(A),s(R)=f(A)
then {eA,µ :A,
Q[k,1], s(A) = s(µ), f (A) = f (µ)} is a set of matrix units for
A' nA1.
DEFINITION 5.5.1 If A C A, g ... C A,,, is a tower of finite-dimensional C*-algebras, we shall use the expression a system of matrix units for A,,, which is compatible with the tower above, to denote any set of the form /-1({eA,,, A, µ E Q.], f (A) = f (p)}), where and {eA,µ A, IL E Q,,1, f (A) = f (IL) } are as in Proposition 5./1.1(v) and (iv). :
:
Thus a consequence of Remark 5.4.2(2) is that any system of matrix units for A,,, which is compatible with the tower {Ak : 0 < k < n} can be obtained from any other by an inner automorphism of A,,, of the form Aduou,...u,,, where
Uk is a unitary element of A'_1 n Ak, for 0 < k < n. If {pa, : A, µ E s(A) = s(µ), f (A) = f (µ)} is a system of matrix units for A,,, which is compatible with the tower A0 C C A,,, and if B[k,l] = {pa,µ : a, µ E Rk,l], s(A) = s(µ), f (A) = f (µ)} is the system of matrix units for Bk n B1 as above, we shall say that the system B[k,l] extends to the system L3a1, or that the system B.a1 restricts to 13[k,1].
We are now ready for the biunitarity condition. (In the following proposition, we shall use the following notation: if A0 C Al C C A,,, is a tower of finite-dimensional C*-algebras, we write to denote the set of oriented paths which was denoted by Q [O,,,] in the last section. We shall also use the convention that the symbol a o 0 will be used to denote concatenation
of the paths a and 0, and it will be tacitly assumed that if this symbol is used, then necessarily s(/3) = f (a).) PROPOSITION 5.5.2 Let
Bo C B1 U Ao
U
(5.5.1)
C Al
be a square of algebras. Denote an element of I (C;Ao;A,;B,) (resp., Q(C;Ao;Bo;B,)) by a triple a = (ao, al, a2) (resp., 3 = (00, N1, /32)) where ao, /do E Q(c;Ao),
etc., so that, for instance, 3 = /50 o /31 0/32.) (a) Let {pa,,,, : a, a' E Q(C;Ao;Ai;Bi), f (a) = f (a')} (resp., {qo,A'
9(c;Ao;Bo;B,), f (N) = f (,3')I) be a system of matrix units for B1 which is compatible with the tower C C A0 C Al C_ Bl (resp., C C A0 C Bo C_ B1). Then there exists a matrix U = ((uO)), with columns (resp., rows) indexed by Q(Ao;A,;B,) (resp., Q(Ao;Bo;B,)), satisfying the following conditions:
(2) ua = 0 unless s(a) = s(/3), f (a) = f (/3)i
CHAPTER 5. COMMUTING SQUARES
82
(ii) for all a, a' E Q(C;A0;A1;B1), we have 01
L
pa,a' =
7/a1 oa27/a
loa2 gaooa,aoo/i'
a,a'E0(A0;B0;B1)
(iii) U is unitary. (b) If U' is another matrix satisfying (a)(i)-(iii), then there exist complex scalars cop, p E'rr(Bi), of unit modulus, such that (u')O = co f(a)uO.
(c) Suppose tr is a faithful tracial state on B1, and suppose t : 7r(C) [0, 1] is the trace vector corresponding to tr1c, for C E {Ao, A1, B0, B1}; define a matrix V with columns (resp., rows) indexed by SZ(Ao;A1)°Q(Ao;Bo) = {al°,Qi
a1 E 0(Ao;A1) 1 E Q(Ao;Bo)} (resp., ,L(A1;B1) ° f2(Bo;Bi) = {a2 0 N2
:
a2 E
Q(A1;B1), /32 E Q(Bo;B1)}) as follows: C
0
fo
) a1°a2
fO(Al1)
t 7(o1) l t
aloa2
0
(s(a2) S(02)) = (f (al), f (01)), )
(5.5.2)
otherwise; erwise;
then (5.5.1) is a commuting square with respect to tr if and only if V is an isometry - i.e., the columns of V constitute an orthonormal set of vectors. Proof.- (a) Let V) (resp., 0') denote an isomorphism of B1 onto the pathalgebra model for B1 coming from the tower A0 C_ Al C_ B1 (resp.) A0 C_
B0 C B1), as in Proposition 5.4.1(v), such that 4'(pa,a') = ea,a' da,a' E Q(C;Ao;A1;B1) (resp., ?/ (ga,a') = ea,a' VQ, /3' E Q(c;Ao;Bo;B1)); and let po (resp.)
po) denote the representation of V)(B1) (resp., 0'(131)) on a Hilbert space 71 E (resp.) 7i') with orthonormal basis { a : a E 0(C;Ao;A1iB1)} (resp.) {rlp 0(C;Ao;Bo;BI)}), as in Remark 5.4.2(1).
Consider the representations p = po o V) and p' = po o 0' of B1. It follows from Remark 5.4.2(1) that these representations are unitary equivalent.
Let U : 7-l -> 7-t' be a unitary operator which intertwines p and p'. The fact that U intertwines the restrictions to A0 of the representations p and p', is easily seen to imply that there exist scalars u,3, a E 0(Ao;A1;B1), E 01
0(Ao;Bo;B1)(8 (a), f (a)) = (s(/3), f (,Q)) (which are essentially just the matrix coefficients of U with respect to the given bases), such that
US" =
E
$Ecz(A0;B0;B1)
5(0)=f(a0)J(0)=f(012)
whenever a = ao o al o a2 E 0(C;Ao;A1;B1)-
uaaloa271aooa
5.5. THE BIUNITARITY CONDITION
Since p(p.,.,) _
83
and p'(gp,p,) _ ( 77p,)770, it follows that
P
UP(pcr,.&)U*
_
P' R u)aloo<2uoo
thereby proving (a). (b) In the notation of the proof of (a), notice that if U' is another matrix as in (a), and if we use the same symbol for the associated unitary operator from 7-l to 7-l' which is easily seen to necessarily intertwine the representations p and p', then U'*U E p(Bi)'; but it is a consequence of Remark 5.4.2(1) that (the representation p is `multiplicity free' and hence) p(B1)' = p(Z(Bi)), which is seen to imply (b). (c) If a, a' E Q(c;Ao;A,) satisfy f (a) = f (a'), define
pa,a' =
pao-2ro1&oa2
Q2E9(A,;B,)
Then, according to the remarks preceding Definition 5.5.1, it is the case that
the set {p: a, a' E Q(c;Ao;A,), f (a) = f (a')} is a system of matrix units for Ai. Let {qp,p, : 0, Q' E Q(c;Ao;Bo), f (/3) = f (Q')} be the system of matrix units for B0 obtained in a similar fashion. Then for fixed a, a' E Q(c;Ao;A,) satisfying f (a) = f (a'), we have U)31
2uaioa24aoop,ooop'
a2ES2(A,.B,)
p,/Q'Ef (A0.B0;B,)
and hence, by Proposition 5.4.3(iii), -Bi
f
uC'loa2ualoC12p202 Bo
2
gooap1,aoopli
a2 E2(A,,Bl)
p,P'En(AO;B0;Bi)
it follows now from the definition of the matrix V that Al
EBo (p.,al) = tf(a)tf(o) 01,010 (BO;a,)
(V *V)c; Q'1 gaoopj,aoopi
(5 .5.3 )
On the other hand, since 7po
that TA,. f
bpi,a
() P oAo
Ao
tf(ao) tAl
gaoopi,ao0pi
6a,*, tf(ao) i3 Eft(A0;B0)
(5.5.4)
CHAPTER 5. COMMUTING SQUARES
84
Since the gaooR1,aoopi's are linearly independent, and since the pa,a,'s form a basis for A1, the assertion (c) follows from equations (5.5.3) and (5.5.4).
REMARK 5.5.3 (1) It must be clear that (5.5.1) is a symmetric commuting square if and only if the matrix V of Proposition 5.5.2(c) is unitary; this is the reason for referring to the conclusion of Proposition 5.5.2(c) as a biunitarity condition. (2) It must also be clear that if G, H, K and L are given rectangular matrices with non-negative integral entries, and if these satisfy the obvious consis-
tency condition GL = KH, then a necessary and sufficient condition for the existence of a commuting square of algebras such as (5.5.1) with the inclusion matrices of Ao C_ A1, B0 C B1, A0 C_ B0 and Al C B1 being G, H, K and L re-
spectively, is the existence of a unitary matrix U as in Proposition 5.5.2 (with S2(Ao;A,) being the set of oriented edges in a bipartite graph with `adjacency' relations described by G, etc.), such that the corresponding V is an isometry.
(3) In the passage from the square (5.5.1) to the `biunitary' matrix U, it was necessary to fix two systems of matrix units for B1 compatible with
the towers C C Ao C Al C B1 and C C A0 C B0 C B1 respectively. It follows from Remark 5.4.2(2) that the `isomorphism class' of a commuting square is described by an equivalence class of `biunitary' matrices U as in Proposition 5.5.2, where two such matrices, say U and U, are equivalent if U = W2UW1, whereW1 (resp., W2) is a unitary matrix with rows and columns indexed by S2(Ao;A,;B,) (resp.) Q(Ao;Bo;B,)) such that the matrix W1 (resp., W2) (resp., has the form (W1)aloa2z = 6(8(al),f(al),f(az)),(s(ai),f(ai),f(az))(al)al(a2)a, z 1 1
(W2)p1op2 = b(S(A1),f(R1),f(Pz)),(S(Pi),f(pi),f(R2))(b1)pi(b2)Q?) where a1, a2, b1, b2 are
suitably indexed unitary matrices. (4) The passage U H V respects equivalences of biunitary matrices (which describe symmetric commuting squares with respect to the Markov trace); i.e., suppose U, U, W1, W2, a1, a2, b1, b2 are as above; suppose V is obtained from U
by the same prescription by which V was obtained from U, then it is easy to see that the biunitary matrices V and V are also equivalent; to be brutally explicit, one has V = W2VW1i where (yV2)& , _ a(S(a2),f(a2),S(pz)),(S(a2),f(ai),S(R2))(CI2)a2(b2)Q',
(Wl)al Rli =
a(s(al),f(al),f(Rl)),(s(ai),f(ai),f(Ri))(a'1)ai(bi)Qi.
We shall say that the commuting square (5.5.1) is described by the biunitary matrix U if the commuting square and the matrix are related as in Proposition 5.5.2. Thus, this happens precisely when it is possible to find two systems of matrix units {pa,a,} and {qp,p'} for B1, compatible with respect to the towers A0 C Al C_ B1 and A0 C B0 C_ B1 respectively, which are related by the matrix U as in Proposition 5.5.2(a). We now want to consider the case where (5.5.1) is a symmetric commuting square with respect to the Markov trace, and consider the `dual' (symmetric) commuting square obtained, as in Lemma 5.3.3, from the basic construction.
5.5. THE BIUNITARITY CONDITION
85
PROPOSITION 5.5.4 Let Ai, Bi, e be as in Lemma 5.3.3. Suppose in addition that (5.3.2) is a symmetric commuting square. Suppose this symmetric commuting square is described by the biunitary matrix U as in Proposition 5.5.2, and suppose V is the matrix associated with U as in Proposition 5.5.2(c). (a) H'
B1 C B2 UK
LU
(5.5.5)
G'
Al C A2, is also a symmetric commuting square, and it is described by the biunitary matrix U11,2) = V;
(b) further, HH'
C
B0
B2
UK
KU GG'
C
Ao
A2
is also a symmetric commuting square,and it is described by the matrix U10,2) defined by (f
K100
ki0Ri0Rz
Aopz
uaio) va2ok2.
L ,2))aloa2ok2 = AES2(Aj;Bi)
Proof: It follows already from Lemma 5.3.3 that the inclusions in the squares (5.5.5) and (5.5.6) are as indicated, and that (5.5.5), and consequently (5.5.6), is a commuting square with respect to the Markov trace. Further, it is obvious that the symmetry of the commuting square (5.3.2) implies that of the commuting squares (5.5.5) and (5.5.6). So we only need to prove the assertions concerning biunitary matrices. We shall use the symbols ctio, ci, cr2, 01, Q2, K1, A and ice (or `primed' versions of them) respectively, to denote the typical oriented path in St(C;Ao), Q(Ao;A1), l(A,;A,), I(Bo;B1), 9(Bi;B2), I(Ao;Bo), 1(Ai;B,) and Il(A2;B2), with the un-
derstanding that a2 (resp., /32) is an oriented path in SZ(A(,;A,) (resp., S2(Bo;BI)) Suppose {raooa,oa,aooaioa'} and {gaooklo,,,aooklopi} denote systems of ma-
trix units for B1 compatible with the towers C C_ A0 C Al C B1 and c C A0 C B0 C B1 respectively, and that the biunitary matrix U is related to the commuting square (5.3.2) via these systems of matrix units; thus, kioPi-iopi
raooaloA,aooaloa' =
ualoA uaioa'gaookloQl,a,okjopi
(5.5.8)
kioQi,kioPi
It is clear from Proposition 5.4.4 and Lemma 5.3.1 that there exists an isomorphism, call it 03, of B2 onto the path-algebra model for B2 coming from the tower C C A0 C B0 C B1 C B2, such that 03 maps the system
CHAPTER 5. COMMUTING SQUARES
86
{gaoOKI001'aoorlo01} of matrix units for Bl onto the standard system of matrix units for the path algebra given by {eaQoKloFjl,aooKiopi}, and such that tf(P1)tf(Pi)
03(e) _
-B0
(5.5.9)
eR10 1,R1OR'1,
ts(pl)
al,ai,s(al)=s(pl)
where, as before, we write t° : 7r(C) - [0, 1] for the trace-vector correspond-
ing to tric, for C E {Ai, Bi : i = 0, 1, 2}. Let 233 = {q«ooK1OR1OR2,a, OKi00104'2 } be the system of matrix units for B2 (which is compatible with the tower c C A° C_ B° C Bl C B2) obtained as the inverse image under 03 of the standard system {eaooKloQ10R2,a'0l orc' op,lop,2 } of matrix units for the path-algebra
model. (It should be clear that this notation is justified - in the sense that this system of matrix units does indeed extend the system {gaooKlOQl,aoo c, o0,1 }
chosen earlier.)
Similarly there exists an isomorphism 0° of A2 onto the path-algebra model for A2 coming from the tower C C A° C_ Al C_ A2, such that W0(raooal,aooal) = eaooal,aooai and such that tf(al)tf(a
'010(e) _
ao
al,ai,s(al)=s(ai)
(5.5.10)
l
t,,(-l)
-, , } be any system of matrix units for Now let B1 = {p° B2 which is compatible with the tower C C A0 C Al C A2 C B2 and which extends' the system (010) 1({eaooa,o62ia o«loa2}); let 41 denote the isomor phism of B2 onto the path-algebra model from the tower C C_ A° C_ Al C_ A2 C B2 which maps p°aooaloa2orc2,aooaloa 2orc2 onto e aooa1OtY2orc2,aooaloa 2o,c2 It should be clear that if we define apoalocy2orc2,a'oal,oa 20K2
KloR1-Ki°Qi
raooaloaoR2,aooaioa'op 2 =
q
ua1oa ualoa'g10OOK1O Rl OP2,1 OK1O/R1 OP'2' K1ORl,?1 OPi
then 232 = {r«0oalOA0Q2,aooalo'\,oa,2} is a system of matrix units for B2 which is
compatible with the tower C C_ A° C Al C B1 C B2 and which extends the already chosen system of matrix units {raooaloa,a0O«1Oa,} for B1. It should be
obvious that we have the relations gaoOCl01oP2,aoOKI013ioP'2
K1o 1 E ualoa v'aiioa'raooaloaoRz,aooaioa'oop 2. (5.5.11)
aloa,ajoA'
The definitions and equation (5.5.10) imply that Ptf(al)tf(ai)
e=
0
paloal,aloa 1. al,al,$(al)=$(ai)
t
(al)
(5.5.12)
5.5. THE BIUNITARITY CONDITION
87
Similarly, it follows from equation (5.5.9), equation (5.5.11), and the definition
of the matrix V (in terms of U), that 001)tf(P,)
e =
Bo P1,Pi,s(P1)=s(P1)
qP1°R1 Pi°R'1
ts(P1)
tf(P1)tf(Pi) 4aoOK1°P1°Rl,Cko°K1°P;°R',
-B p t s(P1)
s(R )=s(Pi)f(K1)=s(P1)
f(P1)tf(P1)1°P1 oA,.,oa'
K1°Pi
,,OA u,iOA, raooaloAO, 1, ..0., Oa'o(i11
Bo powloPl
ts(P1)
s(P1)=s(Pi)
f(Aofi1 .0 ...lopl,pi,.loa,aloa"
A'ofj'l
val°KlVQ'i OKi rip oocl oaoPl ,moo oc'oA'o)j',
-Ao t,(-1)
s(P1)=SW1' )
Let fl and 7-12 denote Hilbert spaces with orthonormal bases {dc, : a E S2(C;Ap;A1;A2;B2)} and fn,
:
ry E Q(c;Ao;A1;B1;B2)} respectively. Let pi (resp.,
P2) denote the representation of B2 on fl (resp., l2) such that pl(p°',C,,) _ (resp., p2(r,y,,y,)
ri.y,)r7,y). Consider the unitary operator V : H,
f2 defined by a°R2 vd\2ok2?7QOOa1OAOP2"
[C
VSQOOCilod2OK2 AoP2
It is clear from the definitions that V intertwines the restrictions to Al of the representations pl and P2, and that Vpi (p°
01 o62oK2,UQOai0a'2oK'2) va42
v
oP'2
02oK2 ab2oK2
*
- ) P2 (r a0oaloAoP2,abOCioA'oP'2
.
(5.5.13)
A 421A/42
It follows quite easily from equation (5.5.12), equation (5.5.13) and the already obtained expression for e in terms of the r.y,,y,'s (after making the change of variables (/32i /K2) to (/31i i1)) that V*pl(e)V = p2(e). Since A2 = (Al, e), this means that the operator V intertwines the restrictions to A2 of Pi and P2. Suppose now that ((wA2pk2)) is the biunitary matrix which describes the commuting square (5.5.5) with respect to the systems of matrix units {po,,,,, } and {r.y,.y, }; thus, if we define an operator W : H1 --> 7-12 by the prescription a°R2
[[
WSio0alOa2oK2 = E wdA2OK2'/apo0JOAO) 2) a°P2
CHAPTER 5. COMMUTING SQUARES
88
then W is a unitary operator which intertwines the representations pl and p2 of B2.
It follows then that W*V E p1(A2)'; on the other hand, it is clear from the definitions that W*V commutes with {pi(f) : f c Z(B2)}; on the other hand, by Remark 5.4.2(1), we have pl(Z(B2)) = pi(B2)'; thus we find that W*V E p1(B2), and that in fact W*V = pl(uo) for some unitary element uo c A2 n B2. It is easy now to see that if we define pa,,,, = uopo,,,uo, then {p,,,,,,} is a system of matrix units (which is compatible with the tower
ccA0CA,CA2CB2)andthat
A'°$'2
Ao32
OA' 0)3'2
Aoa2,A'o$'2
(5.5.14)
This proves (a), while a combination of equations (5.5.14) and (5.5.8) estabEl lishes (b). COROLLARY 5.5.5 Suppose (5.3.2) is a symmetric commuting square with
respect to the Markov trace. Let {B,.} denote the tower of the basic cnstruction applied to the initial inclusion B0 C_ B1, with e,+1 E B,,,+1 implementing the conditional expectation of B,, onto B,,,_1. Inductively define An+1 = (An) en.+1), n > 1.
(i) Then, for every n > 0, B,,,
C Bn+1
U
(5.5.15)
U
An c
A,,,+1
is also a symmetric commuting square,and it is described by the biunitary matrix U(n,n+1), where
U if n
_
is even,
V if n is odd.
U("°"+1)
(ii) For each n > 0, we have isomorphisms A' n B,,, c A' n B,,,+1 U
Z(An)
=
U
c A' n A,,,+1
A;,+2 n B, +2 c A' +2 n B,+3 U
U
Z(An+2)
C
n An+3
Proof: (i) is an immediate consequence of Proposition 5.5.4(a) and the easily verifed fact that if we denote the passage U H V by V = V(U), then V(V(U)) = U. (ii) By definition, we can find systems of matrix units {pak , a, at E :
Q(C;Ak;Ak+,;Bk+,)} and {q Q,
:
0, /3' E Q(c;Ak;Bk;Bk+l)} for Bk+1 (which are
5.6. CANONICAL COMMUTING SQUARES
89
compatible with the towers C C Ak C Ak+l C Bk+l and C C Ak C Bk C Bk+1 respectively, such that pao) ako\k+l,aooak0\k+1 A, °Q
-
(ufk,k+1})akoAk+, (n[k,k+1))aio\k k
k+1 qaa)
\k o Qk,a0 o\k aQ k
\k oQk,\k opk
Since S2(A.n;B,,;B,}1) _
Q(An+2;Bn+2;Bn+3), it follows from the path-algebra
description that there exists a unique isomorphism 0 : (A' n Bn+1) -> n B,,+3) such that Q),\noRn. Since U[n,n+1; qan = U[n+2,n+s;, it (n+2) follows that also 0(p n a)o\n+l,ano\,+l) _ and so the map 4b =p +l,ano\,}1, implements the desired isomorphism.
5.6
Canonical commuting squares
Assume, as usual, that N C M is a finite-index inclusion of Ill factors, that
N=M_1CM=M0CM1C...CMnC... is the tower of the basic construction, and that en+1 denotes, for n > 0, the projection in Mn+1 which implements the conditional expectation of Mn onto Mn_1.
PROPOSITION 5.6.1 Fix n > 0, and let tr denote the restriction, to N' n Mn+1, of
(i) The following is a commuting square with respect to tr:
N'nMn C N'nMn+1 U
U
(5.6.1)
MI nMn C M'nMn+1 (ii) tr is the unique Markov trace for the inclusion (N' n Mn) C_ (N' n Mn+1) .
Proof: Fix x E (M'nMn+1) (resp., (N'nMn+i)); then for arbitrary a E M (resp., N), an application of EMn to both sides of the equation xa = ax yields
(EM, x)a = a(EMJ (x)). Since a was arbitrary, this means that EMJ (M' n Mn+1) C (M' n Mn) (resp., EMJ(N' n Mn+1) C_ (N' n Mn)), which clearly implies the truth of (i). As for (ii), note, as above, that the projection en+2 E (N' n Mn+2) implements the tr-preserving conditional expectation of (N'nMM+1) onto (N'nMn) and that en+2 satisfies the Markov property with respect to (N' n Mn+1) and trMn+2I(N,nM,+2). It follows now from Lemma 5.3.1(b) and Proposition 3.2.3 that tr is a Markov trace for the inclusion (N' n Mn) C (N' n Mn+1). On the
CHAPTER 5. COMMUTING SQUARES
90
other hand, as a result of the bimodule picture of the principal graph, this has already been seen to be a connected inclusion, and the proof is complete. To proceed with the analysis, it is desirable to take a closer look at Lemma 5.1.1 and its consequences. According to that lemma, if N C_ M is a finiteindex inclusion of III factors, then we can find a subfactor N_1 of N such that M is isomorphic to the result of the basic construction for N_1 C N. However, the choice of N_1 is far from unique. For instance, if u is a unitary element of N, it should be clear that uN_lu* works just as well as N_1. (It is a fact - see [PP1] - that this turns out to be the only freedom available in the choice of N_1; but we do not need that fact here.) Exactly as we constructed the tower {M,,,}n>1 of the basic construction, we may also construct a tunnel {N_,,,},,,>1 in such a way that every inclusion of neighbours in (5.6.2) (below) is an inclusion of IIl factors with index equal to [M : N], and such that any string of three equally spaced factors `is a basic construction':
M,,,C....
(5.6.2)
The point, however, is that while the `tower' construction is canonical, the `tunnel' construction is only 'semi-canonical' in the sense discussed in the last paragraph; thus, given N_,,,, the subfactor N_,,,_1 is determined only up to an inner automorphism in N_,,,. In any case, suppose we have chosen a tunnel as in (5.6.2). A little thought should convince the reader that once we have fixed n, the map x --> JMx*JM defines an anti-isomorphism of N_,,, onto Mn+1 which maps N_k onto Mk+1 for 1 < k < n. Thus, we have, for each n, an anti-isomorphism (of commuting squares):
M'nM,, c-M'nMn+1 U
U
MI nMn c_ MinMn+1
anti-isom.
=
MnN'(n_1)-c MnN'n U
U
NnN'(n_l
c_
NnN'n (5.6.3)
It follows, thus, that independent of the manner in which the tunnel {N_n}n>1 was constructed, the following grid of finite-dimensional C* -algebras, equipped with consistently defined traces (given by the restrictions of trM) - or rather, the trace-preserving isomorphism-class of this grid - is an invariant of the inclusion N C M:
c=MnM' c MnN' c MnN11 CMnN12 c_ U U U NnN' C NnN', C NnN'2 c
--
(5.6.4)
DEFINITION 5.6.2 The trace-preserving isomorphism-class of the grid (5.6.4) of finite-dimensional C*-algebras is called the standard invariant of the sub-
5.6. CANONICAL COMMUTING SQUARES
91
factor N C M, and the individual squares in this grid are called the canonical commuting squares associated with the subfactor.
Notice, incidentally, that in view of the anti-isomorphisms given by (5.6.3),
the standard invariant contains the data of the principal and dual graphs of the subfactor. (The latter (resp., the former) describes the Bratteli diagram of the tower contained in the top (resp., the bottom) row in the standard invariant.) Further, it must be clear that the standard invariant would even be a complete invariant for the subfactor N C_ M, provided we could, by making a judicious set of choices, find a tunnel {N_,,} with the (generating) property that Un (M n N' n) is weakly dense in M and U,, (N n N' n) is weakly dense in N. Notice that an obvious necessary condition for this is that N and M be both isomorphic to the hyperfinite III factor. (It is a fact that a finite-index subfactor of R is necessarily hyperfinite.) It is a deep theorem due to Popa ([Pop6]) that a subfactor admits such a generating tunnel if and only if it satisfies a certain property he calls strong amenability. It follows from one of the equivalent ways of characterising this notion of strong amenability that any subfactor of finite depth - i.e., for which the principal (equivalently the dual) graph is a finite graph - is necessarily strongly amenable, and is hence characterised by its standard invariant.
It will be a good idea to consider the finite-depth case a little more closely. Suppose then that N C M is a subfactor of finite depth. It then follows from the definition of the principal and dual graphs, that if n is large enough, then (MI n M.,,_1) C_ (Mi n Ma) C_ (Mi n Mn+1) is an instance of the basic construction, as is (M' n M.,,_1) C_ (M' n M,') C_ (M' n ,nn,), with the conditional expectation being implemented in either case by e,,+1. Thus, in view of Lemma 5.3.3 and Corollary 5.3.4, we may paraphrase the finite-depth case of Popa's theorem on amenable subfactors as follows:
THEOREM 5.6.3 Let N C M be a subfactor (of a III factor) of finite depth. Then, for n sufficiently large,
M'nMn C M'nMn+1 U
U
(5.6.5)
Min Mn C MlnMM+1 is a symmetric commuting square. Further, the isomorphism class of this square of algebras is a complete invariant of the subfactor.
92
5.7
CHAPTER 5. COMMUTING SQUARES
Ocneanu compactness
Suppose, in the opposite direction to Theorem 5.6.3, that we are given a symmetric commuting square (with respect to the Markov trace), say H
Bo C B1 KU
UL
(5.7.1)
G
A0 C A1, where the inclusions are assumed to be connected. Let
Bo cB1CB2CB3c be the tower of the basic construction (applied to the initial inclusion Bo C_ B1), where B,,,+1 = (B,,,, e,,,+1) for n > 1. Define A,,,+1 = (An, en+1) for n > 1.
(Notice that, according to our conventions, the sequence {e} now starts only with e2.) It then follows from Lemma 5.3.3 and Corollary 5.3.4 that Hn
Bn C Bn+1 KnU
UL,
(5.7.2)
An C An+1 is a commuting square for each n > 1, with inclusion matrices as given, where (Gn, Hn, Kn, Ln) is (G, H, K, L) or (G', H', L, K) according as n is even or odd. As has already been observed in §3.2, the algebra Un Bn admits a unique tracial state, and consequently has a copy R of the hyperfinite II1 factor as its von Neumann algebra completion with respect to this trace; for the same reason, the weak closure of U,, A,, is a subfactor Ro of R. Thus each symmetric commuting square of finite-dimensional C*-algebras (with respect to the Markov trace, and such that the inclusions are connected)
yields, by the above iterative procedure, a subfactor of the hyperfinite III factor. The content of Theorem 5.6.3 is that if, in place of (5.3.2), we had started out with the canonical commuting square (5.6.5) associated with a finite-depth subfactor of the hyperfinite II1 factor, then the subfactor Ro obtained by the above prescription would have been 'anti-conjugate' to N in the sense that the inclusions N C M and Ro C R would be anti-isomorphic. Conversely, we can start from an arbitrary symmetric commuting square as above, obtain the subfactor Ro C R, and ask if there is any relation between the initial commuting square and the canonical commuting squares. This need not always be the case, but if we recapture the canonical commuting square, then the commuting square we started with is said to be flat. The first step in investigating flatness or otherwise of a commuting square
is the computation of the higher relative commutants; the answer to this computation lies in the following result due to Ocneanu.
5.7. OCNEANU COMPACTNESS
93
THEOREM 5.7.1 (Ocneanu Compactness) Let A, B,,,, R0, R be as above. Then
RonR=A1nBo.
(5.7.3)
Proof: First observe, in view of the equations A,,,+1 = (An, e,,,+1) and {en+1}' n Bn = Bn_1i that if m > n > 1, then
A' n Bn = Ann {en+i, en+2,
,
e,n} n Bn
An n Bn-1
and we thus find that
A'mnBn=A1nBo Vm>n>1. Since Un An is weakly dense in R0, it follows that
A'nBocRonR. (In the following, we identify a E R with aQR, and thus view R as being metrised by II 112, the metric in L2(Ro).) Let fln denote A2n n B2n+1, viewed as a (finite-dimensional) Hilbert space (with respect to I 12), and let En (resp., Fn) denote the subspace A2n n Ben (resp., A2n+1 nB2n+1). Then, as in the proof of Corollary 5.5.5, we can find an (algebra) isomorphism Ln : Ho -> 7-ln such that Ln(Eo) = E,,- and Ln(Fo) = -
1
I
Fn.
We first establish the following:
Assertion: There exists a constant c > 0 such that c hIIxll
IILn(x)ll C cllxII Vx E Ho, dn.
Proof of assertion: Since Ln is a *-algebra homomorphism, we have to show that the set { tr Ln (x*x) tr x*x
EA'nB1in>11
is bounded away from 0 and oo. Since Ao n B1 is a finite-dimensional C*algebra, it is sufficient to show that for all minimal central projections p E Ao, q E B1 (thus the typical minimal central projection in Ao n B1 is pq), and for any one minimal projection f E (A' nB1)pq, the sequence {trt : n > 1} If is bounded away from 0 and oo. In the following paragraphs, we shall use the notation and terminology of §5.5; in addition, we shall use the following notation: we write n : ir(Ao) --> N for the dimension-vector for A0 defined by nP = dim Aop; further, for C = An
or Bn, we shall write t : ir(C) -+ [0, 1] for the trace-vector describing tr1c; thus, it follows from Corollary 5.3.4 that tB°+2 = A-1tB" and fl-+2 = A-1tA"
CHAPTER 5. COMMUTING SQUARES
94
where A = JIGI12 = I1HI12. Finally we shall write v = to
so that v is
the normalised Perron-Frobenius eigenvector for (GG') corresponding to the eigenvalue A. In the sequel, we shall need the following standard fact from the Perron-Frobenius theory: under the standing assumption of inclusions being connected, it is true that for any vector w of the same size as v, the sequence {(1a )nw} converges to the vector (w,v)v. ko By the construction of Ln, for each n, there exists a system {proQ 0, I' O Q' E Q(Ao;Bo;Bi), f (0) = f (N'), s(k) = s(ic'), f (ic) = s(0), f (c') = s(/3')} ",oQ,) of matrix units for A2nnB2n+1 such that Ln(pKoQ = p oQ k,op,. So we have to show that for each fixed path rc o Q E Q(Ao;Bo;Bi), the sequence {tr p(-) xop'Kop is bounded away from 0 and oo. But we have (n) trpropko
-TB2n+1
f(a) A(B)=S(k) i
-Bi
_ i fP(GA )n(P, s(r,))tf(Q) PE"ir(Ao)
Ga ')nn)s(w)tf(Q)
-' >
B1
(n, v)vs( )tf(R) 0,
thereby proving the assertion.
Before proceeding further, notice that if E, F are subspaces of a finitedimensional Hilbert space 7-l, the two expressions p1(x) = d(x, E n F) and p2(x) = d(x, E) +d(x, F) define norms on f/(E n F) and consequently there
exists a constant k > 0 such that k-lp2(x) < pl(x) < kp2(x) Vx E 7-l. In particular, there exists a constant ko > 0 such that ko 1(d(x, Eo) + d(x, Fo)) < d(x, Eo n FO) < ko(d(x, Eo) + d(x, Fo)) Vx E'Ho. (5.7.4)
Coming back to the proof of the theorem, notice first that (as already shown) En n Fn = A'2 1 n B2n = A'1 n Bo dn. Suppose now that x E RI n R; for all n; it follows then that xn E An n B.,,; i.e., x2n E En and set xn = x2n+1 E Fn. It follows now from the above, the assertion and (5.7.4) that
d(x2n, Al n Bo) = d(x2n, En n Fn) cd(Ln1(x2n), E0 n FO)
< cko(d(Ln l(x2n), Eo) + d(Ln1(x2n), Fo)) < C2ko(d(x2n, En) + d(x2n, Fn))
=
e2 kod(x2n, Fn)
<
C2kollx2n - x2n+l (l
->
0 as n -- oo.
5.7. OCNEANU COMPACTNESS
95
On the other hand, since Zr, -; x (in
I
112) as n , oo, this means that
x E A'1 n B0, and the proof is complete.
REMARK 5.7.2 Theorem 5.7.1 is false if we drop the requirement that the commuting square (5.7.1) is a symmetric commuting square, as shown by the following example: let Al = A0 = C 54 B0 B1; then (5.7.1) is trivially a commuting square, Ro = {e,,, : n > 1}", and it is not always the case that Ro n R = *1}1 n B0 =)BO. (See the construction of the 3 + f subfactor in [GHJ].)
The use of Ocneanu's compactness result lies in the fact that it can be used just as well to compute all the higher relative commutants. In order to see how that goes, we start with a lemma. LEMMA 5.7.3 Suppose (5.7.1) is a symmetric commuting square (with respect to the Markov trace). Then there exist a finite set I, and {A i E I} C Bo, { fi i c I} C Ao such that :
:
(i) each fi is a projection; (ii) EAo(A)}) = aijJi-) (iii) EiEI tr fi = I IKI I2; and Vx E B1.
(iv) x = &EI EA,
Proof: As before, we shall use the notation ir(C) for the set of minimal central projections of a finite-dimensional C*-algebra C; for C E {Ao, A1, Bo, Bl},
we write t : 7r(C) -; [0, 1] for the trace-vector corresponding to tric; we shall also write n : 7r(Ao) -j N for the 'dimension-vector' of A0 defined by nP = dim A0p. Thus, for instance, we have nptAo = 1.
KK'iAo = IIKII2tAo,
(5.7.5)
pE?r(Ao)
Finally, we shall denote the typical paths in I (C:Ao), Q(Ao;Bo) and Q(C;Ao;Bo) by
the symbols a, ,c and /3 respectively. Let {q,3,#,} denote a system of matrix units for B0 compatible with the tower C C A0 C B0. (The following notation is motivated by the fact that the
matrix with all entries equal to 1 is customarily denoted by the symbol J.) For each p E ir(Ao), define ga,al
np
f (a)=f (a')=p 1 gaok,alok.
np
f(a)=f(a')=p
(The normalising constant ensures that jp E P(Ao).)
CHAPTER 5. COMMUTING SQUARES
96
Define I = {(ic, 3) : f (r,) = f (Q)} and
ar,,o = F
gaok,s V (r, 0) E I.
(5.7.7)
a'Aa)=S(am)
It follows directly from Proposition 5.4.3 that -Bo
f(k)-
EA. (an,
ts(k)
Define Bp
2
Ao ns(.) t8( )
f"s = is(k); It is then clear that {Ai, fi
:
a,,Q,
V(n, 0) E I.
i E I} satisfies (i), while (ii) is an immediate
consequence of (5.7.5). Further, it follows easily from the definitions that (5.7.8)
Ai = fiAi Vi E I. Another fairly straightforward computation shows that
x=
EAo (x.\ ).i Vx E Bo.
(5.7.9)
iEI
Now, it follows from Corollary 5.3.4(b)(iv) that B1 is.linearly spanned by A1B0; on the other hand, it is a consequence of (5.7.9) that B0 is linearly spanned by UiE7 Ao.\i; thus B1 is linearly spanned by UiE7 A1.i; also, the commuting square condition implies that E I.
(5.7.10)
It follows therefore that {A1.i i E I} is a pairwise orthogonal (with respect to the trace-inner-product) collection of subspaces of B1 which span B1. Hence, if x e B1, there exist ai E Al, i E I, such that x = EiEI aiAi; it :
follows from (5.7.10) that EA1(xA*) = ajfj, and hence, in view of (5.7.8),
EA1(x. ))i = > aifi)i = x. iEI
iEI
COROLLARY 5.7.4 If Ro C R is constructed as in Theorem 5.7.3, and if {Ai : i E I} is as in Lemma 5.7.3, then (i) {.i i E I} is a basis for R/Ro as in §..3; and consequently, (ii)[R:Ro]=IIK112=IIL112.
5.7. OCNEANU COMPACTNESS
97
Proof Since (5.7.2) is a symmetric commuting square for each n, it follows
(from the same reasoning employed in the proof of Lemma 5.7.3(iv) and induction) that for any n and for any x e B?,, x = F EA. (xA )Ai. iE1
Since
B,1 C R U A,,,
U
C Ro
is clearly a commuting square for each n, this means that
xExo(xA)Ai, VxE iEI
Since both sides
00
UBn.
(5.7.11)
n=1
of this equation vary continuosly with x, we see that
x=J:E&(xaz)Ai, VxER, iEI
thus proving (i); assertion (ii) follows at once from Lemma 5.7.3(iii).
Before we start to interpret Ocneanu's compactness theorem to compute the higher relative commutants (and consequently the principal graph) of the subfactor Ro C R constructed as above (starting from an arbitrary symmetric commuting square), it will be prudent to adopt a slight change in notation. We shall henceforth use the symbols Ak and Ak, respectively, to denote what we have so far been calling Ak and Bk; likewise, we shall write R1 for what was earlier called R; thus Rn = (Uk Ak)", n = 0, 1. Let Ro C Rl C_ R2 C . . C Rn C . . denote the tower of the basic construction. Since the symbol ek+1 has already been reserved for the projection in Ak+1 which implements the conditional expectation of Ak onto Ak_1, for n = 0, 1, we shall use the symbol fn+l to denote the projection in Rn+1 which implements the conditional expectation of Rn onto Rn_1i for n > 1. (Again, the sequence starts only with f2.) For n > 1, inductively define .
Akn+1
= (Akn, fn+l), dk > 0.
PROPOSITION 5.7.5 Consider the grid {Ak : n, k > 0} constructed as above.
Let n> 0. (i)n Ak is a finite-dimensional C*-algebra, for each k > 0, and Rn = M I An) 11
(ii)n If n > 1, then for each k > 0, An
U Akn-1
C Ak+1 U n-1
C Ak+1
(5.7.12)
CHAPTER 5. COMMUTING SQUARES
98
is a symmetric commuting square with respect to trR, IAk}, (which is also the Markov trace).
(iii)n If n > 2, then, for each k > 0,
-C An-1-C An
kAn-z
k
k
is an instance of the basic construction, and the projection fn implements the conditional expectation of Ak-1 onto Ak-z. (iv)n For each k > 1,
Ak-1 C Ak C Ak+1
is an instance of the basic construction, and the projection ek+1 implements the conditional expectation of An onto Ak_1. Proof: We prove the proposition by induction on n. The proposition being
true, by construction, for n = 0, 1, we assume that n > 2 and that the assertions (i)n_1-(iv)n_1 are valid. It follows from (ii)n_1 that An-1
k
U Akn-2
C Rn-1 U
(5.7.13)
C Rn_z
is a commuting square, for each k; hence, by the defining property of fn, we E Ank-1, then find that if ank-1
fnak-'fn =
and hence fn indeed implements the conditional expectation of Ak-1 onto Ak-z; in particular, it follows that the set {ao + >i'' 1 aifna' : m E N, ai, a' E Ak 1} is a finite-dimensional C*-algebra containing (Ak-1 U { fn}), and consequently In
Ak = {ao +
ai fna'i : m E N, ai, a'i EAk-1
(5.7.14)
i-1
thus establishing (i)n. EAk-l(fn) = EAk-1(ER._1(fn)) = T, where T = [R1 Rol-1, the Since equation (5.7.14) implies that EAk-' (Ak) C Ak-1, and consequently (5.7.12) is indeed a commuting square with respect to trR, tAk}1. In particular, since this is valid for all k, it is also true that :
Ak U Akn-1
c c
Rn U ,n_1
is a commuting square with respect to trR .
5.7. OCNEANU COMPACTNESS
99
Let {.\i i E I} C Ao be a basis for R1/R0 as in Corollary 5.7.4(i). It follows from Lemma 4.3.4(ii) that f2Ai : i E I} is a basis for Rn/R,-1. In particular, if k > 1, and ak E Ak, then :
{T-21
fnfn-1
ak = T (n
1)
iEI 1) T (n
ERn-1 (akA f2 ... fn) fn ... f2Ai
EAR-1lanA f2 ... fn) fn ...
(5.7.15)
iEI
Hence
Ak =VAk-1.Ak 1
(5.7.16)
where the symbol V denotes `span'. It follows from Remark 5.3.5 that zAk_1(fn)
= 1, and hence, by Corollary 5.3.2, we have proved (iii),. Notice next that k > 1 r f, f2Ai e An C Ak_1; so, in view of the already established fact that (5.7.12) is a commuting square, we see from (5.7.15) that for arbitrary ak E Ak, .
EAk(an) = T-(n-l)
EAk_, (Q'k Aa f2 ... fn) fn ... f2Ai iEI
on the other hand, since fn ek+l ak ek+l
=
f2Ai E Ao C {ek}', it follows from (iv)n_1 that
T-(n-
EAk-1(anA f2 ... fn)fn .. ' f2Aiek+1 iEI
(5.7.17)
EAk-1(ak)ek+l;
hence ek+1 indeed implements the conditional expectation of An onto Ak_1. Deduce next from equation (5.7.16) and (iv),_1 that An k+1
=
An-iAn k+1
k
n-1
= V Ak ek+1Akn-1 Akn Ak-lek+lAk
=V
C V Akek+iAk,
which implies that zAk}1(ek+1) = 1, which establishes (iv), in view of Corollary 5.3.2. Finally, in view of the already establlished fact that (5.7.12) is a commuting square for all k, we see from the parenthetical remark in (ii)n-1, and from (iv)n-1, that E-''Ak(ek+l) = EAk-1(ek+l) E c1.
I11 view of the already established (iv)n, this means that is a Markov trace for the inclusion Ak_1 C A. The already established fact that (5.7.12) is a commuting square, the identity (5.7.16), and the previous sentence complete
the proof of (ii)n, and with that, the induction step, and consequently the proof of the proposition, is complete.
CHAPTER 5. COMMUTING SQUARES
100
We are finally ready to use Ocneanu's compactness theorem to compute the higher relative commutants. THEOREM 5.7.6 (Ocneanu compactness (contd.)) Let ...
C
U
...
C
U
...
C
...
U U
...
C
...
U
C
...
U
U
U
U
U Rn U
U
C
...
C U
U
U
C Ak C
Ao C Ai C AZ C U
C
C An C
U
U
C
...
U
U
U
Ao C An C AZ C ...
C
R1
-+
Ro
U
U
U
AO C A° C AZ C .
->
C A. C
be as in Proposition 5.7.5. Then, for each n > 1, Ro n Rn = A°' n An
Proof: Fix n > 1 and consider the grid
C An C
An C An C AZ C U
U
U
Ao C A°
C
AZ C
U
U
C A.
Rn U
C
Ro.
By Proposition 5.7.5, this grid satisfies the hypothesis of Theorem 5.7.3, and hence, by that theorem, the desired conclusion follows. We now state a few related problems whose solutions we would like to see.
QUESTION 5.7.7 Given a symmetric commuting square, say (5.7.1), does there exist an algorithm which, in the preceding notation, would: (a) compute dim, (R' n Rn) in polynomial time (as a function of the input data) ?
(b) decide in finite time if the subfactor Ro C R1 has finite depth?
(c) answer (b) in polynomial time?
We should mention here that Ocneanu has shown that if it is already known that the depth is a finite integer, say n, then the dimensions of the higher relative commutants can be computed in polynomial time.
Chapter 6 Vertex and spin models 6.1
Computing higher relative commutants
This chapter is devoted to a discussion of the subfactors arising, as in §4.4, from an initial commuting square which is a'vertex model' in the terminology of §5.2.3. Actually, we shall work with a slight generalisation, as follows. LEMMA 6.1.1 Let [N]
Bo C B1 [k]U
(6.1.1)
U[k]
N
Ao
c Al
be a (clearly symmetric) commuting square of finite-dimensional C*-algebras such that the inclusion matrices are 1 x 1 matrices (as indicated) and where AO = C. Suppose this commuting square is described by the biunitary matrix u (see §5.5), which is a kN x Nk matrix given by
u=((ua)),1
1 = u,a
(6.1.2)
Then the square (6.1.1) is isomorphic to the square w(1
®Mk(c))w*
C MN(c) ®Mk(c)
U
c
U
C
(6.1.3)
MN(c) ®1.
Proof: Let {paa,a'a' 1 < cti, cti' < N,1 < a, a' < k} and {gbp,b'p' 1 < 3, l3' < N,1 < b, b' < k} be systems of matrix units for B1 compatible with the towers Ao c Al c B1 and AO c Bo c B1 respectively, with respect to which the biunitary matrix u describes the given commuting square. :
:
CHAPTER 6. VERTEX AND SPIN MODELS
102
'b
The system {paa,a'a' : 1 < cr, cti' < N,1 < a, a' < k} yields an isomorphism : B1 - MN (C) ® Mk (C) thus: b (paa,a'a') = eaa' ® eaa', where the e's denote
the standard systems of matrix units in matrix algebras. By definition, the algebra Al has a system of matrix units defined by Na' = Ea=1 paa,a'a; hence V(Paa') = eaa' ® 1; i.e., z/b(A1) = MN(C) ® 1. By definition, we have N
k
Paa,a'a' - 1: E 'aaua'a'gb/3,b'/3', 0,0'=1 b,b'=l
`
and hence
N
k
gbO,b'O' = L E wpbwp'b'Paa,a'a', a,a'=1 a,a'=1
and consequently, N k E E w)3bWRibr eaa, ® eaa'
W (gb/3,b'(3' )
a,a'=1 a,a'=1
w(e13,3, ® ebb')w*,
where we view w as an element of MN(C) ® Mk(c) in the obvious manner. But B0 is spanned by {qbb' : 1 < b, b' < k}, where qbb' = Q 1 gb/3,b'/3 It follows
at once that b(Bo) = w(1 ®
Mk(C))w*
as desired. DEFINITION 6.1.2 We shall refer to any commuting square of the form (6.1.1) as a vertex model.
REMARK 6.1.3 We shall need the fact, which the proof of the lemma shows, that conversely if w = ((wpb)) E MN(C) ® Mk(C) is unitary, and if
MN(c) ®1 C MN(C) ®Mk(C) U
U
c
(6.1.4)
C w(1 (D Mk(C))w*
is a commuting square (and is hence a vertex model in our terminology), then the biunitary matrix u which describes this commuting square is given by u = ((ubp )), where ubp = wpb
.
In the rest of this section, we shall start with a vertex model, as given by (6.1.3), think of this as the initial commuting square
Ao C Ai U
U
A0 C A1, 0
0
(6.1.5)
6.1. COMPUTING HIGHER RELATIVE COMMUTANTS
103
and describe the tower {RonRn : n > 1} of relative commutants, as discussed in §4.4. We shall use the notation of that section; in addition, we shall find it convenient to use the notation
Cn=RonRa,n>1. By Theorem 5.7.6, we have
On the other hand, the grid {Ak : n > 0, k = 0, 1} (when flipped over by 90° from the way it looks in the grid in the statement of Theorem 5.7.6) is given by A°
C A',
U A00
C
C (A', f2, fs)
(A1, f2)
U
U
...
C
C
U
(6.1.6)
C Ao C (A', f2) C (A', f2, f3)
C
C
the latter is, by Proposition 5.7.5(iii), the result of repeated applications of the basic construction to the top row of the vertex model given by (6.1.4). According to Remark 6.1.3, the basic (initial) commuting square
A° C Ai U
U
C MN(C) ® Mk(c)
MN (C) ® 1
=
U
U
Aa C Ao
C w(1 (9 Mk(r-))w*
C
of the grid (6.1.6) is described by the Nk x kN biunitary matrix u = ((uba)), where uba = wpb . It follows from Corollary 5.5.5(i) and Proposition 5.5.4(b) that, for each n > 1, the commuting square
A0 1
[C]
[N]U
An 1
U[N] [k'}
AO
C
An
o
is described by the Nkn x k n N biunitary matrix U10,n], which is defined by N
(U[0,n])b -- bnO
uaal
biYi (V (u))ryla2ury2a3 b2Y2 baYa
N
Uaai U Y2a2 U'12aa bi'Yi
b2'Yi
baYa
(n factors)
.. (n factors)
N Waai W'Y2a2 w'Y2aa 'Yibi
ylb2
73b3
.
(n factors).
CHAPTER 6. VERTEX AND SPIN MODELS
104
(We pause to point out that in the n-th factors of the product in each of three preceding lines, the subscript is given by bn,6 in the first two lines and by Qbn in the last line.)
By another application of Remark 6.1.3, we find that we may make the identification A°
C_
U
=
U
U
Ao C Ao
Mk (C)) "'I
MN (C)
MN(c) ®1 C
Ai
U
C W[o,n](1 (10 (071 Mk(C)))W[o
C
(6.1.7)
where W[o,n] = (((W[o,n])pbl....a )) E MNkn(c)(= MN(c) ® (®n Mk(c))) is defined by
W[o,n] Qbl...bn - U[o,n] aal...an bi...b.0 '
Hence it follows that Cn = Ao' n An = (1 ® (®n Mk(C))) n W[o,n](1 0 (®n n1. Thus we see that Cn may be identified with the set of those F = ((Fbl'::bn°)) C ®n Mk(C) for which there exists a G = ((Gal... n)) E on Mk (c) such that 1®F = W[o,n] (1® G)W[o n ; in longhand, this means that 2,
N
k
Fal.... .an{Wacl W72C2w720 Cn
k
71bl
rylb2
y3b3
.
.
.. (n factors)}
N
{waal 71Cl 7l z
73 C3
(n factors) }Gcl cn (6.1.8) bl...bn
We shall now re-write the answer to the computation of the higher relative commutants - as provided by equation (6.1.8) - in a diagrammatical fashion that should begin to convince the reader that these examples do indeed have a connection with the vertex models of statistical mechanics. (The letter w that appears above will, in the latter picture, correspond to Boltzmann weights.) We shall be drawing diagrams as in the theory of knots - more precisely, there will be over- and under-crossings, as below:
Actually, we shall be dealing with such diagrams where the two strands of the crossing are both oriented, and we shall follow the convention of knot theory and refer to the two different possible situations as positive and negative crossings. Also, typically, the four ends of the strands will be labelled, with the ends of the over-crossing labelled by Greek alphabets (which are assumed to vary over the set {1, 2, , N}), while the ends of the under-crossing will be labelled by Latin alphabets (which are assumed to vary over the set
6.1. COMPUTING HIGHER RELATIVE COMMUTANTS {1, 2,
,
105
k}). We assign a 'Boltzmann weight' to such a labelled oriented
crossing by the following prescription:
Positive Crossing
a
Negative Crossing
a
wab
I
b
(One way to remember the convention is as follows: for either type of crossing, the Greek (resp., Latin) indices of w correspond to the over- (resp., under-) strand of the crossing; for a positive crossing, the orientation of the over- (resp., under-) strand is given by reading the Greek (resp., Latin) indices from top to bottom (resp., bottom to top); for a negative crossing, the complex conjugate appears, and the orientations of the two strands are both read in the opposite way to that of the positive crossing.) With the above conventions, equation (6.1.8) may be rephrased as the equality
F (6.1.9)
G
where the vertical strings (on each side of the equality) are alternately oriented upwards and downwards (starting with the one at the extreme left), and the diagram is interpreted as follows: By a state or of a diagram such as the following - where we have chosen
n = 2 for simplicity -
ft
F T
CHAPTER 6. VERTEX AND SPIN MODELS
106
- we shall mean a specification of a Latin (resp., Greek) letter to each segment of a vertical (resp., horizontal) string. For example, o might be specified as follows:
F 07
CI cx
l
aryl
I c2
Q
bil
J b2
By definition, the `energy E(o-)' associated to such a state a is the product of the indicated matrix entry with the product of the Boltzmann weights of all the crossings; thus, in the above example, we have E ( D) =
ala2 "' Qc2 F wylblWyIb2
By a boundary edge of a diagram (such as T above), we shall mean any segment of a strand, at least one of whose edges is `free'; thus, in the preceding example, the edges labelled al, a2, bl, b2, cx and p are the boundary edges.
By a `partial state' will be meant a `state' which has been specified only on the boundary edges. Given a partial state i on a diagram T, define the corresponding to `partition function' ZT to be the sum of the energies all states o- which `extend' K.
We are finally ready to state the meaning of the equality (6.1.9); if the diagrams on the left and right sides are denoted by T and B respectively, we require that ZT = ZB for all partially specified states ic, where we identify the sets of boundary edges of T and B in the obvious fashion. We summarise the foregoing analysis in the next theorem. THEOREM 6.1.4 Consider the vertex model Ao U
C_
All U
Ao C A°
=
w(1 (9 Mk(c))w* C MN(c) ® Mk(c) U
U
C
C
.
(6.1.10)
MN(c) ®1
Let Ro C Rl be the subfactor constructed from this initial commuting square C R,, C .. be the tower of the basic as in §4.4, and let Ro C_ Rl C R2 C construction. Then, for n > 1, Ro n R,, can be identified with the set of those matrices
F = ((Fbll...a,)) E ®' Mk(C) for which there exists a G = ((G61:::bn )) E ®n Mk(C) such that equation (6.1.9) is satisfied (in the sense just discussed). We now show how the computations of this section carry over, with slight modifications, to spin models.
6.1. COMPUTING HIGHER RELATIVE COMMUTANTS
107
Analogous to Lemma 6.1.1, one can prove, quite easily, that if G'
Bo
C Bi
GU
UG'
(6.1.11)
G
A0
C Al
is a (clearly symmetric) commuting square with inclusions as indicated, where G = [1 1 . 1] is the 1 x N matrix with all entries equal to 1, if this commuting
square is described by the biunitary matrix u = ((uj')), and if Ao = C, then (the matrix u is a complex Hadamard matrix - see §5.2.2 - and) the square (6.1.11) is isomorphic to
A C MN (c) U
U
CC
(6.1.12)
UAU*,
where A denotes the diagonal sub algebra of MN(c). Thus, these are precisely what we called `spin models' in §5.2.2. Suppose conversely that we start with a spin model, which we assume for notational reasons is given by wOw*
C MN(C) U
U C
C
(6.1.13)
A;
if we think of this as the initial commuting square (6.1.5), and construct the tower {Rn : n > 0} as in §4.4, then an analysis akin to the one given above for vertex models is seen to show that the higher relative commutants continue to be described by the validity of equation (6.1.9), only the interpretation of this equality is now as follows: (we give the general description first, then illustrate with the cases of the second and third relative commutants, and the reader should then be convinced that this diagrammatical description is much more compact and clean than an explicit one with mathematical symbols; among other things, such an explicit description would require different descriptions according to the parity of n.) To start with, we ignore orientations on the vertical strands. Next, given a diagram - such as the left or right side of (6.1.9) - first colour the connected components of the diagram alternately black and white, in a 'chequer-board' fashion, with the convention that the component at the south-west corner is shaded black. By a state a of such a diagram, we now mean an assignment of a symbol from the set {1, 2, , N} to each of the components coloured black. By a boundary component, we mean one which is unbounded, and by a partially defined state, we mean a state which has been specified only on the boundary components.
CHAPTER 6. VERTEX AND SPIN MODELS
108
Once a state a has been fixed, we assign Boltzmann weights to each crossing (which must clearly be one of the following types) as follows: a
Positive Crossing
>
W
b
a
Negative Crossing
wb b
And we define the energy E(a) of the state a to be the product of the indicated `matrix entry' with the product of the Boltzmann weights of all the crossings. Given a partially defined state is on a diagram T, we define the partition function ZT to be the sum of the energies E(a) of all states a which extend rFinally, the equality (6.1.9), in the context of spin models, means that once the boundary components of the diagrams T and B of the left and right sides of (6.1.9) have been identified in the obvious fashion, then ZT = ZB for all partially defined states rc. Since the precise interpretation of this answer depends upon the parity of the relative commutant in question, we illustrate the preceding discussion by explicitly writing out the second and third relative commutants.
n=2: a
a
F
b2
b1
G b2
This means that Ro n R2 consists of those F = ((Fe)) E MN(C) such that there exists an array of numbers G = ((Gb1b2)) (which should be actually thought of as an element of CN2) such that for all choices of a, b1, b2 E N x=1
xy
aa
a Fxwblwb2 = wb1wb2Gblb2.
6.1. COMPUTING HIGHER RELATIVE COMMUTANTS
109
n=3: a2
a1
a2
a1
F G bl
b2
b2
bl
This means that Ro n R3 consists of those arrays of numbers F =
2 ))
(which must be thought of as an element of a direct sum of N copies of MN ((C))
such that there exists an array of numbers G = ((Gb,b2)) (which should also be thought of as an element of a direct sum of N copies of MN(C)), such that, for all choices of a1, a2, b1, b2 E {1, 2,- , N}, N
N Fala2 xa2
a2 _ x x wbib2wb2 -
al al a2
bix
wb12Ux wx Gbib2.
We summarise the foregoing in the following result. THEOREM 6.1.5 Consider the spin model
Ao C Al u
u
Ao C A°
wLw* C MN(c) =
U C
U
C
(6.1.14)
0
Let Ro C R1 be the subfactor constructed from this initial commuting square C Rn C as in §4..4, and let Ro C Rl C R2 C be the tower of the basic construction. Then, for n > 1, R'0n R. can be identified with the set of those `matrices' F for which there exists a `matrix' G such that equation (6.1.9) is satisfied (in the sense just discussed).
REMARK 6.1.6 In the diagrammatic framework described above - for both vertex and spin models - it should be remarked that the biunitarity condition on the matrix w is exactly equivalent to the requirement that `the partition function given by w is invariant with respect to Reidemeister moves of type
II' - meaning that, no matter how the two strands are oriented, resp., the regions are shaded black and white, we have
CHAPTER 6. VERTEX AND SPIN MODELS
110
The verification that this is indeed the case will be a good exercise for the reader to ensure that (s)he has understood our procedure.
REMARK 6.1.7 It is easily verified that, in the case of a vertex model, the embeddings (Ro n Rn) -+ ®n Mk(C) which we obtained are consistent in the sense that the following is a commutative diagram of inclusions (where we embed ®n Mk(C) into ®n+1 Mk(C) by x H x (9 1):
(Ro n Rn+1) _ ®n+1 Mk(c) U
U
(Ro n &)
®n 11'Ik(C)
`->
Consequently, we may effectively use Theorem 6.1.4 for even computing the principal graph of a subfactor arising from a vertex model. For identical reasons, an analogous remark is also valid for spin models.
REMARK 6.1.8 The purpose of this remark is to point out that if Ro C_ R1 is constructed as above from a vertex (resp., spin) model, then so also is R1 C R2, and to consequently derive some facts concerning the dual principal graph-
(i) To start with, make the obvious observation that if a commuting square
C C D U
U
A C B is described by the biunitary matrix u, then the commuting square
B C D U
U
A C C is described by the biunitary matrix u*. It follows, then, from Proposition 5.5.4(a) that if Ao
KU
C
Ai UL A°
A00
C is a symmetric commuting square which is described by a biunitary matrix U, and if we construct the grid {An} starting from this initial commuting square in the usual manner, then the commuting squares
Ai C AZ LU
UK
A° C A2
Al C AZ
Ao C A21 K'U Ao
UL'
C
Ai
and
L'U
UK'
Ai C AZ
6.2. SOME EXAMPLES
111
are described by the biunitary matrices V,V and U respectively, where V is as in Proposition 5.5.4, and vQa
fix
Notice that U = U, and thus only these four biunitary matrices suffice to describe all the squares in the grid.
(ii) It follows from (i) above and equation (6.1.2) that if the matrix w gives rise to the subfactor Ro C R1 as in Theorem 6.1.4 or Theorem 6.1.5, then the transposed matrix w' gives rise to the dual subfactor R1 C R2-
6.2
Some examples
The input for a vertex model, as in Theorem 6.1.4 for instance, is an Nk x Nk matrix w = ((wpb )), which satisfies the biunitarity condition that w is unitary, as also is the matrix v defined by vbp = wab; this is easily seen to be equivalent to the following condition: Suppose we write w in block-form as w = ((w00))1
,'YN} be any collection of k x k unitary ma-
EXAMPLE 6.2.1 Let {'Yl, rye,
trices, and define waba = J ('Ya)b Q 0
l
if a = Q, } otherwise.
(6.2.1)
It is trivially verified that this w satisfies the biunitarity condition stated above.
(Reason: wp = 0 if a f3, and hence what we have calledw'n' is the same as w.) Notice that the 'diagonal constraint' above says that the Boltzmann weight associated to a crossing is zero unless the two ends of the over-strand have the same label. It follows from Theorem 6.1.4 that C,. = R' nR,, consists of those matrices F = ((F,'.::an )) E ®" Mk(C) for which there exists a matrix G = ((Ga,,', n )) E ®n' Mk(C) such that k 11
Sp
Fc ...c'n (7a)b, (7a)b2 (rya)b3 .
.
k
aQ
(( /a)C,
(' )c2 (/a)C3 ...)G61...6n
CHAPTER 6. VERTEX AND SPIN MODELS
112
for all a1,
, a, b1,
,
b, a, /3, or equivalently,
F'(7a0 'ya®Ya0 ...(77,0 ya07a(9 ".)G Va. n factors
n factors
If we let K be the closed subgroup of U(k) generated by {-ya'yp 1 1 < a, 0 < N}, and if we write 7r for the identity representation of K on Ck, the previous conditions are seen to be entirely equivalent to :
Cn = (7r(9 F 0 7r0 ...)(K) n factors
In view of Remark 6.1.7, a moment's thought should convince the reader that the principal graph of this subfactor has the following description: first form a bipartite graph with the set of even (resp., odd) vertices being given
by g(°) = K x {0} (resp., gll) = K x {1}) - where K denotes the unitary dual of the compact group K - where the number of bonds joining the vertices (pi, i), i = 0, 1, is given by (p0 (97r, pl), the mutiplicity with which pi features in the tensor product po 0 7r; finally, the desired principal graph is the connected component of the above graph which contains the even vertex indexed by the trivial representation of K. Notice next that the transpose matrix w' is given by the matrices 7yi, , 7N in exactly the same way that w is given by the rya's. Thus - in view of Remark 6.1.8(ii) - the dual graph is described by the closed subgroup K' of U(k) - given by K' = {g' : g E K} - in the same way that the principal graph was described
by K. Notice now that K' = {g-1 : g E K} = {g : g E K} = K. Hence the equation b(g) = g defines an isomorphism 0 : K' --> K. Let 0* : K -+ K' denote the (obviously bijective) map defined by 0* (p) = po0. Notice that if we write ir' for the identity representation of K' on Ck, then ir' = 0*(W). Finally, it follows that the principal graph is isomorphic to the dual graph by a graph isomorphism which associates the even or odd vertex in the former which is indexed by p (say) to the even or odd vertex in the latter which is indexed by 0*(P)-
EXAMPLE 6.2.2 Let {c1, c2i
,
ck} be any collection of N x N unitary ma-
trices, and define waa pb
{ (ca)Q 0
if a
b,
otherwise.
} (6.2.2)
It is trivially verified that this w satisfies the biunitarity condition stated above. (Reason: each wp is a diagonal matrix, and hence what we called w'N is nothing but the transpose of w.) Notice that the `diagonal constraint' above says that the Boltzmann weight associated to a crossing is zero unless the two ends of the under-strand have the same label. It follows from Theorem 6.1.4 that Cn = RI0 nRn consists of those matrices F = ((Fbll::bn )) E ®n Mk(C) for which there exists a matrixG = ((Gbi:.. )) E
6.2. SOME EXAMPLES
113
®n Mk (C) such that
-
N Fb11..bn 71,...,7n-1=1
(1Cb171(Cbz)7Yi(Cb3.y3...
n factors
N ((Cal
ci e 72(( )ry11 ((Caz)711Cas)73
a1...an
")Gb1...bn
71,...7 -1=1 n factors
for all a1,
, an, b1, ... , b,,, a, /3, or equivalently a.1...an
Fb1...bn
* * Cblcb2Cb3 ...)a = (Ca1CasCas .. )/3 Cs
n factors
for all a1,
, an, b1,
al...an b1...bn
n factors
, bn, a, /3; since the `variables separate', this shows that
a'...a Cr, = {F = ((F b1...bn )) E
®n
Mk(c) . Fa1...an = 0 unless b1...bn
is a scalar multiple of Cb1 C62 Cb3
Cal cat ca3
}.
After a moment's thought, this is seen to imply that the principal graph of this subfactor has the following description: let G be the (non-closed) subgroup of UN(C) generated by and let G be the quotient of G by the
normal subgroup of those elements of G which are scalar multiples of the identity matrix; form a bipartite graph with the sets of even and odd vertices being both indexed by G, where the number of bonds joining the even vertex indexed by [go] to the odd vertex [g1] is given by the cardinality of the set {1 < i < k : [gl] _ [goci]} - where we have used the notation g i--> [g] for the quotient mapping G -+ G; finally, the desired principal graph is the connected component of the above graph which contains the even vertex indexed by the identity element of G.
Here also, it is true that the dual graph is isomorphic to the principal graph. To see this, first note - in view of Remark 6.1.8(ii) - that w' is constructed out of the ca 's in exactly the same manner that w was constructed out of the ca's. Using natural notations, we see that the dual graph is the Cayley
graph of the group G' = {g'
g E G} modulo scalars, with respect to the 1 < a < N}. The group isomorphism g --> g' establishes the desired graph isomorphism. generators {[c'a]
:
:
EXAMPLE 6.2.3 We now discuss the second relative commutant of a subfactor constructed from a spin model (as in Theorem 6.1.5). Using the notation of that theorem, recall from §5.1 that Ro n R2 consists of those F = ((F° )) E MN(C) such that there exists an array of numbers G = ((Gb1b2)) (which should be actually thought of as an element of CN2) such that for all choices of a, bl, b2 E {1, 2, ..
,
N},
N
F°wx G b1 b2 x b1 wh,= 7Ubb1 wa b2 X=1
(6.2.3)
CHAPTER 6. VERTEX AND SPIN MODELS
114
For each a = (al, a2) E {1, 2,. , N} x {1, 2, , N}, define the vector va E CN by (va)s = walwa2; then equation (6.2.3) says that vb is an eigenvector for the matrix F with eigenvalue Gb1b2. Since Ro fl R2 is generated by self-adjoint elements, and since eigenvectors of a self-adjoint matrix which correspond to distinct eigenvalues are orthogonal, we see that the matrix G must satisfy Gb1b2 = Gaia2 if vb is not orthogonal to va. More precisely, we be the smallest equivalence relation in the set N} x {1, 2, , N} such that a w b if vb is not orthogonal to va. Then any G as in (6.2.3) must necessarily be constant on equivalence classes. Conversely, if G is any matrix which is constant on equivalence classes, it is easily seen that the equation can deduce the following: let {1, 2,
,
N Fa
x=
/
/
Cblb2(vb)a(vb)x b1,b2=1
defines an element F satisfying (6.2.3). It follows that Ro fl R2 is an abelian *-subalgebra of MN(C) with dimenequivalence classes. It turns out - see [JNM] sion equal to the number of - that if C is one such equivalence class, then the number #{a : (a, i) E C} is independent of i; call this number the valency of the equivalence class. It is a fact that if Fc is the minimal projection in Ro fl R2 which is the projection onto the subspace spanned by {vb : b E C}, then the valency of C is precisely the rank of the projection Fc. On the other hand, since trR2 JRonR2 = NTrIRonR2, where TI denotes the (non-normalised) matrix-trace on MN(C), and since the subfactor determines not only the tower {Ro n R,ti : n > 0}, but also the traces {trR,,, IRonR : n > 01, we see that the subfactor given by a spin model as above determines the number of ^-,,, equivalence classes as well as the valencies of each of these components. We wish to make two points through this example: (a) Spin models yield a passage from Hadamard matrices to subfactors via
a commuting square. It is clear that two Hadamard matrices are equivalent - as described in §5.2.2 - precisely when the associated commuting squares are isomorphic (see Remark 5.5.3), and that in such a case, the associated subfactors are conjugate. The point here is that, via the preceding remarks, the subfactor associated to a Hadamard matrix w in the above fashion determines the number of ^ equivalence classes and the valencies of those equivalence classes, and it turns out - see [JNM] - that these suffice to tell the five pairwise inequivalent real Hadamard matrices of order 16 from one another. (b) It is a fact that there exists a 16 x 16 real Hadamard matrix which is not equivalent to its transpose. It follows from (a) above that the subfactor associated to this Hadamard matrix is not self-dual.
6.3. ON PERMUTATION VERTEX MODELS
6.3
115
On permutation vertex models
This section' is devoted to a brief discussion of vertex models for which the underlying biunitary matrix is a permutation matrix - i.e., has only 0's and 1's as entries. Recall - from the second paragraph of §6.2 - that a unitary
matrix w = ((wQa)) E U(Nk) is biunitary precisely when its 'block-wise transpose' 7b, defined by waa = w'b, is also unitary. In the next few pages, we shall write w for the block-wise transpose of w. Also we continue to use the conventions of this chapter concerning the use of Greek and Latin letters for elements of 1N and 1 k respectively, where we write S2, = {1, 2,. , n}. We shall find it convenient to work with an alternative description of such biunitary permutation matrices, which we single out in the next lemma. LEMMA 6.3.1 Let w E MN(C) ® Mk(C). Then the following conditions on w are equivalent:
(i) w is biunitary, and is further a permutation matrix (i.e., is a 0, 1 matrix);
(ii) there exist permutations {pa a E s2k} C S(S2N), {A : a E QN} C S(1k) (where we write S(X) for the group of permutations of the set X), such that: :
(a) the equation 7r (0, b) = (Pb(/3), Al (b))
defines a permutation 7r E S(S2N X ak); and (b)
wOb
6(a,a),'r(Q,b) -
Proof: (i) = (ii): If w is a biunitary 0, 1-valued matrix, then let 7r E S(S1N X S2k) be defined by wpa = 6(a,a),,(p,b)-
Assertion: For any OE 1 N, a E 1 k (resp., a E S2N, b E 1 k), 7r({/3} x 1k) n (S2N x {a}) (resp., 7r(S2N x {b}) n ({a} X S2k)) is a singleton. Furthermore,
ir({/j} x S2k) n (S2N x {a}) = {(¢a(/3), a)}, 7r(QN x {b}) n ({a} x 1k) = {(a,,O,, (b))}, where Oa(0) = PA 1(a)(0) and 0,,, (b) = AP
(b). b
The first (as well as the parenthetically included) statement of the assertion is an immediate consequence of two facts: (i) the hypothesis on w 'This section is a reproduction, almost verbatim in places, of parts of the paper [KS].
CHAPTER 6. VERTEX AND SPIN MODELS
116
implies that the block-transpose matrix w is also a permutation matrix; and (ii) 7r(Q, b) = (a, a) ? wab = 1. The second assertion follows from the definitions. The assertion clearly proves the implication (i) = (ii), while the implica(i) is immediate. tion (ii) We shall find the following notation convenient. DEFINITION 6.3.2 Define PN,k = {7t E S(S2N X Qk) : there exists A : S2N -+ S(S2k), P : S2k --> S(SlN)
such that 7r(/, b) _ (pb(/3), Ap(b)) for all /3 E S1N, b E S2k}
where we write )p (resp.,pb) for the image of /3 (resp.,b) under the map A (resp., p). If 7r, A, p are related as above, we shall simply write it <--+ (p, A) E PN,k .
Thus Lemma 6.3.1 states that there is a bijection between biunitary permutation matrices of size Nk and elements 7r f-+ (p, A) E PN,k, given by wb = S" Pb()3)6a,ap(b)
The following proposition, which is an immediate consequence of the definitions, lists some useful properties of the various ingredients of a biunitary permutation.
(p, A), Oa, 0, be as above. Then, for arbitrary PROPOSITION 6.3.3 Let 7r a E S1k, a E S2N, we have: (i) ca E S(IN), `b E S(Slk); /,-1) E PN,k (meaning, of course, that 7r-'(a, a) _ (ii) 7t-1 y Gal(a), el1(a))) (a). (a)' 0, '(a) = A(iii) gal(a) = (a) (p, A) E PN,k and let A, p, 0, 0 be For the rest of this section, we fix a 7r as above. Thus, if w is the biunitary permutation matrix that corresponds to (p,.)), then wab
6C',Pb(3)6a,Ap(b)'
(6.3.1)
The point of the next lemma is to point out that if wab = 1, then any pair consisting of one Greek letter from {a, /3} and one Latin letter from {a, b} determines the complementary pair. We shall find some of these formulae convenient in subsequent computations. LEMMA 6.3.4 If a, /3 E S1N, a, b e S2k, then the following conditions are equivalent:
(i) wab=1
(ii) a= Pa(/3) and b = .sp(a);
6.3. ON PERMUTATION VERTEX MODELS
(iii) Q =
4'b 1(a)
117
and a = ia'(b);
(iv) /3 = pa 1(a) and b = z(i (a);
( v ) a= Ob(0) and a = A161(b).
Proof: (i) q (ii) by definition. (ii) (iii) by Proposition 6.3.3(ii). (ii) (iv) by the formula for 0-1 given in Proposition 6.3.3(iii). (iii) (v) by the formula for L_1 given in Proposition 6.3.3(iii). We wish to discuss the higher relative commutants C,, = R,,nR,,_1, n > 0, where R, = R_1 C R = Ro C R1 C C R,,, C is the tower associated to the subfactor R,,, C R constructed from the commuting square given by w in the usual manner. Before we get to that, notice the following consequence of the preceding lemma: the Boltzmann weights associated with the two kinds of crossings (as per the prescription of §6.1) are as follows:
+b
Positive Crossing
a
,3
I
F_,
Q,Pa 6a,Ob(Q)6a,api(b)
a
a
Negative Crossing
a
3
a8,Pb(a) aa,a (b)
=
b
a biunitary permutation w and corresponding maps A, p, as above, then for arbitrary n > 1 and a E SZn, we define the alternating products Pa = PalPa2'Pa3 ... Pa and
4'a = Pai Pa21`f'ag ... (p an
We are now ready to introduce certain mappings that will play a central role in the computation of the higher relative commutants.
PROPOSITION 6.3.5 (i) For all n > 1, there exists a mapping QN 3 a -->
CHAPTER 6. VERTEX AND SPIN MODELS
118
Lan) E S(SZ') such that
b
a
L ' ) (b)Q'Pal(a)
a
where the
(6.3.3)
are defined as in (ii) below.
(ii) L(') = (a-'; and if n > 1 and if L(n)(b) = a, then 01
an-11 = L(n-')(bn-1])
(where we have used the obvious notation an-11 to mean (a1, .. ) an-1) if a =
(a,,...,an)); and A;1,
an =
nI
(a)(bn)
if n is even,
(,)(bn) if n is odd.
Proof: The proof is a direct consequence of the prescription, given in equa-
tion (6.3.2), for the Boltzmann weights associated to positive and negative crossings. (For (iii), the two prescriptions given for each kind of crossing must be used in conjunction.) In the following, we fix a biunitary w, with associated A, p, 0, 7P as above,
and let {C} denote the sequence of higher relative commutants for this R. LEMMA 6.3.6 With the identification ®TMk((C) = Matnk (C), we have
Cn = {F = ((Fa)) E Matnn(C) bFa = bPL(n)(g)'PL(n)(b) FL°) L() b k
L(n)(b)
for all a, /3 E SZN, a, b E Slk}.
Proof: In the notation of Theorem 6.1.4, we see, from Proposition 6.3.5,
6.3. ON PERMUTATION VERTEX MODELS
119
that on the one hand, b
b
a
a
a on the other hand, we also find that b
.I3
6 13,L( )(b)(a)Ga ")(b)'
G
a which, in view of Proposition 6.3.5(iii), is seen to be equal to 8p Thus we find that C,-,, consists of those F E ®"' Mk(C) for which there exists a G E ®n Mk(C) such that Pb s/j,Pal(a)1'(L,"))(a)
L(")(b)
Q,0b1( Ga
Using the substitution c = (L(-))-1(a), the last equation may be rewritten - again using Proposition 6.3.5(iii) - in a more symmetric form as L LC)(b)
sR, l(a)Fb =
(6.3.4)
This is easily seen to imply that
Fb _ s' i Ob,0c CL
(b) (6.3.5)
for arbitrary a E 0N, b, c E On, and also (as a result of Proposition 6.3.5(iii))
that
1 b) CTb=6P_iP-1F(L() p))1( b
(LA
)
(e)
(6.3.6)
CHAPTER 6. VERTEX AND SPIN MODELS
120
for arbitrary i3 E QN, b, c E Q'. The proof of the lemma is completed by putting together equations (6.3.5) and (6.3.6) (and using the fact - which is a consequence of Proposition 6.3.5(iii) - that U
b c-1(/-1
). -1 = 6P-1 -1 'PLL(n(b)PL1n)(c)
The next lemma is the final ingredient necessary for the identification in an abstract sense - of the higher relative commutants. LEMMA 6.3.7 Let S2 be a finite set. Suppose we are given an equivalence {0--1 : a E L}. relation ^'o on Q and a subset C C_ S(Q) such that L = G-1 = for all i, j E S2, a E G}, Let A = {x = ((xi)) E Matn(E) x = d[i]n
where [i]0={j E 2:i-off}. o,(i) -o Define the equivalence relation - on SZ by requiring that i ' j a(j) for all o, E G, where G is the subgroup of S(12) generated by L. Then G acts on the set of --equivalence classes (by a [i] = [o,(i)], where of course [i] = {j E S2 i - j}). Suppose the set of --equivalence classes breaks up as a disjoint union of 1 orbits under this action of G. For 1 < p < 1, fix one equivalence class [ip] from the p-th orbit, let Hp = {a E G : a [ip] = [ip]} be the isotropy group of that equivalence class, and let 7rp denote the natural permutation representation of Hp on [ip]. Then :
i
A
®'rp(Hp)'.
P=1
Proof: To begin with, ifa'1i a2 E L, note that, for any x in A, we have x°1°2(2) x°2(Z) V. = di d [i]0 r[71o S[°2(2)]0 r[°2(7)]0 °1°2(7), Since ,C = G-1, clearly G = °20) [
{010-2
r^
, o'r : r > 0, o-1i
A = {x =
, o-r E L
}, and it easily follows now that
bi, j E Q, oa E G}.
E Matn(C) : xi =
Suppose now that {[jlp)],
[jtP)]} is the p-th orbit in the set of -equiva-
lence classes under the G-action, and suppose j(p) = ip. For 1 < s < tp, fix a(p) E G such that c(p) [ip] = [gy(p)]. Assume that the elements of S2 have been so ordered that or(p), as a map of [jip)] onto [j( p)], is order-preserving. It is then fairly easy to see that x E A if and only if x has the block-diagonal form tp
x = ®® x(p) with respect to the decomposition S2 = JJ JJ [j(p)], where p=ls=l
p=1s=1
x(P) 1
-
x(P) E
- tp
7p (H)'. p
El
Putting the previous two lemmas together - by considering the specialisation of Lemma 6.3.7 to the case where S2 = SZ', a -o b PLC°)(a) _ a, p E QN} we can sumand L = {(L( ))-1(L(n)) ba E SZN, PL(n)(b) :
marise the contents of this section as follows:
6.3. ON PERMUTATION VERTEX MODELS
121
PROPOSITION 6.3.8 Let we MN(C)®Mk(C) be a biunitary permutation matrix and let .A, p, 0, 0 have their usual meaning. Let R,,, C R be the hyperfinite (subfactor, factor)-pair corresponding to w, and let Cn = R' fl Rn_1i n > 0,
where R,,, = R_1 C R = Ro C_ R1 C_ R2 C is the tower of the basic construction. Then, for n = 1, 2, , the algebra Cn has the following description: Let L(n) be defined as in Proposition 6.3.5; let Gn be the subgroup of S(s2k) generated by {L (n) 'L(n) : c,, E S2N}; and let `"n be the equivalence relation defined on S2k by
a `"n b
P((.n)(a(a)) = PL(.">(a(b))
Vu E Gn, a E S2N.
Suppose the set of equivalence classes in S2k breaks up into In orbits under the Gn-action; fix an equivalence class [a,] in the p-th orbit of equivalence classes,
and let Hp = {Q E Gn :
Q(CYp) '"n ctip}. If 7rp
representation of Hp on [ce,], then
is the natural permutation
l
Cn ,.' ®7rp(Hp)'. p=1
We now discuss a few simple special examples.
EXAMPLE 6.3.9 (a) First consider the trivial case A - idok, p - idoN. In this most trivial example, 7r(Q, b) = (0, b) and in this case, the subfactor R,,, -
00
of R = Mk(C) ® ((& MN(C)) may be identified with 1 ® "'2 'Y" MN (C), and the n=1
n=1
principal graph consists of two vertices with k bonds between them. (b) Let A - idnk, and let p : S2k -> S(S2N) be an arbitrary map. Then 7r(Q b) (pb(/3), b), which clearly defines a permutation of S2N X S2k; i.e.,
7r ' (p, t) E PN,k. Then observe that (a) = Pa.l(a)(a) =
and thus n > 1,
= p,
_ A. It follows from Proposition 6.3.5(ii) that, for all
L(n +1) (a a
A -1(0)(a) = a = A, (a)
a) = (L (n) (a), A-', C,
(n)
a)
(a)) = (L (a), a),
for all a E 1N, a E S2k, a E s2k, n > 1; hence, inductively, we find that L(n) = ids. for all n > 1 and for all a E 1 N. In this case, the equivalence classes of 1 k are the sets EQ = {a E S2k : pa = u}, as o- ranges over the group Go generated by {pi : i E S2N}. (Actually, Eo is empty unless a has the form Pal Paa1Pa3
....)
In fact, the hypothesis implies that wpb = 6a b6, pb(p) and hence this case comes under the purview of the vertex models discussed in Example 6.2.2. (c) Let A : S2N -+ S(S2k) be an arbitrary map and let pa = idcN for all a; thus, 7r(Q, b) = (0, Ab(8)), which is again clearly a permutation of S2N X S2k,
CHAPTER 6. VERTEX AND SPIN MODELS
122
whence 7r
<-->
(p, A) E PN,k. Observe again that qa(a) = pa-1(a)(a) = a =
pa(a) and that Eli,,(a) = '\ 1( )(a) = A a(a), so that 0 = p, 0 = A. It follows, again from Proposition 6.3.5(ii), that L(") = Aa l x A- 1 x ... x A- 1.
We assume, for simplicity, that Al = id. Then, if G1 denotes the subgroup of S(S2k) generated by {Aa : a E QN}, we see, in the-notation of Proposition 6.3.8, that Gk = {o- x 9 x X a : 9 E G1}, that the equivalence relation .
on Stn
is the trivial one (a - 3 for all a, 0) - as a result of the triviality
of the p2 's - and if 7r denotes the natural representation of G1 on CN, then
Ck = (7r0 7r®...(9 x)(Gl)'. Again, in this case, we have wpb = 6 ,pba,ap(b), so this case falls under the purview of Example 6.2.1. (d) We may obtain the tensor product of cases (b) and (c) above, by the following device: if P) : 1 N1 -3 S(S2k1) and p(2) : 1 k2 -3 S(S2N2) are arbitrary maps, set N = N1N2i k = k1k2, and define ., pa by
A.(a) = (A(l)(al), a2), Pa(a) = (al, P(2)(a2)) a2 ell (We have made the obvious identification S2N = S2N1 XSLN2, S2k = S2k, XS2k2,
and denoted a typical element of S2N (resp., 0k) by a = (a1i a2) (resp., a = (a,, a2))-) Lest the reader should get the wrong impression that examples obtained from permutation vertex models are all `trivial' in some sense, we should mention that already when N = k = 3, there exists a permutation vertex model whose associated subfactor is irreducible, has infinite depth and has no `intermediate subfactors'. The interested reader may consult [KS] for the details. In fact, the reason for including this section here is that these permutation models are a potential source of interesting subfactors.
6.4 A diagrammatic formulation In this section, we discuss a diagrammatic formulation - along the lines of our discussion of vertex models and spin models earlier in this chapter - that is valid for a general non-degenerate commuting square (with respect to the Markov trace). We assume throughout this section that L
B0 c B1 UH
GU
K
Ao c Al
(6.4.1)
6.4. A DIAGRAMMATIC FORMULATION
123
is a non-degenerate commuting square with respect to the Markov trace, with inclusions as indicated, where we assume that all the inclusions are connected. Further, we shall use the notation 9,H, K and G to denote the Bratteli diagrams encoded by the inclusion matrices G, H, K and L, respectively; also, as before, we shall denote typical edges in the graphs 9,?I, /C and G by a, ,Q, K and .\ respectively.
Let us write Ao = Ao, A° = Bo, Ao = A1, and Ai = B1, so that the commuting square (6.4.1), when `transposed', looks like this: A10
KU
C
Al UL
(6.4.2)
G Ao C Al. A01. o
Let {An : n > Of be the tower of the basic construction for the initial inclusion Ao C AI, with the projection implementing the conditional expectation of Al onto An_1 being given, as usual, by en+1, for n > 1. As usual, let A° = (A°_1, en), n > 1.
If Ro C R1 is the subfactor constructed out of the commuting square (6.4.1) by iterating the basic construction in the usual fashion, then it follows from the analysis of §5.7 that the higher relative commutants are given by
R0' nRn=A"nA0,n>0. Once and for all, let us fix a biunitary matrix u = ((u A)) which describes the commuting square (6.4.2). Let us simply write I for the trace-vector on each A', 0 _< n, k < 1. (Thus i Un k_o ir(An) --> [0, 1], H'H(tJ7r(Ai)) _ :
1IHII2I r(Ai), etc.) We begin by describing how to represent elements of A°, n > 0. Actually,
since we are only interested in relative commutants, we shall only discuss elements of Ao' n A°. We shall think of a typical element of this relative commutant as a `black box' with two sets of n strands, thus:
Such a black box is thought of as a scalar-valued function on the set of possible states, where a state (for such a simple diagram) is an assignment of vertices (from Uk-o 7r (A0)) to the regions, and edges (from 9) to the strands
of the box, in such a way that (a) the assignment is a 'graph-theoretic homomorphism' (meaning that if a strand is labelled by an edge a, and if the two regions adjacent to the strand are labelled by vertices v1i v2 of 9, then a
CHAPTER 6. VERTEX AND SPIN MODELS
124
must be an edge from vl to v2 in 9), and (b) the region at the extreme left is labelled by a vertex from ir(A°0). (We think of the Bratteli diagram as an unoriented graph in this section.) Thus a state - when n = 2 - may be given thus:
a:l
v°0
v°0
c1
V0
a2
where v., vo E ir(Ag), v°, vl E 7r(A°), ctrl (resp., al) is an edge joining vo to v° (resp., v°) and cti2 (resp., a2) is an edge joining v° (resp., V10) to V, The description of elements of Ao' n An' is similar except for the following
variations: here, there are two sets of n + 1 strands, and a state should label the first strands (from the left) by edges in the Bratteli diagram K, and all subsequent strands should be labelled by edges from the Bratteli diagram 7-l; as to the regions, the region on the extreme left should be labelled by a vertex from ir(Ao), and subsequent regions should be labelled by vertices from U1 _07r(A'), and the `homomorphic property' should be preserved; thus, a state, in this case, might look like this (when n = 1):
v0
v° 0
k
vo
where, of course, vo E ir(Ao), vo, vol E ir(Ao), v1 E ir(AI), ,c (resp., k) is an
edge joining vo to 41 (resp., vo) in K, and a (resp., ,Q) is an edge joining vo (resp., va) to v1 in H.
6.4. A DIAGRAMMATIC FORMULATION
125
The inclusion of (Ao' n An) in (Ao' n Ak+1) is given by the obvious identification:
The (sometimes more complex) diagrams we shall be working with will have two other components in addition to (zero or one or many) black boxes, these being: (a) local extrema; and (b) crossings. Before we get to discussing
these, we pause to mention a few features of the diagrams that we shall encounter:
(i) all the strands in the diagram will be oriented; (ii) all the strands connected to either side of a black box will be oriented alternately in opposite directions (as in §6.2); further, the orientation in a strand will be unaffected in passage through a black box; (iii) a'black box' with two sets of n strands will denote an element of Ao'nA° or A"' n A'n_1, depending on whether the first strand from the left is oriented upwards or downwards;
(iv) the curves described by the strands will be smooth; (v) if a strand is the over-strand at some crossing, it will be the over-strand at all crossings it features in; further, as one proceeds along a strand,
the parity of the crossings that one comes across will be alternately positive and negative; and finally,
(vi) at a crossing, neither of the strands is allowed to be horizontal. A state on one of these (possibly complicated) diagrams is an assignment of a vertex from U; k=o ir(Ak) to each region in the diagram, and of an edge from 9 U 7-L U 1C U L to each segment of each strand in the diagram, such that:
(i) the region on the extreme left is labelled by a vertex from ir(A'); (ii) the usual `homomorphic property' is satisfied;
(iii) at a crossing, the regions surrounding the crossing will be indexed as follows, according to whether the crossing is positive or negative -
(*)
CHAPTER 6. VERTEX AND SPIN MODELS
126
- thus, at a positive (resp., negative) crossing, the region enclosed by the two 'out-arrows' (resp., 'in-arrows') is labelled by a vertex from ir(Ag), and as one proceeds from this vertex in the anticlockwise (resp., clockwise) direction, one will encounter, in order, vertices indexed by ir(A°), 7(Ai) and 7(Ao);
(iv) the labelling of the regions and strands incident on a black box is consistent with the requirement (determined by the orientation of the strands going into that box) imposed by demanding that that black box is supposed to represent an element of an appropriate relative commutant (see item (iii) in the earlier description of features of our diagrams).
By a `partial state', we shall mean a state which has been prescribed only on the `boundary' of the diagram - by which we mean the unbounded regions
and unbounded segments of strings. A general diagram is thought of as a function - the `partition function' - on the set of partial states, as follows: if D is a diagram and if -y is a partial state on the diagram, then the the value ZD is defined to be the sum, over all states o which extend ry, of the value of the diagram D on the state ry. In order to evaluate a diagram on a state, we form the product of all the `local contributions' (coming from black boxes, local extrema and crossings); the local contribution coming from a black box is determined as before. We now describe how to determine the `local contribution' coming from (a) a local extremum, and (b) a crossing. Local extrema: These are, obviously, configurations of either of the following types: or Ve
If the `interior' and `exterior' regions of such a local extremum are labelled as indicated above, then the `local contribution' of such an extremum is defined to be
Crossings: To determine the `local contribution' coming from a crossing, there are two points to bear in mind: (a) the assignment is invariant under isotopy; (b) neither strand is horizontal.
6.4. A DIAGRAMMATIC FORMULATION
127
We postulate that at a positive or negative crossing of the following form, the associated Boltzmann weight is as indicated. Ic
Positive Crossing
Negative Crossing
For a crossing which is not in this `canonical form', use isotopy invariance, as illustrated by the following example:
a
Ic
Hence, in accordance with our convention for extrema, the `local contribution'
of the positive crossing given on the left (in 'non-canonical form') is given by a
kop
ao
a
I/tvo
f
\vl
Ctvo
(Note that the above expression is what, in the notation of §5.5, would have been denoted by vaop. We shall use this notation in the immediate sequel, for typographical convenience.)
CHAPTER 6. VERTEX AND SPIN MODELS
128
In an entirely similar fashion, it may be verified that the prescription for assigning Boltzmann weights to the various possible configurations of crossings is as follows:
uaoA
uaoA
a 0
It should be fairly clear, from the nature of our prescription for evaluating diagrams on states, that isotopic diagrams yield identical partition functions.
Alternatively, we could have just defined the Boltzmann weights for all possible crossings (in all possible 'non-canonical forms') by the preceding prescription and then verified that this was an isotopy-invariant prescription. In the rest of this section, we give some indications of the sort of advantages that this formalism has over the corresponding formulations with formulae.
(1) To start with, it is a pleasant exercise to check that the matrix u satisfies the biunitarity condition precisely when diagrams related by Reidemeister moves of type II yield identical partition functions. (2) Next, the inclusion of Ao" n A° into Ao n An is given by the following identification, as can be verified by another pleasant little exercise:
6.4. A DIAGRAMMATIC FORMULATION
129
H
(3) The projection en is represented by the following picture
-
- with the understanding, of course, that if e7,, is viewed as an element of Ao' n A° (resp., Ao' fl An), then there are n (resp., n + 1) strands going through the above `black box'. It follows - from this prescription, and the equation en+lxen+l = (EAk_1x)en Vx E An - that the conditional expectation of Ao' fl An+1 onto Ao' fl An is given thus:
H
CHAPTER 6. VERTEX AND SPIN MODELS
130
(4) It must not be surprising now to find that the relative commutant Al' n A' consists of black boxes of the following form -
- and that the conditional expectation of Ao' n A°+1 (resp., A0 n An) onto A°' n A0 (resp., Ao' n A') n is given thus:
fl H
(5) What is customarily referred to as `flatness of the Jones projections' is an immediate consequence of (1), (2) and (3) above: it is just the asserion
that
(6) Finally, the description of the higher relative commutants is exactly as in the case of vertex models, except that the diagrams are now interpreted
6.4. A DIAGRAMMATIC FORMULATION
131
according to the prescriptions of this section; we state this explicitly as a proposition, whose proof we omit since that is also exactly as for the case of vertex models.
PROPOSITION 6.4.1 If n > 0, then Ro n R, consists of precisely those F E A°°'nA° for which there exists a G E Ao'nA' such that the following equation holds:
F (6.4.3)
G
Appendix
A.l
Concrete and abstract von N eumann algebras
We used the word ‘ concrete' in the opening paragraphs of the first section of this book , to indicate that we were looking at a concrete realisation or representation (as operators on Hilbert space) of a more abstract object. The abstract notion is as follows: suppose M is a C飞 algebra - i. e. , a Banach *algebra , where the involution satisfies I 怡、 11 = IIxll 2 for all x in Mj suppose further that M is a dual space as a Banach space 一i. e. ， there exists a Banach space M* (called the pre-dual of M) such that M is isometrically isomorphic , as a Ban础 spa吼 to the dual Banach space (M川 let us temporarily call such an M an 'abstract von Neumann algebra'. It turns out - cf. [Takl ], Corollary III. 3.9 - that the pre-dual of an abstract von Neumann algebra is uniquely determined up to isometric isomorphismj hence it makes sense to define the σ -weak topology on M asσ (M， M*) ， the weak* topology on M defined by M*. The natural morphisms in the category of von Neumann algebras are *homomorphisms which are continuous relative to the 俨weak topology (on range as well as domain) j such m呻s are called normal homomorphisms. The canonical commutative examples of abstract von Neumann algebras turn out to be L∞ (X ， μ) ， while the basic non-commutative example isι(衍) . (The pre-dualι(冗L is the space of trace-class operators on 衍， endowed with the trace-norm - the duality being given byι(7í.) xι(7í.)* :;) (x , p) 1-+ tr(ρx).) It is not hard to see that if M is an abstract von Neumann algebra , so is N , where N is a町r O"-weakly closed self-adjoint subalgebra. In particular , a O"-weakly closed self-adjoint s由algebra ofι(7í.) is an abstract von Neumann algebra. It follows from the double同commutant theorem that any ‘concrete' von Neumann algebra is an ‘ abstract' von Neumann algebra. The (artificial) distinction between the 丑otions of abstract and concrete von Neumann algebras may (and shall henceforth) be dispensed with , in view of the following theorem (see [Takl ], Theorem III.3.5): anv abstract von Nι
APPENDIX A.
134
mane algebra admits a normal *-isomorphism onto a concrete von Neumann algebra.
A.2
Separable pre-duals, Tomita-Takesaki theorem
As was remarked in §1.2, any separable Hilbert space which carries a normal representation of a von Neumann algebra is expressible as a countable direct sum of `GNS' representations. This fact leads to the following fact. PROPOSITION A.2.1 The following conditions on a von Neumann algebra M are equivalent:
(i) M admits a faithful normal representation on separable Hilbert space; (ii) M* is separable. (ii): If it : M -> L(7i) is a faithful normal *-representation, Proof. (i) it is a fact - cf. [Takl], Proposition 111.3.12 - that 'ir(M) is a von Neumann subalgebra of L (R) and that it is a a-weak homeomorphism of M onto 7r(M). Hence M* = 7r(M)*, but ir(M)* = G(7-l)./7r(M)1i where ir(M)1 = {p E 'C (R). : (7r (x), p) = 0 Vx E M}, whence ir(M)* inherits separability from
(ii) = (i): In general, if X is a separable Banach space, then ball X* (the unit ball of the Banach dual space X*) is a compact metric space (w.r.t. the 00
2-njcp(x,,,)-4/I(xn)I where {x,}°°_1 is a dense
distance defined by d(o,V) _ n=1
sequence in ball X) and hence separable in the weak*-topology. It follows that X*(= U n(ballX*)) is also weak*-separable.
In particular, if M* is separable, then there exists a sequence {x,}11 in M which is or-weakly dense in M. So, if it is a representation on 7-l with cyclic 1 is a countable dense vector , then 7-l must be separable since set in R. In particular, if M. is separable, then R. is separable for every cp in M* (where (7-l,, irw, ,) is the GNS triple for so). If {0n}n 1 is a dense sequence in M, clearly {Vn}°°_1 separates points in M. Now each On is expressible - cf. [Sakl], Theorem 1.14.3 - as a linear combination of four normal states on M. Hence there exists a sequence {cpn}n 1 of normal states on M which separates points in M. Let (7-ln, ten, l,n) be the 9rn. If Sn denotes GNS triple associated with cpn, and let 7-l = ® 7-ln, it n
the vector in 7-l with n in the n-th co-ordinate and zero in other co-ordinates,
it is clear that (i) 7-l is separable (by the previous paragraph); and
A.3. SIMPLICITY OF FACTORS
135
(11) (ir(x)., en) = V.(x) Vx E M, n = 1, 2, ... .
In particular, ir(x) = 0 implies cp,,(x) = 0 Vn, whence x = 0; i.e., 7r is a 0 faithful normal representation of M on the separable Hilbert space 1. REMARK A.2.2 If M satisfies the equivalent conditions of Proposition A.2.1,
then M admits a faithful normal state W. (Reason: if ? l is separable, then G(?i) admits a faithful normal state, for instance p = E(1/2')(., t;,,)t;,,, where {tn} is an orthonormal basis for fl.)
Assume for the rest of this section, that M is a von Neumann algebra with separable pre-dual. Thus, by the preceding remark, we may always find
a faithful normal state , say cp on M. Let l = LZ(M, cp). It is true, as in the case of finite M, that ?t admits a cyclic and separating vector Il. What is different is that vectors of the form xfZ, x E M, are, in general, no longer right-bounded. This and other such problems can eventually be overcome, due to the celebrated Tomita-Takesaki theorem. In the following formulation of this theorem, we identify M with its image under the GNS representation ir,. THEOREM A.2.3 Let M, cp be as above. Then the mapping x) i-* x1l, defined on the dense subspace MSt, is a conjugate-linear closable operator So. Let S = JA 2 denote the polar decomposition of the closure S of the operator S0. Then
(i)JMJ = M'; and (ii)Ado:,(M) = M, Vt E R.
0
One consequence of (i) of this theorem, and Proposition 2.1.2, is that isomorphism classes of separable modules over an arbitrary von Neumann algebra M with separable pre-dual are in bijective correspondence with Murrayvon Neumann equivalence classes of projections in M 0G(['').
A.3
Simplicity of factors
The purpose of this section is to prove the following result. PROPOSITION A.3.1 A factor contains no proper weakly closed ideal. A finite factor contains no two-sided ideal.
Proof: Suppose I is a two-sided ideal in a factor M. Polar decomposition shows that x E 14* JxJ E I. This observation, together with the functional calculus, shows that if I 0 0, then there exists a non-zero projection p E I. (Reason: Write 1r for the indicator function of the set (r, oo); if 0 ,-b x E I, pick r sufficiently small to ensure that p = 1r(Ixl) 36 0. Now define the (bounded measurable) function g by g(t) =
I i ift>r, 0
otherwise,
and note that tg(t) = 1,,(t) Vt E R, whence IxIg(JxJ) = p.)
APPENDIX A.
136
Since we can find partial isometries {ui
:
i E I} in M such that 1 =
>iEI uipui - with I finite if M is finite - the proof of the proposition is complete.
COROLLARY A.3.2 A normal homomorphism of a factor is either identically zero or injective.
A.4
Subgroups and subfactors
We first observe that for an automorphism of a factor, the condition of being free - see Definition 1.4.2 - is equivalent to not being an inner automorphism. (It is obvious that an inner automorphism is not free. Conversely, suppose 0
is an automorphism of a factor P and that there exists an element x E P such that xy = 0(y)x, Vy E P; this is seen to imply that both x*x and xx* belong to the centre of P, and an appeal to polar decomposition suffices to show that either x = 0, or 0 is inner.) In this section, we assume that a : G --> Aut(P) is an outer action of a finite group on a III factor P - i.e., we assume that a is an action such
that if GE) t 0 1, then at is not an inner automorphism of P. Since P is a factor, it follows from the last paragraph and Proposition 1.4.4(i) that
P' n (P x,,, G) = C, and in particular, the crossed product P1 = P xa G is also a II, factor. It turns out that P1 admits a natural action on L2 (P) thus: since the trace on P is unique, it follows easily that there is a unitary representation t H ut of G on L2(P), such that utx1l = at(x)Q; an easy computation shows that utxut = at(x) Vx E P, t c G. This implies that there is a natural homomorphism of P x, G onto (P U {ut : t E G})", where, of course, we regard elements of P as left-multiplication operators on L2(P). Deduce now from Corollary A.3.2 that this homomorphism must be an isomorphism. Hence we assume, in the sequel, that Pi = P U fit, t E G}". In this section, we shall prove the two succeeding propositions and compute the the principal and dual graphs for the inclusion K C_ P1i where K is as in Proposition A.4.2 below. (We continue to use the preceding notation in the next two propositions.) :
PROPOSITION A.4.1 If Po = PG is the fixed-point algebra for the G-action,
then P0 is an irreducible subfactor of P such that the result (P, ep,,) of the basic construction for the inclusion PO C P coincides with P1. (Thus, forming the crossed product is `dual' to taking the fixed-point algebra.)
PROPOSITION A.4.2 The passage H H K = P x,,, H establishes a bijective correspondence between subgroups H of G, and *-algebras K satisfying P C
A.4. SUBGROUPS AND SUBFACTORS
137
Proof of Proposition A./t.1: If J denotes the modular conjugation operator on L2 (P), note that J commutes with each ut - since automorphisms preserve adjoints - and so
JP'J = J(P, {ut}tEG)'J = P n {ut}tEG = PO Thus Po is indeed a 11, factor, and P1 is the result of the basic construction for the inclusion P0 C P. Further,
J(PonP)J=P1nPI =c, and so Po is, indeed, an irreducible subfactor of P, and the proof is complete. 11
Before proceeding further, notice that {ut : t E G} is a basis for P1/P in the sense of §4.3.
Proof of Proposition A.4.2: A moment's thought shows that, since G is finite, it suffices to prove the following assertion:
Assertion: If P C K C P1 is an intermediate *-subalgebra, if 0 x = >tEG atut E K, at E P, and if S = {t E G : at(= Ep(xut)) 0} (denotes the support of x), then there exists t E S such that ut E K. First notice that the assumption P C K implies that EP(Kus) is a twosided ideal in P, and must hence be one of the trivial ideals. Hence if x, S are
as in the assertion, then EP(PxPus) = P for all s E S. We prove the assertion by induction on the cardinality of S. If S = {s} is a singleton, the last observation shows that there exists ai, bi E P such that Ep(Ei aixbius) = 1, which implies, together with the assumption S = {s}, that us = Ei aixbi E K. Suppose then that ISI > 1; fix s E S. Argue as above to find z E PxP such that Ep(zus) = 1. Note that the support S, of z (as described in the statement of the assertion) is contained in that of x, i.e., Sz C S. Further, s E S,z, by construction. If Sz = {s}, we are done by the last paragraph. If not, suppose z = EkES akUk, with at ; 0, t s. Since ats-1 is free, we
can find a unitary element v E P such that vat : atats-i(v); this means that the element y = z - vzas-1 (v*) is a non-zero element of K such that Sy C (S \ {s}), and an appeal to the induction hypothesis completes the proof.
Assume, in the rest of this section, that N C M is an inclusion of Ill factors with [M : N] = d < oo. Once and for all, fix an integer n > d, set I = { 1, 2, . , n}, fix a basis {.\i i E I} for M/N, in the sense of :
§4.3, and consider the (co-finite morphism of M into N given by the) map 0 : M -+ MI(N) defined by O3(x) = EN(AixA ). Let us write 7-l for L2(M), when viewed as an N-M-bimodule. In the language of §4.1, and with the preceding notation, we then have 7I = Ho(=
APPENDIX A.
138
M1x.(L2(N))B(1)). (In the following, if 0 : Q -j M1(P) is a co-finite morphism, we shall write 7-i to denote the P-Q-bimodule Ml,,l(L2(P))0(1).
Recall the fact - which we shall use below - that in this case, we have pLQ(7-1O) = Ml(P)m(1) n q(Q)'.) Also, we assume that Mk, k > 1, are the members of the tower of the basic construction as usual. LEMMA A.4.3 With the foregoing notation, for each positive integer k, define B(k) : M ---> MIk (N) thus: if i = (i1, ... , ik), J = (j1, ... , jk) E Ik, then ..))
Bald (Oi2J2 (... Bik7k (x)
BIj
= EN(Aj1EN(Ai2EN(. .
...)a;2)A,1)'
. EN(Aikxa,k)
(i) Then B(k) is a co-finite morphism of M into N for each k > 1, and in (A.4.1) fact, NL'(Mk_1)M = HOW = Mlxnk(L2(N))B(k)(1);
consequently, we have an isomorphism of inclusions: 0(k+1)(N)' n MIk+l(N)O(k+l)(1)
N' n M2k+1 U
U
N' n M2k
0(k+1) (M), n MIk+1(N)O(k+1)(1)
(A.4.2)
(ii) Let O(k) = B(k), viewed as a map from M into MIk (M). Then ¢(k) is a co-finite morphism of M into M for each k > 1, and in fact, ML2(Mk)M - Ho(k) = MlXnk(L2(M))B(k)(1);
(A.4.3)
consequently, we have an isomorphism of inclusions: B(k)(N)' n MIk(M)e(k)(1)
M' n M2k+1 U
M' n M2k
=
U
(A.4.4)
B(k)(M)' n MIk (M)O(k)(1)
Proof: First note that equation (A.4.1) implies the co-finiteness of B(k) as well as (A.4.2) (in view of Proposition 4.4.1(i) and the parenthetical remark preceding the statement of this lemma); so we only need to prove (i), which we do by induction on k. The case k = 1 is valid, by definition; the implication (A.4.1) k = (A.4.1)k+1 is a consequence of Proposition 4.4.1(ii) and the fact (already mentioned at the end of §4.1) that Ho ® 7-I = Hoop. The proof of (ii) is similar.
We shall now apply the preceding lemma to compute the principal and
dual graphs for the subgroup-subfactor N = (P x H) C (P x G) = M. So assume, for the rest of this section, that P, G, ut are as in the first part of this section; assume further that H is a subgroup of G. To be specific, suppose [G : H] = n, and suppose G = jl 1 Hgi is the partition of G into the distinct
A.4. SUBGROUPS AND SUBFACTORS
139
right-cosets of H. It is clear then that {v,9i 1 < i < n} is a basis for M/N, and yields the co-finite morphism 0: M --- M,(N) given by :
91.9(x) = EN(ug;xug- 1), dx E M.
Notice, in particular, that (A.4.5)
9i9(r) = bijcg,(r), Vr E P, and ugigg
Bi7(ug)
1
l0
if gig E H97, } dg E G. otherwise,
(A.4.6)
In order to discuss 9(k), k > 2, we shall find it convenient to use the following , ik) E Ik by i; if notation: let I = {1, 2, , n}; denote the element (i1) i2i i E Ik, define ngi = gilgi2 .. ' gik. Also, we shall write g F--> Og for the action of G on the set Ik defined thus: R9 (j) = i
ga,gj,+1 ... gjkg 1 E Hgilgit}1 ... gik for
1<1
we shall also, later, write /Qk for the associated permutation representation of G on Cnk.
With the preceding notation, the definitions imply that, for i, j E Ik, k > 1, we have (A.4.7) bjjcing;(r), Vr E P, and 1(ng,)g(ngs)-1 S
l0
if i = 0g (j), l Vg E G. otherwise, J
(A.4.8)
Now fix X = ((xii)) E MIk(M); the fact that P' n M = C is seen to imply that X E 0(k) (P)' if and only if there exist scalars Cij E C such that xii = Ciiu(ng,)(ng9)-1 di, j E Ik.
(A.4.9)
Another easy computation shows that X E 9(k) (M)' if and only if X is given by equation (A.4.9), where the scalars Cij satisfy the relations Cij = Cp,(i),p,(j) Vg E G, i, j E Ik. 9 9
(A.4.10)
It follows immediately, from Lemma A.4.3, that we have an isomorphism of inclusions: M' n M2k+1 /3klx(H)' U
M' n M2k
=
U
(A.4.11)
Qk (G)'
Thus, in the Bratteli diagram for the inclusion (M' n M2k) C (M' n M2k+1), the central summands of M' n M2k (resp., M' n M2k+i) are indexed by those irreducible representations of G (resp., H) which feature in the representation
APPENDIX A.
140
13k (resp., NlI H), and it is clear that the number of bonds joining a vertex
indexed by a suitable 7 E G to a vertex indexed by a suitable p E H is precisely the multiplicity with which p occurs in 7TIII. g E H, we find that Now, suppose X = ((xij)) E MIk(N); since u9 E N X E 9(k)(P)' if and only if there exist scalars Cil E C such that xij
if (ngi)(n9i)
1 Ciiu(n9;)(n9s)-1 0
1E
otherwise,
=l
H,
(A.4.12) }
Thus, the relative commutant 9(k)(P)I n MIk(N) gets identified with the set Ck = {((Cij)) E MIk((C) : Ci; = 0 if (ngi)(ng;)-1 V H}. Consider the mapping
®P
MIk-1(C) 1
D
(D- ((Ci,1),)) H ((Co)) E Ck i
defined by
if ngi,ng,EHgp,1Gp
G,,_ j C,PJ_ l0
if n gi, ng; belong to distinct cosets,
ik) for i E Ik. A simple verifiwhere we have used the notation i_ = (i2, cation shows that this mapping is an isomorphism of *-algebras. MIk-1(C). If the element Thus, we find that (O(k)(P)' n MIk(N)) X E (B(k)(P)' n MIk(N)) corresponds to the element ®P_1((C;,?,)) under this ismorphism, another simple computation shows that X commutes with ,
®P=1
B(k)(u9) if and only if C03" (A) R9
k
(1)"Q9
1
(J)
-
by E I, i, j E Ik-1.
(A.4.13)
Let us temporarily write K = P x Go, where Go is a subgroup of G, which we will later choose to be G or H. The preceding analysis shows that X E (8(k)(K)' fl MIk(N)) if and only if the corresponding C(il)'s satisfy equation (A.4.13) for all g E Go. It follows readily from this description that, if the g(l) is a set set I breaks up into 1 orbits under the action /31IG0, if g(1), containing one element from each of these distinct orbits, and if Go') is the isotropy subgroup of Go corresponding to the point g(i), then ,
(9(k)(K)' n MIk(N))
c-®1'=1
Qk-1(G(z))'.
(A.4.14)
When Go = G, since the action j31 of G is clearly transitive, we have 1 = 1, so we may choose g(1) = 1, in which case we find that Got) = H.
When Go = H, the orbits under the action all H of H correspond to the double cosets of H in G, and so if l is the number of double cosets, and if
A.5. FROM SUBFACTORS TO KNOTS
141
G = 11i=1 Hg(')H is the partition of G into double cosets of H, we find that Got) = H n gli>-1Hgli> = Hi (say). It follows, from Lemma A.4.3, equation (A.4.14) and the last two para-
graphs, that in the Bratteli diagram for the inclusion (N' n M2k) C (N' n M2k+1), the central summands of (N'nM2k) (resp., (N'nM2k+1)) are indexed by those irreducible representations of H (resp., Hi for any i) which occur in ,Qk-1IH (resp.,
and that the vertex labelled by a suitable 7 E H is connected to the vertex labelled by a suitable p c Hzj 1 < i < 1, by m bonds, if m is the multiplicity with which p occurs in irlH,. A moment's thought about the nature of the principal and dual graphs should convince the reader that we have proved the following. PROPOSITION A.4.4 Let H be a subgroup of finite index in a discrete group
G. Suppose G acts as outer automorphisms on the II1 factor P. Then the principal graph 9 and the dual graph 7il for the inclusion N = P x H C P x G = M have the following descriptions. Let G = 11'=1 Hg(')H be the partition of G into double cosets of H, and let Hi = Hngli>-'Hg(i). First define a bipartite graph 0 as follows: let 010> = (1IZ_1(HZ x {i})) x {0}, g(') = H x {1}; join the even vertex ((p, i), 0) to the odd vertex (7r, 1) by m bonds, if m is the multiplicity with which p occurs in 7rIH;. Then 9 is the connected component in 0 which contains the odd vertex (1, 1) which is indexed by the trivial representation of H. Define the bipartite graph 7-l as follows: let I-l(°) = G x {0},7-1(1) = k x {1 }; connect the even vertex (71, 0) to the odd vertex (p, 1) by as many bonds as the multiplicity with which p occurs in 11tH. Then 1-1 is the connected component in 7-l which contains the even vertex (1, 0) indexed by the trivial representation of G.
A.5
From subfactors to knots
In this section, we briefly sketch the manner in which the initial contact between von Neumann algebras and knot theory was made; to be precise, we outline the construction and some basic properties of what has come to be known as the one-variable Jones polynomial invariant of links. The starting point is Artin's n-strand braid group, which we briefly describe. Fix a positive integer n - which should be at least 2 for anything interesting to happen. Consider two horizontal rods with n hooks on each of them, and suppose n strands, say of rope, are tied with one end to each of the rods, in such a way that no hook has more than one strand tied to it. In order to avoid pathologies, we assume that the two rods are placed horizontally with one vertically above the other, and that the passage from the top rod to the bottom is not allowed to `double back', meaning that at any intermediate height, there is exactly one point of each of the n strands.
APPENDIX A.
142
An n-strand braid is an equivalence class of such arrangements, where two such arrangements are considered to be equivalent if it is possible to continuously deform the one to the other. We shall think of the strands of a braid as being oriented from the top to the bottom. Consistent with this convention, we define the product of two n-strand braids by concatenation, as follows:
c/3
It is fairly painless to verify that this definition endows the set B,, of nstrand braids with the structure of a group. (The inverse of a braid is given by the braid obtained by reflecting the given braid in a mirror placed on the horizontal plane through the bottom rod.) We shall only consider tame braids, by which we mean one which admits only a finite number of crossings. It then follows from our definition that B. is generated, as a group, by the set {61, , o-n_1}, where cr is a braid with only one crossing, which is between the
(i - 1)-th and i-th strands and which is positive according to the convention adopted in §6.1. Thus, for instance, when n = 2, the generator and its inverse are given as follows:
Q
0
1
A result due to Artin establishes the precise relations between these generators. (The reader will find it instructive to draw some pictures to convince herself that these relations are indeed satisfied.)
THEOREM A.5.1 The braid group has the following presentation (in terms of generators and relations): Bn
/
o-zc, = c cx if ji - il > 1, (b2)
o-joj+1o2 = ui+iaio-i+1
11
, g._i are arbitrary elements In other words, the theorem says that if gi, in a group G, then a necessary and sufficient condition for the existence of a
A.S. FROM SUBFACTORS TO KNOTS
143
homomorphism 0 : B, -- G such that q (Qi) = gi Vi is that the gi's satisfy the so-called braid relations (bl) and (b2), and in this case such a 0 is unique. Two consequences are worth singling out.
REMARK A.5.2 (1) There exists a unique epimorphism 7r B,,, -* S,,, such that lr(o-i) is the transposition (i, i+1). (It should be clear that in the language of rods and strands, the braid a is such that the strand which ends at the i-th hook on the bottom rod starts at the (7r(cx))(i)-th strand of the top rod.) (2) There exists a unique homomorphism 0,,, Bn -+ B,,,+1 such that 0,(,(n)) = O-(n+1) 1 < i < n - where we have used the notation to denote the i-th generator of Bk. We shall, in the sequel, write :
:
a(n+1) =
Vn(c (n)),
b'a(n) E B.
(A.5.1)
Alternatively, we can see that
(3) If we set a- = 071072
a-,,,_1 E Bn, it follows immediately from the braid
relations that co-i = o-i+1Q for 1 < i < n - 1; in other words, the generators o-i are pairwise conjugate in B.
In view of the striking similarity between the braid relations and the relations satisfied by the en's, it is natural to try to use the latter to obtain a representation of the former. The simplest way to obtain an invertible element from a projection is to form a (generic) linear combination of the projection and the identity. In view of Remark A.5.2(3), we wish therefore to set
gi = C{(q + 1)ei - 1},1 < i < n,
(A.5.2)
where C and q are non-zero scalars. Recall that the ei's come with a parameter T; an easy computation shows that the gi's as defined above satisfy the braid relations precisely when the parameters T and q are related by the equation T-1
= q + q-1 + 2.
(A.5.3)
Notice that T-1 = 4 cosh2 z G* q = exp(±2z).
Hence, as the parameter q varies over the set {exp(2, n 1) : n = 3, 4, } U (0, oo), the parameter T ranges over all possible index-values of subfactors. Further, the values of q for which the gi's (with the normalisation C = 1) afford a unitary representation of the braid group are precisely the roots of unity. For convenience of reference, we paraphrase the foregoing remarks in the following proposition.
144
APPENDIX A.
PROPOSITION A.5.3 Let q E {exp(2"-1)
n = 3, 4,
} U (0, oo), let T
be defined by equation (A.5.3), and let {en,}°n°__1 be the sequence of projections
associated to this T. Then, for any C 0 0, there exits a unique homomorphism irn of Bn into the group of units of R such that
C{(q + 1)ei - 1} for 1
It should be clear that the closure of a braid is an oriented link. (Recall that a link is a homeomorphic image of a disjoint union of circles, and that a link is said to be oriented if an orientation has been specified in each component. Alternatively, an oriented link is a compact 1-manifold without boundary, with a distinguished orientation. A knot is a link with exactly one component - i.e., a knot is just a homeomorphic image of the circle.) A moment's thought should convince the reader that the number of components of & is precisely the number of disjoint cycles in the cycle-decomposition of the permutation 7r(a) - see Remark A.5.2(1).
Two links are considered to be equivalent, or the same, if the one can be continuously deformed to the other. Precisely, this means that there is a continuous map F : IR3 x [0, 1] --k IR3 such that if we write F(x, t) = ft(x), then fo = id'R3, each ft is a homeomorphism of IR3 onto itself, and fl maps the first link onto the second. (We only consider links in IR3 here.) Two oriented links are said to be equivalent if they are equivalent as above in such a way
that fl preserves the orientations. We shall use the symbol G to denote the class of oriented links (in IR3).
Recall that a link is said to be tame if it is equivalent to one which is a smoothly embedded submanifold of IR3.
THEOREM A.5.4 (Alexander) Every tame link (in IR3) is equivalent to the closure of some braid (on a possibly large number of strands).
The final ingredient for us to make the connection between subfactors and links is a result due to Markov which explains precisely how two different
braids (on possibly different numbers of strands) can have equivalent link closures. In order to describe this result, some terminology would help.
A.5. FROM SUBFACTORS TO KNOTS
145
DEFINITION A.5.5 Let a(n) E B,, 0(m) E B,,,,. The braids a(n) and 0(') are said to be related by a Markov move of (a) type I, if n = m and if a(') and 0(') belong to the same conjugacy class in B,,,; and (b) type II, if either (i)
m = n + 1 and
o(n+1) = a(n+1)(07nn+1))±1
or (ii) m = n - 1 and j3(n) _
a(n) (or(n)1)t1.
A couple of diagrams should convince the reader of the sufficiency of the following condition.
THEOREM A.5.6 (Markov) In order that two braids a(n) E Bn and 0('n) E B,,,, have equivalent link-closures, it is necessary and sufficient that there exist braids a(n) = c , ai, , ak = 13('n) such that ai and ai+l are related by a Markov move (of either type), for 0 < i < k.
An invariant of oriented links (which takes values in some set, say S) is an assignment L E) L - PL E S with the property that L -_ L' PL = PL,. An immediate consequence of Theorems A.5.4 and A.5.6 is that in order to define an invariant L i--> PL of tame oriented links taking values in S, it is necessary and sufficient to find functions Pn : Bn -* S, n > 2 such that Pn is a class function on Bn, Vn,
(A.5.4)
and Pn+1(a(n+1)(07(n+1))±1)
= Pn(a(n)), ba(n) E Bn Vn;
(A.5.5)
when this happens, we have
p(-- = Pn(a(n)),
Va(n) E Bn Vn > 2.
Since class functions are usually obtained by taking the trace of a representation, it is natural to seek a link invariant by considering the functions Qn(a) = trR(7rn(a)),
where irn is as in Proposition A.5.3. (Thus the condition (A.5.4) is automatically satisfied because of the trace.) + As for the condition (A.5.5), note, to start with, that gi 1 = 1)ei - 1}, and hence gn 1 satisfies the Markov property with respect to the C-1{(q-1
algebra generated by 9r,,,(Bn). Hence, if there is any hope of the Qn's satisfying (A.5.5), it must at least be the case that tr gn = tr gn 1; a little algebra shows that this happens precisely when we make the choice C = q 2 , in which case,
we find that
trg±1=-(q2+q 2)
and consequently, that
{-(q2 +q a)-1}Qn(a(n)) for any a(n) E Bn.
l
APPENDIX A.
146
A moment's thought shows that if we define P,, (c,) = {-(q2 +q 2)}"-'Q,, (oz),a E B,,,,
then the P,,,'s do satisfy both the conditions (A.5.4) and (A.5.5). We have thus proved the following: THEOREM A.5.7 Let q, Y, 7r,,, be as in Proposition A.5.3, where we assume
that C = q2; then there exists a complex-valued invariant of oriented links, which we shall denote by L D L H VL(q), such that, if cti E B,,, then
{-(q2 + q2')1n-1tr7rn(a). Most properties of this invariant are consequences of the fact that it satisfies the so-called skein relations. To see what these are, it will be convenient to use the point of view of link diagrams. The fact is that the image of a tame link in 1R3 under the projection onto a generic plane in 1R3 will have only
double points - meaning that the inverse image of a point in the plane will meet the given link in at most two points. In order to fully recapture the link from the projection, it is necessary to indicate, at each crossing, which of the two strands goes `over' the other. For instance, a diagram representing the so-called right-handed trefoil knot (with an orientation indicated) is
We shall henceforth identify the class L with the class of all oriented (tame) link diagrams. We shall say that three link diagrams L+, L_, Lo are skein-related if they are identical except at one crossing, where they have the following form:
L+
L_
Lo
Before stating the next result, we recall that the unlink UU with c components is nothing but 1,, where 1, denotes the identity element of B,.
147
A.5. FROM SUBFACTORS TO KNOTS
PROPOSITION A.5.8 The invariant defined in Theorem A.5.7 satisfies the following relations: (i) Vujq) = {-(q2 +q-12 )}" (ii) if L+, L_ and Lo are skein-related as above, then q-1VL+(q) - gVL_(q) = (q2 - q 2)VLo(q)
Proof: Assertion (i) is an immediate consequence of the definitions. (ii) Begin by observing that gi=gei-q2(1-ei),
and hence gi satisfies the quadratic relation
(gi -
g32)(gi
+
q12)
= 0,
which may be re-written as 4
1gi-ggi1=(q2-q
2 )1.
The desired conclusion is a consequence of the definition of VL, properties of the trace, and the fairly obvious observation that the assumed skein-relation
between L+, L_ and Lo amounts to the existence of a positive integer n, elements cti and a of B,,, and an integer 1 < i < n such that - thinking of these link diagrams as links - we have
The following elementary lemma will be very useful in deducing properties of the invariant VL from Proposition A.5.8.
LEMMA A.5.9 Let L be an oriented link diagram; there exists a subset of the set of crossings in L such that, if we change all these crossings (from an over- to an under-crossing and vice versa), the resulting diagram represents an unknot with the same number of components as L. Proof: Arbitrarily label the distinct components of the diagram 1, 21- , c, and fix a reference point on each component, which is not a double point. Given any crossing, change it if (and only if) one of the following things happens: either (a) the crossing involves two different components, and the component with the larger label crosses over the one with the smaller label; or (b) the crossing involves only one component, and in travelling along that component from the chosen reference point in the direction specified by the orientation, the first time you come to the crossing, you find yourself going along the over-strand of the crossing.
148
APPENDIX A.
A moment's thought should convince the reader that this algorithm yields a proof of the lemma.
For instance, if L is the link diagram given earlier to depict the righthanded trefoil, our algorithm would yield the following diagram (with the reference point as indicated):
Given a link diagram L, define its `knottiness' to be the ordered pair (n, k),
where n denotes the number of crossings in the diagram and k denotes the minimum of the cardinalities of subsets of the set of crossings which satisfy Lemma A.5.9; we shall say that a link diagram L2 is more knotty' than a diagram Li if the knottiness (ni, ki) of Li precedes the knottiness (n2, k2) of L2 in the lexicographic ordering - i.e., if either ni < n2, or ni = n2, ki < k2. With respect to this `ordering', the least knotty diagrams represent unlinks, and the preceding lemma has the following pleasing consequence: given any link diagram which does not represent an unlink, there is a triple (L+, L_, Lo) of skein-related diagrams such that two things hold: (a) L is either L+ or L-, and (b) L is more knotty than the other two diagrams in {L+, L_, Lo}.
Since the set Z+ x Z+ is well-ordered with respect to the lexicographic order, the preceding considerations allow us to prove facts about VL using a `knotty induction'. By such a process, it is easy (and amusing) to prove the following facts. (The strategy of proof is: first prove it for unlinks; then assume the result for Lo and L+ (resp., L-) and use Proposition A.5.8(ii) to deduce the result for L_ (resp., L+).) PROPOSITION A.5.10 (1) VL(q) is a Laurent polynomial in q2; more precisely, if L has an odd number of components, then VL(q) is a Laurent polynomial in q, while if L has an even number of components, then VL(q) is q2 times a Laurent polynomial in q. (2) If L denotes the mirror-reflection of L, then
Vj(q) =
VL(q_i).
(3) Properties (i) and (ii) of Proposition A.5.8 determine the invariant VL uniquely (via the process of `knotty induction' discussed earlier).
A.5. FROM SUBFACTORS TO KNOTS
149
We close by illustrating (3) above with the already mentioned example of the right-handed trefoil.
L+ = T+ = right-handed trefoil
L_ = Ul = unknot
Lo = H+ = Hopf link
It follows that VT+(4) = 4{4Vu1(4) +
(VI-q-
-
I
)VH+(4)}.
(A.5.6)
Next, in order to determine VH+, notice the following skein-related triple
APPENDIX A.
150
of links:
L+=H+
L_=U2
Lo = Ui
This implies that VH+(q) = q{gVv2(q) + (VI-q- -
I )Vu1(q)}.
(A.5.7)
Putting equations (A.5.6) and (A.5.7) together, we find that VT+(q) = q + q3 - q4,
which implies that if T_ = T+ - in the notation of Proposition A.5.10 - so that T_ denotes the so-called left-handed trefoil - then VT_ (q) = q-1 + q-3 - q-4.
The invariant VL is quite good at detecting knots from their mirror-images in this fashion.
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BIBLIOGRAPHY
Bibliographical Remarks
§1.1: Almost all the material in this section was first established in the seminal paper [MvNl]. The only exceptions to this blanket statement is the existence of a tracial state on a finite factor (Proposition 1.1.2) - which was established in [MvN2] - and the fact (briefly alluded to) about disintegration of a general von Neumann algebra (with separable pre-dual) into factors which was established in [vN3].
§1.2: The fundamental GNS construction first made its appearance in [GN], and appears later in polished form in [Seg], while the basic facts contained in this section concerning the so-called standard module of a finite factor were all known to the founding fathers - see [MvN2] (although they did not quite use the same terminology as here). §1.3: The notion of a discrete crossed product - at least when the algebra
that is being acted upon by the group is commutative - first appears in [MvNl]; already in this paper, they use this construction to give examples of II, factors, and they identify ergodicity as the crucial property of the group action to ensure factoriality of the crossed product.
§1.4: The definition given in Definition 1.4.2 is from [Kal]. The typeclassification given in Theorem 1.4.5 is again from [MvNI]. The Powers factors made their appearance in [Pow], while the description of the model for the hy-
perfinite III factor coming from the crossed product L' (R2, B, p) x SL(2, z) (in Example 1.4.8) is due to [Aub]. Infinite tensor products were first treated in [vN2], and while the uniqueness statement concerning approximately finitedimensional II, factors was first proved in [MvN3], the ultimate classification of (all types of) approximately finite-dimensional factors was completed - ex-
cept for one case, the so-called III, case - in [Conl]; the outstanding III, case was finally disposed of in [Haa].
§2.1: The classification of all possible modules over a factor goes back to [MvN3].
§2.2: The importance of bimodules was first recognised by Connes [Con2]. The coupling constant was introduced in [MvNl]. All the assertions in Proposition 2.2.6 appear in [Jonl] although many of these can also be found in the papers of Murray and von Neumann.
BIBLIOGRAPHY
157
§2.3: This entire section is from [Jonl].
§3.1: The basic construction appeared in [Ch] and [Sk], although it was not really exploited in the manner discussed here until [Jonl]. Further, the index of a subfactor was first considered in [Jonl], and indeed, most of this section is also from that source.
§3.2: The notion of a Bratteli diagram was first systematically used in [Bra]. Most of the discussion in this section is `folklore'; thus, for instance, Lemma 3.2.2 may be found (in possibly different pieces) in [GHJ]. On the other hand, facts concerning the Markov trace - such as Proposition 3.2.3 and Corollary 3.2.5 - occur in [Jonl]. §3.3: But for the reference to Kronecker's theorem concerning integral matrices of small norm - which may be found in [GHJ] - all of this section is also from [Jonl].
§4.1: The importance of bimodules was identified and underlined in [Conl]; their significance for subfactors was recognised in [Ocn]; the treatment given here may be found in [Sun3]. §4.2: The importance of the principal graph invariant of a subfactor at least in the `relative commutant formulation' - was already recognised in the first paper [Jonl] on subfactors, where it was also shown that the A,,, diagrams all arose as principal graphs. The `bimodule formulation' of the principal graph is due to Ocneanu ([Ocn]). The fact that the principal graph had to be one of the Coxeter diagrams was recognised in [Jonl]. It was in [Ocn] that it was stated that E7 and D2n+1 could not arise as the principal graph invariant of a subfactor; (independent) proofs of this fact were furnished in [Izl] and [SV]. It was later shown, in [Kaw], [B-N) and [Iz2] respectively,
that all the graphs D2ni E6 and E8 did in fact arise as principal graphs. The principal and dual graphs of the `subgroup-subfactor' - referred to in Example 4.2.3(iii) were explicitly computed in [KY]. The real significance of the usefulness of the so-called `diagonal subfactor' discussed in Example 4.2.4 has been brought out by Popa (see [Pop6]), who, incidentally, also computed the principal graph of this diagonal subfactor.
§4.3: Everything in this section is from [PP1], the only exception being
158
BIBLIOGRAPHY
Proposition 4.3.6 which is from [PP2]. (As explained in the text, we have worked here with a marginal variation of what they term an `orthonormal basis' - in that we replace the requirement of `orthonormality' by `linear independence'.)
§4.4: Everything in this section is contained in the work of Ocneanu and Popa (if not in such explicit detail).
§5.1: Practically everything in this section is from [PP1]. The notion of a commuting square, however, appeared much earlier in the work of Popa see [Popl] and [Pop2]. §5.2: Much of this section is motivated by considerations in [Jon3] and [JonS]. Hadamard matrices originated in [Had]; also see [SY]. §5.3: Lemma 5.3.1, Corollary 5.3.2 and Lemma 5.3.3 are from Wenzl's thesis - see [Wen]. The terminology `symmetric commuting square' (in the case of finite-dimensional C*-algebras) goes back to [HS]; the terminology 'non-degenerate' to describe the same notion (but for more general inclusions of von Neumann algebras) is due to Popa - see [Pop6], for instance. §5.4: The contents of this section were independently obtained in [Ocn] and [Sun4].
§5.5: Essentially all of this section is contained - although in a seemingly different form - in [Ocn]. The formulation contained here - at least as far as Proposition 5.5.2 is concerned - is explicitly worked out in [HS]. §5.6: The basic theorem stated in Theorem 5.6.3 was announced, without proof, in [Ocn]. Subsequently, this was proved in full detail in [Pop4]; the ultimate formulation of this theorem - in terms of (strong) amenability - is in [Pop6].
§5.7: Ocneanu's compactness result - both parts of it, as stated here - is from [OK]. The assertion of Corollary 5.7.4(ii) was first established in [Wen], although the proof given here, using bases, is different (and perhaps simpler).
BIBLIOGRAPHY
159
§6.1: The diagrammatic formulation of the higher relative commutants, for vertex and spin models, is due to the first author (unpublished notes). §6.2: The computations of the principal graphs of the examples considered in Examples 6.2.1 and 6.2.2 were independently performed in [BHJ] and [KSV]. These sort of Cayley graphs had, of course, been obtained in more general situations, in considerations of `diagonal subfactors' in [Pops] (also see [Pop6]) and of subfactors arising as fixed-point algebras of compact group actions - see [GHJ] and [Was]. §6.3: The contents of this section come from [KS].
§A.1: The equivalence of the abstract and concrete notions of a von Neumann algebra was established by Sakai in [Sak2]. §A.2: For the Tomita-Takesaki theorem, see [Tak2].
§A.4: The use of the co-finite morphism and the bimodule calculus for computing the principal graph of the subfactor N C N x G was demonstrated in [OK]. The computation of the principal and dual graphs for the subfactor N x H C N x G was carried out in [KY].
§A.5: Most of this material appeared first in [Jon4] - with the obvious exception of the results of Artin, Alexander and Markov, which appeared first in [Art], [Alex] and [Mark], respectively; for these results, also see [Bir].
Index action 7 ergodic - 13 free - 13 outer - 29 Ad u 29 AFD algebra 18 Alexander's theorem 144
Coxeter graph 43 crossed product 7 crossing 104 positive - 105, 108, 127 negative - 105, 108, 127 cyclic vector 4
& 144
d, 25, 53
ctio/381 Artin's theorem 142
diagonal subfactor 54
AutM7
dimM7-l 25 dimM_7-1 28
automorphism 7 free - 13
dim-MR 28
B, 142 basic construction 33 basis for M/N 55 bimodule 23 bifinite - 30 biunitarity condition 84 black box 123 Boltzmann weight 105, 108, 127, 128
dimension vector 36 double commutant theorem 1 dual graph 50 EM 34
e - f (rel M) 2 eN 33 e[k,lJ 58
equivalent biunitary matrices 84
ergodic action 13
bounded vector 48
left - 6 right - 6 braid group 70, 142 Bratteli diagram 37
central support 72 closure of a braid 144 co-finite morphism 47 commuting square 69 canonical - 91 non-degenerate - 77 symmetric - 77 completely positive 66 conditional expectation 33 connected inclusion 41 contragredient 48 coupling constant 25 covol(F) 25
Fn, 12
factor 2
- of type 12 - of type In33 - of type 112
-oftype Ill3 - of type III 3
-oftypeIII2
finite depth 91 finite factor 2 finite projection 2 fixed point algebra 13 Frobenius reciprocity 52, 62
generating tunnel 91 GNS construction 4 group von Neumann algebra 10
INDEX 7-l ®P K 49
7t, 48 710 28, 47 7-h 48
)748 Hadamard matrix 71 complex - 71 real - 71 hyperfinite III factor 18
ICC group 11 inclusion matrix 37 index of a subfactor 29 infinite tensor products 17 irreducible bimodule 28 irreducible subfactor 29 JM 6
knot 144
L2(M,T) 5 L2(X, v) 5 L°° (X, µ) 1 LG 10
161
M« 13 [M : N] 29 (M, eN) 33 M-dimension 25 M-module 19 M-valued inner product 49 M - N-bimodule 23 Markov 144 - move 145 - property 34 - theorem 145 - trace 41
Mat,,,, 24 matrix units 80, 81 modular conjugation operator 6
module 19 left - 23 right - 23 Murray-von Neumann equivalence 2
n-strand braid 141 normal homomorphism 133
£2(G;7-1) 7
Ocneanu compactness 93, 100
'C (-H) 1
I
,C(-H). 133
(Ao;Ai;...;A.) 81
£Ar(7-l, )C) 24
PG 53
M,C(H) 19
2(M) 2
MG (7-l, K) 19 M.C.N(71, K) 24
partition function 106, 108,
AB 37 A (B, A) 67
path algebras 77 Perron-Frobenius theorem 40,
\(G) 11 link 144 - diagram 146 - invariant 145 local extrema 125, 126
Mx,G7
126
52
ir(A) 77 it, 5, 24 7r,. 5, 23
Pimsner-Popa inequality 66 Powers factors 15 principal graph 50
M00 (M) 24
M. 133 Md(N) 59
Qn(T-1) 44, 53
M°p 23 Mp 3
R 18 R(N) 17
INDEX
162
RA 16
Reidemeister move of type II 109, 128 p(G) 11
tensor product - of bimodules 48 - of co-finite morphisms 50 - of maps 61
90 S' 1 S,,, 12
separating vector 5 a-weak topology 133 Skau's lemma 53 skein relation 146 spin models 71, 107 standard invariant 90 state 4 - of a diagram 105, 107, 125 faithful - 4 normal - 4 partial - 106, 107, 126 tracial - 2 strong amenability 91 subfactor 29 diagonal - 54 irreducible - 29
50
Tomita-Takesaki theorem 135 Tr 3 trace 3
semifinite - 3 - vector 38
U(M) 7 VL(q) 146
vertex models 71, 102 permutation - 115 von Neumann algebra 1 abstract - 133 concrete - 1 finite - 9 OP q 49 "1)M 49
T®pS61
Z(M) 2
T 15
ZT 106 ZM(e) 72
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Edited by PROFL-SSoR J. W. S. CASSELS
Department of Pure Mathematics and Aaihematical Statistic 16 Mill Lane, Cambridge, CB2 1 SB, England
with the assistance of G. R. Allan (Camhridge) P. M. Cohn (Londtur) J. X1. E. Hvland (Camhridge) N. J. Ilitchin (Cambridge) C. B. Thomas (Cambridge)
The London Mathematical Society is incorporated under Royal Charter
Introduction to Subfactors V. Jones and V. S. Sunder
Subfactors have been a subject of considerable research activity for about 15 years and arc known to have significant relations with other fields such as low-dimensional topology and algebraic quantum field theory. These notes give an introduction to the subject suitable for a student who has only a little foniliarity with the theory of Hilbert space. A new pictorial approach to suhfactors is presented in the final chapter.
CAMBRIDGE UNIVERSITY PRESS
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Fbr a list of hook available in this series see page i
