Introduction to Operator Space Theory (London Mathematical Society Lecture Note Series)

104

Introduction to Operator Space Theory

0'1: E1

1---+ B(H1 EB H2 EB H 3) and 0'2: E2 notation, as follows:

-+

B(H1 EB H2 EB H 3) using matrix

0 fil~,) ~0 a+»0!1 (Xl)

0 0

0'1(X1) =

)

f32(X2)

u,(x,)

0 0

Then, by (ii), we have

hence 0'1,0'2 have commuting ranges and are complete contractions. Therefore, if we let W: HI EBH2EBH3 --+ 1t be the projection onto the first coordinate and V: 1t --+ HI EB H2 EB H3 be the isometric inclusion into the third coordinate, then we obtain (iii). Finally, the implication (iii) implies (i) is obvious by the very definition of E 1 ® f.I. E 2 • •

Proof of Theorem 5.17. By duality, it clearly suffices to show that, for any linear map cp: E1 ® E2 --+ B(1t), the norms lIifllcB(El®"E2 -+B(1tll and IIcpQllcb are equal. But this is precisely the meaning of the equivalence between (i) and (ii) in Theorem 5.18. Thus we conclude that Theorem 5.18 implies • Theorem 5.17.

u:

Remark. Let X = R EB C. Let E1 --+ E2 be a finite rank map with associated tensor u E Ei ® E 2 • Using Proposition 5.15, it is easy to check that Theorem 5.17 implies (5.23)

Moreover, it is easy to see that the identity of X factors completely contractively through K, and hence /1(.(u) ~ /x(u).

(5.24)

Corollary 5.19. ([OP]) Let E be an n-dimensional operator space. Let iE E E* ® E be associated to the identity of E and let

5. Tlle Haagerup Tensor Product

105

Then (5.25)

Moreover, /1(E) = 1 iff eitller E = Rn or E = C n (completely isometrically). Proof. Note that (5.25) clearly follows from (5.23). Assume that IL(E) = 1. Then, by Theorem 5.17 and Proposition 5.16 (using also an obvious compactness argument), we have a decomposition IE = Ul + U2 with (5.26) In particular, this implies that 12(IE) = 1 (where 12(') denotes the norm of factorization through Hilbert spacej see, e.g., [P4, chapter 2] for more background), whence that E is isometric to C~ (n = dimE). Moreover, for any e in the unit sphere of E we have

Therefore we must have (5.27) Let ai = Iluil!, so that (by (5.26» al + a2 = 1. Assume that both al > 0 and a2 > O. We will show that this is impossible if n > 1. Indeed, then Ui. = (ai)-lui (i = 1,2) is an isometry on e~, such that, for any e in the unit sphere of E, we have e = al U l (e)+a 2 U2 (e). By the strict convexity of f~, this implies that Ul(e) = U2 (e) = e for all e. Moreover, by (5.27) we have IR(Ut} = 1 and Ic(U2 ) = 1. This implies that E = Rn and E = C n completely isometrically, which is absurd when n > 1. Hence, if n > 1, we conclude that either al = 0 or a2 = 0, which implies either IC(IE) = 1 or IR(IE) = 1, or, equivalently, either E = C n or E = Rn completely isometrically. The remaining case n = 1 is trivial. • Remark. Here is an alternate argument: If /1(E) = 1, then, using (5.23), (5.24) and the reflexivity of E, we see that, for both E and E*, the identity factors through K**j therefore E is an injective operator space as well as its dual. Now, in [Ru2], Ruan gives the complete list of the injective operator subspaces of finite-dimensional C*-algebras (see also [EOR] for more on this theme). Running down this list, and using an unpublished result of R. Smith which says that a finite-dimensional injective operator space is completely contractively complemented in a finite-dimensional C* -algebra (see [B2]), we find that Rn and C n are the only possibilities. For the isomorphic version of the last statement, we will use the following result from [01]:

106

Introduction to Operator Space Theory

Theorem 5.20. ([01]) Let E be an operator space SUell that IE can be factorized completely boundedly through the direct sum X = He EBl Kr of a column space and a row space (i.e., there are c.b. maps u: E ---+ X and v: X ---+ E such that IE = vu). Then there are subspaces El C He and E2 C Kr SUell that E is completely isomorpllic to El EBl E 2 • More precisely, if we have Ilullebllvileb ~ c for some number c, then we can nnd a complete isomorphism T: E ---+ El EBl E2 such tllat IITllebllT-1lleb ~ f(c), where f: 1R+ ---+ 1R+ is a certain function. Theorem 5.21. ([OP]) The following properties of an an operator space E are equivalent: (i) For any operator space F, we have F ®min E = F ®,.. E isomorpllically. (ii) E is completely isomorphic to the direct sum of a row space and a column space.

Proof. The implication (ii) =} (i) is easy and left to the reader. Conversely, assume (i). Then, a routine argument shows that there is a constant K such that for all F and all u in F ® E we have lIull,.. ~ Kllullmin. Let SeE be an arbitrary finite-dimensional subspace, let is: S ---+ E be the inclusion map, and let ts E S* ® E be the associated tensor. Then we have by (5.23) sUPs 1'RE!)lc(is) = sUPs IItsll,.. ~ K. By a routine ultraproduct argument, this implies that the identity of E can be written as in Theorem 5.20 with c = K; thus we conclude that E ~ El EBl E 2, where El is a row space and E2 is a column space. Note that we obtain an isomorphism T: E ---+ El EBl E2 such that IITllebllT-11leb ~ f(K). In particular, if E is finite-dimensional, we find T such that

•

Complex interpolation. The Haagerup tensor product also has a nice "commutation property" with respect to complex interpolation (defined in §2.7) that can be briefly described as follows. Let (Eo, E 1 ) and (Fo, F 1 ) be two compatible pairs of operator spaces. Then the couple (EO®hFO, El ®hFt) can be viewed as compatible. Indeed, assume that (Eo, E 1 ) (resp. (Fo, Ft}) are continuously injected into a Banach space £ (resp. F), which allows us to view the couple as compatible. Then, it is easy to check that the tensor product of the suitable injections defines a continuous injection from Eo ®h Fo into the v

injective tensor product £ ® F. Similarly, El ®h Fl is continuously injected into the same space £ F. (Note: The injectivity of the tensor product of two injective mappings can sometimes fail, but not for tensor norms as nice as the injective one or the Haagerup one. See [PI, Theorem 6.6] for a general result related to this technical point.) Thus we may view the couple (Eo ®h Fo,El ®h Ft) as compatible. Then we have (cf. [PI, Theorem 2.3]):

0

5. The Haagerup Tensor Product

107

Theorem 5.22. Let (Eo, E l ) and (Fo, F l ) be two compatible couples of operator spaces. Then, viewing the couple (Eo Q9h F o, El Q9h F l ) as compatible as explained above, we llave a complete isometry

Exercises Exercise 5.1. Let E l , E2 be operator spaces such that Ei Q9h E2 does not contain a subspace isomorphic (as a Banach space) to co. Then

Exercise 5.2. Prove (5.16) and (5.16)'. Exercise 5.3. Prove (5.17) and (5.18). Deduce from (5.18) that C Q9h E = C Q9min E and E Q9h R = E Q9min R for any operator space E. Use this to give an alternate solution to Exercise 5.2. Exercise 5.4. Let Ei C B(Hi } (i = 1,2) be operator spaces. Then any x in El Q9h E2 can be written as a series of the form L~ an Q9 bn , where an E El bn E E2 with the series L~ ana~ and L~ b~bn converging in norm in B(Hd and B(H2 }, respectively. Moreover, in that case, the series L~ a n Q9bn converges in El Q9h E 2 , and, if Ilxlih < 1, we can choose (an) and (b n ) so that

Exercise 5.5. Show that for any operator space E we have completely isometrically

Exercise 5.6. Show that if E is finite-dimensional, we have completely isometrically (K Q9min E)** C:::' K** Q9min E. Moreover, if A is any C*-algebra, we have (K

Q9min

A)**

C:::'

K**Q9A**.

Exercise 5.7. Let E l , E2 be maximal operator spaces. Then for any x in El Q9 E2 we have

Introduction to Operator Space Theory

108

where the infimum runs over all n and all possible ways to decompose x as n

X=

L

aijXi ®Yj'

i,j=l

Exercise 5.8. Let A, B be arbitrary C* -algebras and let A*B be their (nonunital) free product. For any k ;::: 1, let T2k = A ®h B ®h ... ®h A ®h B (where A and B each appear k times) and T2k+1 = T2k ®h A. For any d > 1, let 1/Jd: 7d ---+ A*B be the completely contractive map induced by the product map in A*B. Show that 1/Jd is a completely isomorphic embedding of 7d in A*B. l\.fore precisely, the range 1/Jd(7d) is the closed subspace Wd C A*B spanned by all the "words" of length d that begin in A, and we have

lI1/Jdi!vJcb:::; (d _1)d-l. Exercise 5.9. Fix an integer n. The aim of this exercise is to produce explicit completely isometric embeddings of C n and Rn into .e~ ®h.e~. We will denote by (ei) the canonical basis of .e~. Let W = (Wik) be an n x n matrix with unimodular entries (Le., IWikl = 1 for all i, k) such that

Iln- 1/ 2 wIIMn = 1. In

.e~ ®h.e~

we consider the vectors Xi

= ei ® L

Wikek

k

Yi

=L

Wikek ® ei'

k

(Note that Yi = t Xi .) Let Ex c .e~ ®h.e~ (resp. Ey C .e~ ®h .e~) be the operator subspace spanned by {Xi I i = 1,2, ... , n} (resp. {Yi Ii = 1, ... , n}). Prove that Ex :::; C n (resp. Ey := Rn) completely isometrically.

Exercise 5.10. The aim of this exercise is to produce an explicit completely isometric embedding of Mn into .e~ ®h .e~ ®h .e~. Let w, Wi be two n x n matrices with the same properties as in the preceding exercise. We introduce . the following elements of the operator space .e~ ®h.e~ ®h i'~: Zij

= ei ®

(~WikW~jek) ® ej.

Then the span of {Zij I i, j = 1, ... , n} in .e~ ®h .e~ ®h .e~ is completely isometric to Mn. Let H be an infinite-dimensional Hilbert space. Show that a matrix (aij) in Mn(B(H» is in the unit ball iff it can be factorized as a matrix product Dl(n-l/2w)D2(n-l/2w')D3, where Db D 2, D3 are three diagonal matrices in the unit ball of B(H).

Chapter 6. Characterizations of Operator Algebras In the Banach algebra literature, an operator algebra is just a closed subalgebra of B(H). A uniform algebra is a closed unital subalgebra of the space C(T) of all continuous functions on a compact set T, which is (usually) assumed to separate the points of T. In the 1970s, the theory of uniform algebras gave birth to the notion of Q-algebras (quotients of a uniform algebra by a closed ideal) and ultimately uncovered some surprising stability properties of the class of operator algebras. Specifically, in the late 1960s, B. Cole (see [We]) discovered that Q-algebras are necessarily operator algebras, and soon after that, G. Lumer and A. Bernard proved that the class of operator algebras is stable under quotients (see Theorem 6.3). One of the natural problems considered dming that period was to characterize the Banach algebras that are isomorphic to a uniform algebra, or a Q-algebra, or an operator algebra. Among the many contributions from that time, those that stand out are Craw's Lemma characterizing uniform algebras and Varopoulos's work on operator algebras ([Vl-2]). Varopoulos discovered that, if the product mapping PA: x Q9 y ---4 xy of a Banach algebra A is continuous on the tensor product A Q9 A equipped with the ')'2-norm, then A is an operator algebra. (From this he deduced, using Grothendieck's inequality, that any Banach algebra structure on a commutative C* -algebra is necessarily an operator algebra.) Several authors then tried to characterize operator algebras by a property of this type but with a different tensor norm than the ')'2-norm. Variations on this theme were given by P. ChaTpentier and A. Tonge (see [DJT, Chapter 18]) until K. Carne [Cal somewhat closed that chapter by showing that operator algebras cannot be characterized by the continuity of the product map on a suitable tensor product. More precisely, he showed that there is no reasonable tensor norm ')' such that the continuity of PA: A Q9')' A ---4 A characterizes operator algebras. In sharp contrast with Carne's result, it turns out that in the category of operator spaces the situation is much nicer. We have Theorem 6.1. ([BRS]) Let A be a unital Banad1 algebra witll unit 1A given witll an operator space structure sucll tllat IlIA II = 1. TIle following are equivalent.

(i) TIle product map is completely contractive. (ii) Tllere is, for some Hilbert space H, a completely isometric unital 110momorpllism j: A ---4 B(H). Remark. Let us denote here simply x . y for PA(X, y). Note that, by the definition of ®h, (i) is clearly the same as:

110

Introduction to Operator Space Theory

(i)' The natural matrix product

ab =

(L.:

aik . bkj) ij

k

defines a unital Banach algebra structure on Mn(A) for all n In other words, (i) holds iff Ilell = 1 and

~

l.

The implication (i)' (¢:> (i)) ~ (ii) shows that this implies that, for any unital operator algebra B, B ®min A is equipped with a natural unital operator algebra structure corresponding to the tensor product of PB and PA. Curiously, the isomorphic version of this theorem resisted until Blecher [B4] recently proved the following statement. Theorem 6.2. ([B4J) Let A be a Banach algebra given with an operator space structure. The following are equf.,lalent. (i) The product map PA: A ®h A ---- A is completely bounded. (ii) There is, for some Hilbert space H, a homomorphism j: A ---- B(H), which is a complete isomorpllism from A to j(A). Remark. Note that (i) in Theorem 6.2 holds iff K[A] = K ®min A is a Banach algebra up to a constant for its natural matrix product, in other words, iff there is a constant C such that Ilabll ::; Cllalilibil for all a,b in K[A]. The original proofs ofthese theorems do not use the results from the 1970s, and can be used to give new proofs of the latter. However, we have recently found a new approach: We will deduce the preceding two theorems from the stability under quotients of operator algebras. Note, however, that, both for Theorems 6.1 and 6.2, some trick evocative of the original proofs of [BRS, B4] still remains. Our approach is based on the following statement (due to A. Bernard, to G. Lumer independently in some form, and originally due to Cole when A is a uniform algebra). Theorem 6.3. Let A c B(H) be a closed subalgebra of B(H) and let I c A be a closed ideal. Then there is, for some Hilbert space 1-1., an isometric (and completely contractive) homomorphism cp:

AI I

----+

B(1-I.).

In other words, the class of operator algebras is stable under quotients. Proof. We essentially follow Dixon's exposition in [Dix1]. Consider x E AII with IIxll = 1. We claim that there is, for some ii, a homomorphism

111

6. Characterizations of Opera. tor Algebras

'Px: All

--+ B(ii) with II'Pxllcb ::; 1 and II'Px(x)11 = Ilxll = 1. From this claim the conclusion of Theorem 6.3 follows immediately: We simply let 'P = EBx 'Px (with x running over the unit sphere of A). We now briefly justify this claim. Consider ~ in the unit sphere of (AI1)* such that ~(:l:) = 1. Let us denote by q: A --+ AJI the quotient map and let [ E A* be defined by [(a) = ~q(a», so that II~IIA* = 1. Then there is a representation 7r: B(H) --+ B(H) and elements s, t in the unit ball of ii such that

[(a) = (7r(a)s, t).

(6.1)

\;fa E

A

(This is entirely elementary and well known, but we can view it as a very special case of Theorem 1.6 since II[IIA' = 1I€'IlcB(A.q.) Now l('t E1 C ii and E2 C E1 be defined by

E1 = span[s, 7r(a)s I a E A] E2 = span[7r(i)s liE I]. Then E1 and E2 are clearly invariant under 7r(A); therefore the subspace

is semi-invariant with respect to 7r(A), so that (by Sarason's well-known ideas, see, e.g., [PlO, Theorem 1.7]) the compression ii': A --+ B(E) defined by

is a homomorphism. Indeed, for a, bE A, we have PE7r(a)PE2 = 0, and hence ii'(a)ii'(b) = ii'(ab). lVloreover, ii' is clearly contractive (actually completely contractive). Observe that since [ vanishes on I, we have t E E;}-. Thus, it is now easy to check that (6.1) yields

[(a) = (ii'(a)PEs, t),

(6.2)

and also, since 7r(I)E1 C E 2 , that ii': A --+ B(E) vanishes on I. Hence, if we let 'Px: AI I --+ B(E) be defined by ii' = 'Pxq, we have (take a with q(a) = x in (6.2» 1 = IIxll = ~(x) = ('Px(x)PEs, t), whence 1 ::; II'Px(x)ll. Since (by (2.4.2» II'Pxllcb claim.

=

11ii'llcb ::; 1, this proves our ..

Corollary 6.4. In the situation of Tlleorem 6.3, there is, for some Hilbert space 11., a completely isometric homomorphism

1/;: AI I

--+

B(11.).

112

Introduction to Operator Space Theory

Proof. Two simple proofs of this can be given. A first proof consists of applying Theorem 6.3 for each integer n to the algebra Jl.Jn(A)/Mn(I) ~ Mn(A/ I). Assuming A and r.p: Mn(A)/ Mn(I) --+ B(1i) unital, it is easy to check (by elementary algebra) that r.p is necessarily of the form Iu" ® '¢n for some homomorphism '¢n: A/I --+ B(1i). A direct sum argument (considering '¢ = EEln'¢n) then completes the proof. A second (and better) proof consists of repeating the proof of Theorem 6.3, but this time with in the unit sphere of CB(A/I,B(L» for some Hilbert space L. The same proof (using the fundamental Theorem 1.6 to factorize [: A --+ B(L) instead of (6.1» then yields Corollary 6.4. •

e

Remark. The analogs of Theorems 6.1 and 6.2 for dual operator algebras are proved in [LeM6] (see also [B5]). Remark. See [LeM2] for an extension of the preceding two results to the case when the class of Hilbert spaces is replaced by that of subspaces of quotients of Lp, with p =I 2. The idea of the proofs of Theorems 6.1 and 6.2 is very natural: We will represent A as a quotient of the "free operator algebra" generated by A. The latter is defined as follows. We will distinguish between the unital and nonunital cases. Let E be an operator space. Let T(E) (resp. Tu(E» be the tensor algebra (resp. the unital tensor algebra) over E, that is, T(E)

= E EEl (E ® E) EEl .. ·

and

We view T(E) as a subspace of Tu(E). Then T(E) (resp. Tu(E» is an algebra (resp. a unital algebra) containing E as a linear subspace and characterized by the property that any linear map v: E --+ B from E into an algebra (resp. a unital algebra) B uniquely extends to a homomorphism (resp. unital homomorphism) T(v): T(E) --+ B (resp. Tu(v): Tu(E) --+ B). We will denote the free operator algebra associated to E by OA(E), and we define it as follows: Let C = {v: E --+ B(Hv) I IIvllcb ~ I}, where Hv is an arbitrary Hilbert space with dimension at most, say, the density character of E. We define a linear embedding J: Tu(E)

--+

ffiB(Hv) C B (ffiHV) vEe

vEe

by setting J(x) = ffiTu(v)(x). vEe

6. Characterizations of Operator Algebras

113

Note that J is a unital homomorphism. We define OAu(E) (resp. OA(E)) as the closure of J(Tu(E)) (resp. J(T(E))) in B (ffiVEC Hv). Thus OAu(E) is a unital operator algebra and OA(E) C OAu(E) is a closed subalgebra. It is easy to check that OAu(E) is characterized by the following universal property: For any map v: E ---+ B(H) with Ilvllcb :=:; 1 there is a unique unital homomorphism OAu(E) ---+ B(H), extending v, with Ilvllcb :=:; 1.

v:

The elements of Tu (E) can be described as the vector space of formal sums:

where each Pj (1 :=:; j :=:; N) is "homogeneous" of degree j, that is, Pj is of the form

with Ao(j) E C and e'j'(j) E E (1 :=:; i :=:; j). Similarly, any element Q of T(E) can be written as Q = PI + ... + PN . For any v E C, let

v(Pj and let v(P)

)

= L Aa(j)v(er(j))··· v(ej(j))

= AoI + v(P1 ) + ... + v(PN )

and v(Q)

= v(P1 ) + ... + v(PN ).

Clearly v(P) and v(Q) are nothing but Tu(v)(P) and T(v)(Q). Then we have 1IPIIoA,,(E) = sUPvEC IIv(P)11 and IIQlloA(E) = sUPvEC Ilv(Q)II· Explicitly, this means that N

AoI + L

IlPlloA,,(E) = sup vEe:

Ao(j)v(er(j))v(e~(j)) ... v(ej(j))

L

j=l

a

B(H,.)

l\'1ore generally, let us now assume that the coefficients Ao, Ao(j) are all in K (or even in B(f2)) and let P be an element of K 0 Tu(E) of the form N

P

= Ao 01 + LLAa(j) 0er(j) 0··· 0ej(j). j=l

a

Then the above definition of OAu(E) means that

N

= sup

vEC

Ao 0 I

+L j=l

Aa(j) 0 v(er(j)) ... v(ej(j))

(6.3) K®miuB(H,,)

Of course a similar formula holds for OA(E) with the constant term omitted.

114

Introduction to Operator Space Theory

Remark 6.5. Let E be any operator space. Consider the iterated Haagerup tensor product X = E ®h E . .. ®h E (N times). Let x be arbitrary in IC ® E ® ... ® E. Then x can be written as a finite sum

x= L

Ai ®

xI ® ... ® xi"

i

with Ai E IC and

xl, ... ,xi" E E.

By Corollary 5.3(iii), we have

where the supremum runs over all possible choices of H and of complete contractions 0'1: E --+ B(H), ... ,aN: E --+ B(H). We claim that this supremum is actually attained when 0'1, ..• , aN are all the same; more precisely, we have

where the supremum runs over all possible H and all complete contractions --+ B(H). Indeed, this follows from a trick already used by Blecher in [B1] and which seems to have originated in Varopoulos's paper [V2]. The trick consists of replacing 0'1, ..• ,aN by the single map v: E --+ B(H EB ... EB H) ~ N+l times of the form v: E

o v(e) =

o

o

(More precisely, v(e) is the (N + 1) x (N + 1) matrix having (al(e), ... , aN (e)) above the main diagonal and zero elsewhere.) Then it is easy to check that

Ilvllcb = SUPj lIa j IIcb, 'v'xl, ... ,x N

E

E,

From this our claim immediately follows. Proposition 6.6. Let E be any operator space. Fix N ~ 1. Let EN = E ® '" ® E (N times). Consider EN as embedded into OA(E). Then the identity mapping on EN extends to a completely isometric embedding of

6. Characterizations of Operator Algebras

115

E ®h ... ®h E (N times) into OA(E). l\!oreover, there is a completely tractive projection from OA,,(E) onto the range of this embedding.

COll-

Proof. For simplicity let us again denote X = E®h·· ·®hE (N times). Since the algebraic tensor product E®·· ·®E is dense in X, to show the completely isometric part it suffices to prove that, for any element G in K ® E ® ... ® E, we have

IIGI1K0IHinX = IIGI1K0IHiuOA(E). But this is immediate by (6.3) and (6.4). We now show that we have a "nice" projection QN from OA(E) onto the closure of EN in OA(E). Let x be a typical element of T(E), of the form x = L::=o Xn with Xn E En viewed as included in OA,,(E) (so J is viewed as an inclusion) and (say) N ~ 1n. Note that by (6.3) we have

IlxIIOA(E) = for any real t and XN =

IlL eintXn I

OA(E)

(6.5)

f e-iNt(L: k eiktxk)dt/27f, which implies (6.6)

This proves that the projection defined by QN(X) = XN is contractive. But by an obvious modification (using operator coefficients instead of scalars) one easily verifies that it actually is completely contractive. • Let E be an operator space equipped with a Banach algebra structme. Let p: E ® E --+ E be the linear map associated to the product map. Then, by associativity, p uniquely extends to a mapping

p:

T(E)

--+

E,

which is the identity on E and satisfies for any N ;::: 2 and any Xi in E

To prove Theorem 6.2, we will need the following simple lemma (already used in [B4]). Lemma 6.7. For any z in C, let us denote by a z: T( E)

--+ T( E) the homomorphism that, wIlen restricted to E, is equal to z IE. Then, if the product map extends to a mapping on E ®h E with IlpllcB(E01,E.E) = c, we 118ve, for any z with Izi < lie,

~ II Izl I paz cb:S 1 -Izlc

(6.7)

116

Introduction to Operator Space Theory

Proof. By an obvious iteration p defines a product mapping from E ®h ... ®h E (n times) into E with c.b. norm::; cn-l. By Proposition 6.6, the natural embedding of E ®h ... ®h E (n times) into OA(E) is a complete isometryj hence the restriction of p to E ® ... ® E c T(E) c OA(E) has c.b. norm ::; cn - l . We have

and hence by (6.5)

or, equivalently,

IipO'z11 ::;

",00

L.."l

Izlnen-l =

1-

Izl . Izle

By a simple modification (left to the reader) we obtain the same bound for the c.b. norm of PO'z, whence (6.7). •

Proof of Theorem 6.2. The implication (ii) ::::} (i) is immediate by Corollary 5.4. We now prove the converse. Assume (i). We denote by E the operator space underlying A. Fix z with Izl < l/e. Clearly pO'z is a surjective homomorphism from OA(E) onto E. Let I be its kernel and let w: OA(E)/I --+ E be the associated isomorphic homomorphism. By (6.7) and the definition of the quotient operator space structure (cf. §2.4), we have Ilwllcb ::; l~i;,c' Let i: E --+ OA(E) be the canonical inclusion and let q: OA(E) --+ OA(E)/ I be the quotient map. Clearly we have O'zi(x) = zi(x)j hence pO'zi(x) = zx or (wq)i(x) = zx, so that (l/z)qi = w- l . Whence

Thus if we choose for instance the value z = 1/2e, we find a homomorphism liw-llicb ::; 2c and liwllcb ::; l/e. By Corollary 6.4, this completes the proof. • w: OA(E)/ I --+ E with

Paradoxically, in our approach the proof of Theorem 6.1 is a bit more complicated. The crucial point is the following.

Lemma 6.B. Let E be an operator space that is also a Banach algebra with a unit e. Let p: E ® E --+ E be associated to the product map. Then, if lieli = 1 and if p defines a complete contraction from E ®h E into E, the mapping p introduced before the preceding lemma extends to a completely contractive unital homomorphism Pu from OAu(E) onto E.

6. Characterizations of Operator Algebms

117

Proof. Let Eo = C and Ej = E®·· ·®E (j times). Let x be a typical element in K®Tu(E). Then;1: can be written as;r = L;l=O Xj with ;1:0 = t ® 1 E K®C and with each x j expanded as a finite sum Xj = L::>j(O:)0ef® .. ·®ej a

with tj(O:) E K, ej E E, 1 :::; j :::; n. For el,"" ej E E, we denote simply below by ele2 ... ej their product in E. Note that h:®pu(x) = Lj La tj(a)® ele2' ... ej, so that the content of Lemma 6.8 is the inequality

L L tj(O:) ® (ele2' ... ej) j

(6.8)

a

This is somewhat reminiscent of von Neumann's inequality for polynomials in a contraction. Observe that (as in the proof of Theorem 6.2) we clearly have (6.8) if x is reduced to a "homogeneous" term of some degree, that is, if x = Xj for some j. Indeed, by iteration the product map p defines a complete contraction el ® ... ® ej ~ ele2 ... ej from E 0h ... 0h E into E, and E ®h ... ®h E = E ® f ... ® f E clearly naturally embeds completely isometrically into 0 Au (E). Thus we immediately get (6.8) in the homogeneous case. For the general case, the idea will be to replace a general element. x = L x j as above by another one y(N) that is homogeneous of degree N and satisfies moreover Ily(N)llmin :::; Ilxllmin and limN---+oo 11(h: 0 pu)(y(N))llmin = 11(1}( IZl pu)(x)llmin. The conclusion will then be easy to reach. Note that we will denote simply by II Ilmin the norms appearing on both sides of (6.8). This should not bring any confusion. Note that K ®min OAu(E) is an operator algebra embedded in the unit.al operator algebra B = B(£2) ®min OA,,(E). For convenience we introduce the following elements of B: We denote ~k

=

1 B (£2) ®~.

k-times Then we define l:!!.j(N)

=

1 N-j N- '+1 L~k'Xj'~N-k-j, J k=O

where the dot denotes the product in B. Observe that (6.9)

Introduction to Operator Space Theory

118

~

We claim that we have, for all N

n, (6.10)

To verify this let N

cp(t)

= (N + 1)-1/2 L

N

1jJ(t)

and

eikt(,k

= (N + 1)-1/2 L

k=O

Note that since B(H), we have

II!

Ilell =

cp(t)cp(t)* ::

1 we have

t(H)

e-ilt(,N_e.

(=0

$ 1

II(,kll

and

=- 1 for all

II!

k; hence, if we view B c

1jJ(t)*1jJ(t) ::

t(H)

$ 1.

(6.11)

Let us compute

(6.12) A simple verification shows that

y(N) = (N + 1)-1 k+j=e

O::;j::;n, O::;k,e::;N

(6.13)

=

t ~j(N) .(N;!: 1) . J=O

The preceding formula shows that y(N) is "homogeneous of degree N," that is, we have y(N) E K Q9 EN. As mentioned above after (6.8), Pu is clearly completely contractive when restricted to K Q9 EN. Therefore we have (6.14) and by (6.12) we have (by a variant of (1.12»

whence by (6.11) and (6.5) (6.15)

6. Characterizations of Operator Algebras

119

Finally, we have by (6.9) and (6.13)

(h @Pu)(y(N»

=~ n

(N-j+1) N + 1

(h @Pu)(Xj);

hence by (6.14) and (6.15) we obtain our claim (6.10). Then, from (6.9) and (6.10), we have

:s; lim sup lI(h @pu)(y(N»lImill :s; Ilxllmill' N->oo

min

This concludes the proof of (6.8) and hence of Lemma 6.8.

•

Proof of Theorem 6.1. Again (ii) =} (i) is immediate by Corollary 5.4. We turn to the converse. We denote by E the operator space underlying A. The proof is analogous to that of Theorem 6.2, but instead we need to know that, if (i) holds, the product map el @ ••• @ en ----> el . e2 ... en extends to a completely contractive unital map from OAu(E) to E. This is provided by Lemma 6.8. Given this fact, we conclude as before: Assume (i). The map Pu: OAu(E) ----> E yields after passing to the quotient by its kernel I a unital bijective homomorphism w: OAu(E)/ I ----> E with Ilwllcb :s; 1. As before, we have w- 1 = qi, where i: E ----> OAu(E) is the natural injection and q: OAu(E) ----> OAu(E)/I is the natural quotient map. Hence we conclude that IIw- 1 ll cb :s; IIqllcbllillcb = 1, that is, w is a completely isometric isomorphism from OAu(E)/I onto E. By Corollary 6.4, E is unit ally and completely isometrically isomorphic to a unital operator algebra .

•

Remark. We refer the reader to [BLM] for a detailed study of various possible operator algebra structures on "classical" examples such as £p or Sp (the Schatten p-class). Operator spaces that are also modules over an operator algebra (in other words, "operator modules") can be characterized in a similar way (see leES, ER9], and see also [Ma1-2] for dual modules). Let X c B(H) be an operator space and let A, B be two subalgebras of B(H). If X is stable under left (resp. right) multiplication by elements of A (resp. B), then X can be viewed as a concrete submodule of B(H) with respect to the actions of A and B by left and right multiplication. The next statement characterizes the "abstract" bimodules that admit such a concrete realization in B(H).

Theorem 6.9. (fCES}) Let A, B be C* -algebras and let X be an (A, B)bimodule, that is to say, X is both a left A-module (witl} action denoted

Introduction to Operator Space Tbeory

120

(a,x) ---4 a.x) and a rigbt B-module (witb action denoted (x, b) ---4 x.b), so tbat we bave a "module multiplication" map m: A x X x B ---4 X defined by m(a, x, b) = (a.x).b = a.(x.b). For simplicity we write m(a, x, b) = a . x . b. We assume tllat tbe sets {a.x I a E A,x E X} and {x.b I x E X,b E B} are dense in X. Tben, given an operator space structure on X, tbe following are equivalent: (i) m defines a complete contraction from A Q9h X Q9h B to X. (ii) Tbere exists a completely isometric embedding j: X ---4 B(11.) and representations PI: A ---4 B(11.) and P2: B ---4 B(11.) sucb tbat, for all x in X, a in A, and b in B, we bave j[m(a,x,b)] = PI(a)j(x)p2(b). Proof. Assume X C B(H). Applying Lemma 5.14 to the trilinear map m: A x X x B ---4 B(H), we obtain a complete contraction 0': X ---4 B(11.); representations 11"1: A ---4 B(11.), 11"2: B ---4 B(11.); and contractive maps V, W: H ---4 11. such that (\Ix E X, a E A, bE B) (6.16) Let PI (resp. P2) be the orthogonal projection onto span [11"1 (A)V H] (resp. span[1I"2(B)WH]). Let s = PlO'(a· X· b)P2 and t = pl11"1(a)O'(x)1I"2(b)P2' We claim that s = t. Indeed, it suffices to check that

for all

0:

in A, f3 in B, and h, k in H. But this equality is the same as

and indeed by (6.16) both sides are equal to (o:a· our claim that

X·

b(3)(h), k), which proves (6.17)

Note that PI (resp. P2) commutes with 11"1 (resp. 11"2) so that Pl(-) = PI1I"1(-) and P2(-) = 1I"2(')P2 are representations. Setting j(x) = PlO'(X)P2, we have IIjllcb ~ 1 and we obtain by (6.17)

j(a· X· b) = pl(a)j(x)p2(b). It remains to show that j is completely isometric. In the unital case this is immediate because (6.16) then allows us to write x = V*j(x)W, and since V, Ware contractive (and IIjllcb ~ 1), j must be completely isometric. In general, our density assumption guarantees that if (ai) (resp. (bk» is an

6. 911aracterizations of Operator Algebras approximate unit in the unit ball of A (resp. B), then ai . X· bk by (6.16) we have

121 ---->

x. Hence,

and hence we have IIxll ::; Ilj{x)1I and similarly for the matrix norms, so again we conclude that j is completely isometric. • Suitably modified versions of the Haagerup tensor product are available for operator modules (see [Bl\·IP, ChS, Ma2, AP, Pop]). Operator modules playa central role in [Bl\lP], where the foundations of a Morita theory for non-self-adjoint operator algebras are laid. 'See [BMP], [B4], Blecher's survey in [Kat], and references contained therein for more on all of this. Exercises Exercise 6.1. Let A be an operator algebra. Let E 1 ,. , • , EN be operator spaces. Then the product map A x ... x A ----> A induces a complete contraction

A fortiori, for N = 3, the product map PA: A 0h A contraction

---->

A defines a complete

Chapter 7. The Operator Hilbert Space Hilbertian operator spaces. In operator algebra theory and in quantum physics, numerous Hilbertian operator spaces have appeared. Let us say that a subspace E c B(H) is Hilbertian if it is isometric (as a Banach space) to a Hilbert space. We have already met the spaces Rand C, for instance. A more sophisticated example is the linear span of the Clifford matrices or, equivalently, of the generators of the Fermion algebra (this is sometimes called a spin system). Essentially the same example appears with the linear span of the creation (or annihilation) operators on the antisymmetric Fock space (see §9.3). A different example is the linear span of the generators of the Cuntz algebra 0 00 (see §9.4) and also the generators of the reduced C*-algebra C>:(Foo ) on the free group with infinitely many generators (this one is only isomorphic to £2; see §9.7) or the "free" analog of Gaussian variables in Voiculescu's "free" probability theory (see §9.9). Actually it is possible to show that there is a continuum of distinct (Le. pairwise not completely isomorphic) isometrically Hilbertian operator spaces. Furthermore (we will come back to this in Chapter 21), if n > 2, it can be shown that the set of all Hilbertian operator spaces of a fixed dimension n equipped with the "complete" analog of the Banach-Mazur distance is not compact, and not even sepamble. As mentioned in Chapter 3, Blecher and Paulsen [BPI J observed that any separable infinite-dimensional Hilbertian operator space 1{ sits "in between" the extreme cases min(l'2) and max(l'2), that is, we have isometric and completely contractive inclusions

Moreover, the interval between the two extremes is in some sense very "broad." Existence and unicity of OH. Basic properties. Despite the multiplicity of (different) examples of Hilbertian operator spaces, it turns out that there is a centml object in this class, that is, a space that plays tlle same central . role in the category of operator spaces as Hilbert space does in the category of Banach spaces. To motivate the next result, we recall that, if H is a Hilbert space, we have a canonical (Le. basis free) identification H=H*. Moreover, this characterizes Hilbert spaces in the sense that, if a Banach space E is such that there is an isometry i: E -+ E* that is positive (Le., i(x)(x) ~ 0 \Ix E E), then E is isometric to a Hilbert space. In the category of O.s. this becomes (see [PIj) Theorem 7.1. Let J be any set. Then there is an operator space OH(J)

7. The Operator Hilbert 8pace

123

tlwt is isometric (as a Banach space) to £2 (J) and which is such that tIle canonical identification £2(1) = £2(1)* induces a complete isometry from OH(I) to OH(1)*. AIoreover, OH(1) is the unique operator space up to a complete isometry witIl this property. Let K be any Hilbert space and let (li)iEl be any Ol·tllOnormal basis of 0 H (1). Tllen for allY finitely supported family (x;);El in B(K) we have (7.1 )

Notation. If J = N, we denote the space OH(1) simply by OH (we call it "the operator Hilbert space"). If J = {l, 2, ... , n}, we denote the space OH(1) simply by OHn . Note that any n-dimensional subspace of OH is completely isometric to OHn . To prove Theorem 7.1, it suffices to produce a norm 0: on K>g;£2(1) with a suitable self-dual property. The formula is extremally simple: \:Ix = LiE! Xi Q9 e; E K Q9 £2(1) (finite sum) with X;. E K, we set

o:(x) = IILXi Q9Xillll2 'El

mm

The proof that this is indeed an operator space structure on £2(1) with the required property is based on a version of the Cauchy-Schwarz inequality due to Haagerup ([HI, Lemma 2.4]), as follows:

(7.2)

valid for all Xi, Yi E K (or, more generally, for all Xi, Y;. E B(£2)). These forn1Ulas are one more illustration of the idea of "quantization" (see the discussion in §2.1l).

Proof of (7.2). By Proposition 2.9.1 we have (7.3)

where the supremum runs over all a, b in the unit ball of 8 2 8 2 ((2, (2) and where ( , ) denotes the scalar product in the Hilbert space 8 2, that is, (x;ayi, b) = tr(xiaYib*). Note that lIall2 S 1 iff a can be written as a = ala2 with trlall 4 S 1 and trla21 4 S 1, and similarly for b. Using this we can write

124

Introduction to Operator Space Theory

Hence, by Cauchy-Schwarz

but by Proposition 2.9.1 again I : Ilbixia111~

= I:tr(bixialaix;bl) = I:tr(Xi a l ai x ;b1bi)::; III:Xi ®Xillmill'

and similarly for the other term. Hence we finally derive (7.2) from (7.3) . •

Remark. The preceding proof also establishes the following two additional formulas: (7.3)'

III: Xi ® Xillmill

= sup {I: tr(xiax;b) I a ~ 0, b ~ 0, tra 2 trb 2

(7.3)"

III:Xi ® Xillmin

::;

::;

1,

1}

= sup { (I: lI,8xioll~) 1/2 I trlol 4

::;

1, trl,81 I}. 4 ::;

Proof of Theorem 7.1. We first show the "existence" of the space OH(I). Let K be a fixed infinite-dimensional Hilbert space. Let B = B(K) (actually, the proof works equally well if we set B = IC). Let us denote by C the set of all finitely supported families Y = (Yi)iEI in B such that IIEYi ® Yillmin ::; 1. Let Hy = K for all Y in C. For any fixed i in I, we define

Then, for any finitely supported family (Xi)iEI in B, we have

But by (7.2) we have (7.4)

since the supremum is attained for

125

7. The Operator Hilbert Space Thus we obtain

(7.5) In particular, if e E B is any element with Ilell supported familly of scalars (ai)iEI

= 1, we have for any finitely

Hence if we let E

= span[Ti liE JJ,

then E ~ €2(I) isometricaHy. We claim that E ~ E* completely isometrically. To verify this it suffices to check that, if (~i)iEI is the basis of E* biorthogonal to (Ti)iEI, we have for any x = (Xi)iEl as before (7.6) But by the definition of the complex conjugate (see §2.9) we have

where u: E ---> B is the mapping defined by u(x) = L (or, equivalently, U(Ti) = Xi for all i).

~i (x )Xi

for all x in E

By (2.1.9) (and by a density argument) we have lIulicb = sup {IILYiQ9 U(Ti) II}, where the sup runs over all finitely supported families (Yi )iE I in B with IILYi Q9 Tillmin ::; 1. Equivalently, lIullcb = sup {IILYi

Q9

Xiii

I YE C}.

Hence, by (7.4),

Thus (7.7) and (7.5) give us (7.6). This proves the existence ofa space with the properties in Theorem 7.1. We now address the unicity. Let F be an operator space isometric to €2(I) and such that F ~ F* completely isometrically. Let (Oi)iEI be any orthonormal basis in F. We will show that, for any (Xi)iEI as before, we have necessarily

This will show that the correspondence Ti ---> Oi is a completely isometric isomorphism, thus establishing the announced unicity. Let (T/i)iEI be the

126

Introduction to Operator Space Theory

basis of F* biorthogonal to (lJi)iEI. Reasoning as above, we see that F implies that, for all (Xi)iEI as before, we have

~

F*

where the supremum runs over all finitely supported families (Yi )iEI in B such that IIL:Yi ~ lJdlmin :::; 1. In particular, we must have by homogeneity

Taking Xi = Yi, this implies

In particular, IIL:Yi~lJillmin:::; 1 implies (Yi)iEI E C, and hence (7.8) and (7.4) imply the converse inequality

Thus, we conclude that for any finitely supported family (Xi)iEI in B we have

= B(I<)

Of course, this holds a fortiori when I< is finite-dimensional. Thus if we set OH(I) = E, we have checked all the properties in Theorem 7.1. • Definition. Let E be a (complex) vector space. We will say that a linear mapping J: E --+ E* (resp. i: E* --+ E) is ''positive definite" if it satisfies

'V.xEE

J(X)(X) ;::: 0

(resp.

'VeE E*

Remark. It is easy to see that a bounded linear operator J: E --+ E* (resp. i: E* --+ E) is "positive definite" iff there is a Hilbert space H and a bounded operator v: E --+ H (resp. v: H --+ E) such that J = tv v: E --+ H ~ H* --+ E* (resp. i = v tv: E* ~ H* ~ H --+ E). In the category of Banach spaces, the analog of Theorem 7.1 is the following obvious fact: If a Banach space E admits a positive definite isometry J: E --+ E* (resp. i: E* --+ E), then E is isometric to a Hilbert space, and J (resp. i)" is then surjective and coincides with the canonical (Le. basis free) correspondence between E* and E. Thus Theorem 7.1 says that the operator space E = OH(I) is characterized by the property that there is a positive definite map J: E --+ E* (resp. i: E* --+ E) that is a complete isometry. The next statement is mainly a useful reformulation of (7.1).

127

7. The Operator Hilbert Space

Proposition 7.2. (i) Let E be any operator space, I any set, and let v: OH(I) -+ E be a c.b. map. Let tv: E* -+ OH(I)* be its adjoint. Consider tile composition vtv: E* -+ OH(I)* -:.::= OH(I) -+ E.

We have then (7.9) (ii) Moreover, if (Ii) is an orthonormal basis of 0 H (I), we have

Ilvll~b = sup {II~ v(Td 0 V(Ti)11 tEJ

(iii) For any v: OH(I)

.I

J c I,

IJI < co} .

(7.10)

mm

-+

OH(I') (I, I' arbitrary) we have

IIvllcb = Ilvll.

Proof. (i) (Finite Case) Assume first that I is finite. Let ~i E OH(/)* be the biorthogonal basis to (Ti)' so that the mapping v corresponds to L ~i 0v(T;). Then, by (2.3.2) and (7.5), we have

Ilvllcb = 1II:~i 0 = III:Ti

0

tin 1II:~i v(Ti)tin V(Ti) tin = III: V(Ti) V(Ti)lI~i: .

V(Ti)

0

=

0

On the other hand, for any ~ in E* we have tv(~)

=

L(~,v(Ti))~;. Hence

and hence

which means that the map vtv: E* -+ E corresponds to the tensor L v(T;) 0 veT;). Thus, by (2.3.2) again we obtain

Ilvtvlleb =

III: V(Ti)

0 V(Ti)tin '

and we conclude by the first part of the prpof

= IIvll~b· (ii) Now let I be an arbitrary (a priori infinite) set. Let J C I be a finite subset and let E(J) = span[Ii liE J]. By a simple density argument, we have Ilvlleb = sup IlvIE(J) Ileb. J

Hence, by the first part of the proof we obtain (7.10).

128

Introduction to Operator Space Theory

(i) (General Case) Assume now that I is infinite but that E is finitedimensional. Then the series EiEI ~ Q9 v(Ti) and EiEI v(T;) Q9 v(T;} are convergent, respectively, in OH(I)* Q9 E and in E Q9 E, so that the preceding proof of (7.9) is easily extended to this case. Finally, in the general case, we can use Exercise 1.4 to show that (7.9) reduces to the case when E is finite-dimensional, which we just treated. (iii) By density, it suffices to check this when I is finite. Then, applying (i) with E = OH(I') we are reduced to showing Ilvtvllcb = IIvtvll. But now, since w = vtv is a non-negative Hermitian finite rank operator, it is diagonalizable; hence, for some orthonormal basis (Ti)iEI' in OH(I'), w is associated to a tensor of the form iEI'

with (Ai)iEI' finitely supported. The inequality to the following claim:

for any finite subset J

IIwllcb ::; IIwll

is thus reduced

c I'. But then, by (7.3)", we have

and by (7.10) applied to the identity map on OH(I') we know that

•

so we obtain the announced claim.

Corollary 7.3. Let I and J be arbitrary sets and let OH(I) and OH(J) be the corresponding operator Hilbert spaces. Then OH(I) Q9h OH(J)

~

OH(I x J)

completely isometrically.

Proof. For simplicity, we may and will assume that I and J are finite sets. It is easy to reduce the general case (by density) to that case. Let E = OH(I) and F = OH(J). Then, by Corollary 5.8, the following complete isometries hold: (E Q9h F)* '::! E* Q9h F* '::! E* Q9h F* ~ EQ9h F.

7. Tile Operator Hilbert Space

129

Clearly, the corresponding mapping E ® F --+ (E ® F)* is positive definite (by the same argument as the one showing that the tensor product of two scalar products is also one). Thus, by t.he preceding remark, we conclude that E ®h F is completely isometric to OH(I') with the cardinality of I' equal to dim(E ®h F), so that we can set I' = I x J. •

Remarks (i) By (7.1), the o.s.s. just defined on OH(1) is dearly independent of the choice of an orthonormal basis.

(ii) Thus, given a Hilbert space H, we may equip H with the o.s.s. corresponding to OH(I) with the cardinalit.y of I det.ermined by t.he Hilbertian dimension of H. Thus, Theorem 7.1 admits the following basis free ("canonical") reformulation: For any Hilbert. space H, there is a unique o.s.s. on H for which the canonical isometric identification H ~ H* becomes a complete isometry. \Ve will sometimes denote by Hoh the resulting operat.or space. (iii But actually, we will rarely lise the notation Hoh in the sequel. We prefer to make the following convention: We consider that t.his o.s.s. is the "nat.ural" one on a Hilbert space, so we always assume (unless explicitly stated otherwise) that. H is equipped with the o.s.s. of H oh .

(iv) In particular, if H ~ 1'2(1), t.he Hilbert. space S2(H)oh can be identified with OH(I x I), allowing us to view S2(H) as an operator space from now on. (v) It will be convenient to record here the following formula describing this o.s.s. on S2(H). For any x = L eij ® Xij in 1I/n (S2(H)) (Xij E S2(H)), we have (7.11)

where the supremum runs over all a, (3 in lIIn wit.h trlal 4 < 1 and trl(31 4 :::; 1. Indeed we have

i.,j

p,q

p.q

with Ypq = Li,j eijXij(P, q). Hence, since {e pq I p, q E I} is an orthonormal basis of S2(H), we have by (7.3)" .

Ilxllmin

= sup

(2: Ilaypq(3I1~)

1/2

p,q

= sup Iia . x

. (3lls2(e~02H)'

Introduction to Operator Space Theory

130

•

where the supremum is as above.

Lemma 7.4. Consider a, bin MN. We denote IIall4 = (trlaI 4)1/4 and IIbl14 = (trlbI 4)1/4. Let Ma,b: MN --+ sf)! be the operator defined by 1\fa,b(X) = axb, and let v: OHn --+ MN be any map. We llave tllen (note: here by sf)! we mean (Sf)!)oh) (7.12)

IIMa.bVIIHS :::; IIvll cbll a l1411bl1 4 IIMa,bllcb :::; IIal1411blk

(7.13)

More generally, the same estimates hold on any infinite-dimensional Hilbert space H, with Ma,b: B(H) --+ S2(H) defined in the same way with a, b in B(H) such tllat trlal 4 < 00, trlbl 4 < 00, and for any c.b. map v: OH --+ B(H). Proof. Let (Ti) be an orthonormal basis in OHn . We then have

IIMa,bVllHS = (l: IIMa,bV(Ti) 112) 1/2 = (l:

Ilav(Ti)bll~) 1/2.

Hence, by (7.3)" and (7.10), :::; IIall411bll411l: V(Ti) ®

V(Ti)ll~i: : :; Ila11411b11411vllcb.

This establishes (7.12). To verify (7.13), consider x = L, eij ®Xij in Mn(MN). We have (I®Ma,b)(X) = L,eij ®axijb. Hence, by (7.11), II (I ® Ma ' b)(x)IIMn (SN) = sup {11(a ® a) . ' " eij ® Xij . (/3 ® b)11 2 ~ S2(t 2ot f) x IIall4 :::; 1, 11/3114 :::; 1 } , which yields

:::; IIa®aI1411l:eij®Xijll Thus we obtain (7.13).

Af.,0mln M N

II/3®bI14:::; IIxILu,,(MN)II a ll41I blk

•

Finite-dimensional estimates. By a classical result due to Fritz John (1948), for any n-dimensional Banach space E, there is an isomorphism u: e~ --+ E such that IIuli = 1 and Ilu-111 :::;..;n. Equivalently, d(E, e~) :::; ..;n. In John's original proof, the mapping u is selected so that u(Be2 ) is the maximal volume ellipsoid included in BE. One can also majorize the projection constants: If E is an n-dimensional subspace of a Banach space X, it is known that there is a linear projection P: X --+ E with norm IIPII :::; n 1/ 2 (Kadec-Snobar, 1971; see IKTJl-2] for more references and more recent information) . In modern Banach space theory (see IP8, TJ1]), the following lemma due to Dan Lewis has turned out to be a very useful generalization of John's idea.

7. The Operator Hilbert Space

131

Lemma 1.5. Let E be an n-dimensional Banach space and let 0: be any norm 011 the space L(£2' E) of all linear maps from £2 to E. For any v: E ---> 1'2' we define o:*(v) = sup{ltr(vw)11 w: c~ ---> E, O'(w) ~ 1}.

Then there is an isomorpllism 11: Proof. Choose u:

£2

I det(u)1

--->

£2

--->

E with n (u)

E sucll that 0:( 11) = 1 and 0:* (11- 1 ) = n.

= 1 such that

= sup{1 det(w)11

'W:

1'~ --->

E. n(w)

~

1}.

Clearly det(u) -:I 0. By homogeneity, for any z in C and for any with DeW) ~ 1, we have

11':

f2

--->

E

Idet(u + zw)1 ~ Idet(u)I(1 + Izl)n. Hence, letting Izl

--->

0, we find

Idet(I + zu-lw)1 ~ (1 + Izl)n

=

1 + nlzl + o(lzl),

and since det(I + zu-l.w) = 1 + z tr(u-lw) +o(lzl), we obtain Itr(u-lw)1 ~ n .

•

Remark. If 0: is invariant under the unitary group U(n) (i.e., if we have O'(uw) = O'(u) for any n x n unitary matrix w), then the isomorphism u appearing in Lemma 7.5 is unique modulo U(-n); that is, if u\: f2 ---> E is another isomorphism with the same property, we must have u - \ U 1 E U (n). (In geometric language, the corresponding ellipsoid is unique.) Indeed, let U- l 11l = b. Let b = wlbl be the polar decomposition. By the choice of u ("maximal volume") we know that Idet(udl ~ Idet(u)I and hence I det(b)1 = Idet(lbl)l ~ 1. Therefore the eigenvalues AI, ... , An of Ibl satisfy IIAi ~ 1. Bllt, on the other hand, b- l = u 1l u, and hence L Ail = tr(b- l ) ~ 0'(u)a*(u 1l ) ~ n. Hence we must have (since the geometric mean of Ail is equal to its arithmetic mean) Ai = 1 for all i, which means that U-l.Ul is unitary. • Theorem 1.6. For any n-dimensionaJ operator space E and any E > 0, tIl ere is an isomorphism u: OHn ---> E with 1i11llcb = 1 such tlwt u- l : E ---> OH" admits for some integer N a factorization of tIle form E ~ ~MN ~O H n with Ilvlllcb = ..;n and IIv2licb < 1 + c. Proof. By the preceding lemma, applied with O'(u) = 1111llcB(OH".E)o there is an isomorphism u: OHn ---> E with Ilulicb = 1 and 0'*(u- 1 ) = n. In other words (see Chapter 4), the o.s. nuclear norm of u- 1 is equal to n. Therefore,

132

Introduction to Operator Space Tl1eory

for any 10 > 0 there is an N and a factorization of u -1 of the form u -1 as follows: VI II T sN W OHn u -1 : E -----+ 1. N -----+ 1 -----+

= WTV1

with IIv111cb = n 1/ 2, Ilwllcb < n 1/ 2(1 + c), and with T: MN ~ sf defined by T(x) = axb with a, b in the unit ball of sf. If we wish, we may as well assume (by the polar decomposition) that a ~ 0 and b ~ O. Then, we define T 1: MN ~ sf and T 2: sf ~ sf by T 1(x) = a 1/ 2xb 1/ 2 and T 2(y) = a 1/ 2yb 1/ 2, so that T = T 2T1. Consider the composition T1V1: E ~ sf. By Lemma 7.4, we have (7.14)

Let E2 = T1V1(E) c sf. Clearly, dimE2 = n (since u bijective implies T1V1 injective). Thus we may (and will) identify E2 with OHn . Let P: sf ~ E2 be the orthogonal projection. Clearly (see Proposition 7.2(iii)), 1IPlicb = IIPII = l. Recapitulating, we have

Now let W: E2 ~ OHn be the restriction of wT2 to E2 and let V PT1V1: E ~ E2. We may then write

u- 1 = WV, and V, Ware both invertible. By Lemma 7.4 (note that, with the proper identification, T:; = T 1 ) we have

and by (7.14) we also have

We claim that this implies IIW* - VullHS < fn(c)

(7.15)

with fn(c) ~ 0 when 10 ~ O. Indeed, using the scalar product of the HilbertSchmidt norm, namely, (x, y) = tr(xy*), the identity tr(WVu) = tr(u- 1u) = n means that (W*, V u) = n. Therefore we have IIW* - Vullhs

= IlVullhs + IIW*lIhs -

2 Re((W", Vu})

< n+n(l +10)2 - 2n,

from which the above claim becomes obvious. Since W(Vu) = I, (7.15) implies

7. The Operator Hilbert Space

133

hence, a fortiori IIWW* - III < f~(c:) with f~(c:) ----+ 0 when c: ----+ o. Therefore, a look at the polar decomposition of lV shows that there is a unitary operator U: E2 ----+ OHn such that

IITV - UII < f~(c:) with f;:(c:)

----+

0 when c:

----+

o.

We can now finally conclude: We define V2: AIN ----+ OHn by '1)2 = lVPTI. Note u- 1 = V2Vl. By Lemma 7.4, we have IITlllcb ~ 1; on the other hand, by Proposition 7.2(iii) and the preceding bound,

IITVllcb = IIWII < 1 + f~(c:)· Thus we finally obtain

• Corollary 7.7. Let E c B(H) be any n-dimensional operator space. There are an isomorphism u: OHn ----+ E with Ilulicb = 1 and Ilu- 1 l1cb ~ fo alld a projection P: B(H) ----+ E with 1IPlicb ~ fo. 111 particular,

Moreover, for any c: > 0, there is an integer N and a subspace that dcb(E, E) ~ (1 + c:)fo.

E cAIN such

Proof. By the preceding statement, for any c: > 0, there is a map Ue: OHn ----+ E with Iluelicb = 1 and lI(u e)-lllcb < fo(l+c:). By norm-compactness, the net (u e ) admits a cluster point U when c: ----+ o. Clearly, U has the desired property. Note that, by the extension property of A[N, the mapping VI appearing in Theorem 7.6 admits an extension 11t: B(H) ----+ MN with IIvdlcb = Ilvll1cb. Therefore, we may assume that (u e )-1 admits an extension Ve: B(H) ----+ OHn with IIvelicb < fo(l + c:). The mapping P e = UeVe: B(H) ----+ E is then a projection with IIPelicb ~ fo(l + c:). Taking a cluster point of (Pe ) for the pointwise convergence on B(H), we obtain a projection P with the desired property. Finally, with the notation in Theorem 7.6, let E = 'lh(E) C M N . Then the restriction of UV2 to E is the inverse of VI: E ----+ E, and hence dcb(E, E) ~ IIvIilcblluv211cb ~ IIVl11cbllv211cb ~ (1 + c:)fo. •

Remark. Fritz John's result can be recovered from Corollary 7.7 as a special case: Indeed, for any n-dimensional normed space E we have d( E, e~) ~ dcb(min(E), OHn) ~ fo.

134

Introduction to Operator Space Tlleory

Remark. For any operator space X C B(H), we define the c.b. projection constant of X as follows:

Acb(X) = inf{llPllcb

1

P: B(H)

-+

X, projection onto X}.

By Corollary 1. 7, this is invariant under a completely isometric isomorphism. With this notation, Corollary 7.7 implies that any n-dimensional operator space E satisfies Acb(E) :S In particular, Acb(OHn ) :S Surprisingly, however, this estimate is far from optimal. Indeed, Marius Junge [J2] proved quite recently that Acb(OHn ) was O( VnjLog n) when n -+ 00, and the paper [PiS] already showed that Acb(OHn ) was not asymptotically smaller. Thus, there is a constant J( > a such that, for any n, we have

,;no

,;no

See [KT Jl-2] for recent results on the Banach space analog of n-dimensional projection constants. Theorem 7.8. Let Tn: Rn -+ C n be tlle linear mapping taking For any n-dimensional operator space E, tllere is a factorization

eli

to

eil'

,;no

of Tn tllrougll E witllllwllcb = 1 and IIvllcb = Moreover, tllis is unique in tlle sense tllat if VI, WI is anotller sucll factorization, tllere is a unitary U on e~ c::: Rn sucll tllat WI = wU. Proof. We will apply Lemma 7.5 with o(w) = ilL eil 0 w(ei)lIcn®h E' Given in E*, let v: E -+ e~ be the mapping defined by v(e) = L~ ~i(e)ei' Then, by (5.14), since C~ c::: R n , we have o*(v) = ilL eli 0 ~iIlR"®I,E" By definition of 0h (and by homogeneity), o*(v) :S n iff we can write

6, ... '~n

,;n

with aik E C, T/k E E* such that (Li k laik 12) 1/2 :S and ilL ekI 0 T/k II min = Now let u: e~ -+ E and u . .\ e~ -+ E be as in Lemma 7.5. Let w: Rn -+ E be defined by w(eli) = u(ej). By (2.3.2) and Exercise 5.3, we have IIwllcb = o(u) = 1. Let~; E E* be such that u-I(e) = L~i(e) 0 ei' Let a = (aik) and (T/k) be as above. We then have

,;no

Vi

~i = Lk aikT/k·

By polar decomposition, we may assume without loss of generality that a is hermitian 2': O. Let v: E -+ en be defined by v(e) = LT/k(e)ekI (e E

7. The Operator Hilbert Space

135

E).

Then Ilvllcb = IILek101Jkllmin =.;n. Hence, by (1.5), Ilvwllm, = IlvwllcB(R",C,,) ~ IIvllcbllwllcb ~ .;n. Note that u-1u = I implies ~i(Xj) = Oij' Hence Lk aik1Jk(Xj) = oij. Let bkj = 1Jdxj). We then have b = a-I but also n = tr(ab) = (a*,b) and IIul12 ~.;n, IIbl1 2 ~ .;n. Thus n- 1/ 2b norms n-l/2a* in S"2. Hence we must have b = a*. Since b = a-I, a must be unitary, and since a ~ 0, we must have a = I. Thus we conclude that r,dxj) = Okj, so that vw = Tn. The unicity follows from the rem,ark appearing after Lemma 7.5 . •

Notation. In the direct sum RnEBCn , we consider the vectors c5i =. eliEBe;l. In the style of §2. 7, we will denote by Rn n C n the linear span of {c5i 11 ~ i ~ n}. More generally, let E be any operator space. We will denote by E n EOP the operator subspace of E EB EOP formed of all vectors of the form e EB e (e E E). Similarly, we may consider the mapping Q: E EBl EOP -+ E taking (x, y) to x + y. We will denote by E + EOP the operator space (E EBI EOP)/ ker(Q). Corollary 7.9. ([Jl]) For any n-dimensional operator space E, dcb(E, Rn n Cn) ~ 2.;n. Proof. We apply Theorem 7.8 to the operator space E factorization· Tn: Rn~E+EoP~Cn

+ EOP:

We have a

with Ilwllcb = 1, Ilvllcb = .;n. Since E -+ E + EOP and EOP -+ E + EOP are completely contractive, we have Ilv: E -+ Cnllcb ~ .;n and Ilv: E -+ Rnllcb = IIv: EOP -+ Cnllcb ~ .;n and hence Ilv: E -+ Rn n Cnllcb ~ ..;n. In the preceding, we of course think of R n , C n , and Rn n Cn as having £2 as commOI1 underlying vector space, via the obvious identifications. On the other hand, let w(eli) = Xi E E. Since Ilw: Rn -+ E + EOPllcb ~ 1, we can write Xi = ai +b; (ai, bi E E) with IlL eil 0 aillc,,0m;n E ~ 1, ilL: eil 0 bill c,,0n,;u E "" ~ 1. Hence, if we assume E c B(H), we have IIL:a;adI 1/2 ~ 1 and IIL:bibiII1/2 ~ 1. Therefore, for any Ci in J( we can write

hence, by (1.11),

~ IlL aiai 1 1/211L c;cill '/2+ IlL bibiII1/21ILaia;II'/2 ~ 211Lc5; 0 Cill, and we conclude IIw: Rn n C n -+ Ellcb ~ 2. Thus if we define u(8;) = w(eh), we find lIu: Rn n Cn -+ Ellcb ~ 2 and we have Ilu-l: E -+ Rn n Cnllcb = II v: E -+ Rn n Cnll cb ~ .;n, so that finally Ilullcbliu-11lcb ~ 2.;n. •

Complex interpolation. After we observed the existence of OH(l), we developed in [PI] the theory of these spaces in analogy with what is known in Banach space theory. In these developments, complex interpolation has played a crucial role. For instance, we have

136

Introduction to Operator Space Theory

Theorem 7.10. Let E be an arbitrary operator space equipped with a continuous injection i: E* -+ E with which we may view the couple (E*, E) as compatible. Assume tllat i is ''positive definite," that is to say, such that i(~)(e) ~ 0 for all in E*. Then (E* , E)! is completely isometric to 0 H (1) for some index set I.

e

Proof. In the first part of the proof we treat E as a Banach space. We denote Eo = E* and El = E. Note that, by Lemma 2.7.2, we have an isometric identity (7.16) Let Fo

= E and Fl = E*.

We apply Lemma 2.7.5 to the sesquilinear mapping

u: Eo n El x Fo n Fl

-+

C

defined by u(~, i(r;» = i(~)(1J) or, equivalently (since ~ and i(~) are identified by the "compatibility"), u(i(~),1]) = i(~)(1J). Thus,

lu(i(~), 1])1

::;

Ili(~)IIEII1JIIE·.

Moreover, by the polarization identity, positive definiteness implies selfadjointness, that is, we have i(~)(1J) = i(r;)(e). It follows that, if x, y belong to Eo n El = Fo n F 1 , we have simultaneously

(Note that we do not assume that i is contractive.) Let X = (Eo,Elh/2 = (E*, Eh/2. Note that by symmetry X = (Fo, F 1 h/2. By Lemma 2.7.5, u . extends to a contractive sesquilinear form (still denoted by u) on X x X. Hence we have v x,y E X I'l/(x, y) 1::::; Ilxlix lIyllx. But for all x = i(~) in E* we have u(x, x) = i(~)(e) ~ o. Hence, since i(E*) is dense in X, we have u(x,x) ~ 0 for all x in X, so that u is a scalar product on X. Taking x = y, we obtain VXEX

U(X,X)1/2::;

IIxllx.

(7.17)

We now claim that we have a "natural" isometric embedding X -+ X*. Indeed, by (2.7.6), we have an isometric embedding of (Eo,Eih/2 into X* or, equivalently, of (Eo, Eih/2 into X*. But Eo = E** and Ei = Eo; hence (7.16) ensures that (Eo, Eih/2 = X and we indeed obtain Xc X* isometrically. In particular,

v x = i(~) E i(E*) IIxlix = sup{l(e,y}11 y E i(E*) IIYllx ::; I}.

7. The Operator Hilbert Space

137

Hence (7.17) implies by Cauchy-Schwarz 'V x E i(E*)

IIxllx::; sup{lu(x,y)11 u(y,y) ::; 1} ::; U(X,X)I/2.

Thus we conclude finally that IIxllx

= U(X,X)I/2,

which shows that X is isometric to a Hilbert space. \Ve now turn to the operator space case. By Theorem 2.7.4 (and the above observation (7.16) that takes care of the bidual E** and replaces it by E) the inclusion X ---+ X* is completely isometric, and since X is Hilbertian by what precedes, it is actually surjective. Thus, by the unicity part of Theorem 7.1, we conclude that X must be completely isometric to OH(I) for a suitable • set I. Remark. See [Watl-2] for a more general discussion of the complex interpolation space (with () = 1/2) between a Banach space and its antidual. Corollary 7.11. We have completely isometrically (R,Ch2

= OH

Remark. Let H be a Hilbert space. Recall (see (1.7)) that we have denoted by He (resp. Hr) the space H equipped with the column (resp. row) O.S.s. In the same vein, we sometimes denote by Hoh the space H equipped with the o.s.s. corresponding to the OH-space of the same (Hilbert ian) dimension as H. \Vith this more canonical notation, the first part of Corollary 7.11 says that we have a complete isometry

More explicitly, we have Corollary 7.12. Fix n with the norm

~

1. Let Ao = B(H)n (resp. Al = B(H)n) equipped

lI(xl> ... ,xn)llo = IIEx;x:III/2 (resp.lI(xI,oo.,xn)III

=

IIEx; Xi

l

l

/

2 ).

Then (Ao,Ad! coincides with B(H)n equipped with the norm

Introduction to Operator Space Tlleory

138

Proof. It is easy to reduce to the case H = 1!.2. Then, by (2.7.3), Corollary 7.11 implies (K ®min Rn,K ®min Cnh/2:= K ®min OHn· On the other hand, for any finite-dimensional operator space E, by Exercise 5.6 we have (isometrically)

(/C ®min E)** := B(1!.2)

®min

E.

Hence, by Theorem 2.7.4, we have isometrically

(B(1!.2)

®min

Rn, B(1!.2)

®min

Cnh/2 := B(1!.2)

which is the content of Corollary 7.12.

®min

OHn,

•

This has been generalized to the case when B(H) is replaced by a general von Neumann algebra ([P7]). These ideas have applications to the study of completely bounded projections P: M -+ N from a von Neumann algebra M onto a subalgebra N C M. See Chapters 15 and 23. Vector-valued Lp-spaces, either commutative or noncommutative. Let (0, J.L) be a finite measure space. The natural inclusion map Loo(J.L) -+ Ll (J.L) allows us to view the pair (Loo (J.L), Ll (J.L)) as compatible in the most classical sense. We feel it is worthwhile to first explain why the classical isometric identity (L oo (J.L),L 1 (J.L))1.2 := L 2(J.L) follows from Theorem 7.10. For any y in Loo(J.L), let
In the operator space case, exactly the same argument gives us (using Exercise 3.2) (min(L oo (J.L)),max(L 1 (J.L))h/2:= L 2(J.L)oh. We now turn to the noncommutative case. Let M be a von Neumann algebra, equipped with a faithful, normal, finite trace r. The space Lp(r) (1 ~ p < 00) is defined as the completion of M with respect to the norm IIXllLp(r)

= (r(lxIP»l/p.

7. The Operator Hilbert Space

139

In the case p = 1, it is well known that Ll (r)* ::: ]I.{ (isometrically) with respect to the duality (y,x) = r(xy) (x E L 1 (T),y E 1II). For any y in 1\.1, let t.py E Ll(T)* be the functional now defined by t.py(x) = T(XY). Let i: Ll(r)* -------> Ll(r) be the linear mapping defined by i (t.py)

= y*.

Again i is clearly positive definite, so that, if the "compatibility" is defined using i, we have

Moreover, since the (linear) mapping Uo: Ll (r)* ~ 111 defined by uo( t.py) = y* is clearly isometric, we may invoke Remark 2.7.8 again and we obtain

where the compatibility of the pair (]I.{, Ll (r)) is now defined using the natural inclusion of 111 into Ll (r). More generally, we have

Clearly, we can extend all this to operator spaces as before. There is however an important point that must be emphasized. \Ve must equip Ll (r) with the appropriate o.s.s., namely, the one for which Uo is completely isometric. Unfortunately, this point is insufficiently explained in [P2] (actually page 37 in [P2] is definitely misleading) and I am very grateful to M. Junge and Z. J. Ruan for pointing this out to me, thus allowing me to correct this "loose end" here. So we want Uo: Ll(T)* ~ 111 to be completely isometric. Equivalently, we want Uo: Ll(r)* ~ 111 (defined by uo(t.py) = y*) to be completely isometric, and since lIJ ::: lIfO P completely isometrically (via the map y ~ y*), this means that we want Ll (r) * ::: lIIop, completely isometrically, via the mapping taking t.py to y. In other words, we must equip Ll(r) with the o.s.s. of (M*)OP. Provided we make this convention (and we will stick to it throughout) everything ticks just as before and Remark 2.7.8 yields

(M, L 1(T)h/2 ::: L 2(r)oh

(completely isometrically).

Although we assumed r finite (for simplicity) in the preceding discussion, the situation is the same if r is only assumed semi-finite, which means that the subspace V = {x EM I T(lxl) < oo} is weak-* dense in AI. The space L 1(T) can then be defined as the completion of V with respect to the norm x ~ r(lxl). Let P be the collection of all projections in M with finite trace. We can still view the couple (wI, Ll (r)) as compatible using the natural inclusions

Me EBpMp pEP

and

Ll(r)

C

EBL1(TlpJ\;/p)· pEP

140

Introduction to Operator Space Theory

Then the preceding discussion remains valid (see [Watl-2j for more information). Consider in particular the case when M = B(£2) equipped with the usual (semi-finite) trace denoted simply "tr". We prefer to denote the resulting noncommutative L 1-space by 8 1 . This is the "trace class," formed of all the compact operators x on £2 such that !lxllsl = tr(lxl} < 00. More generally, if 1 ::; p < 00, when r = tr the space Lp(r) is the Schatten class 8 p of all compact operators x such that trlxl P < 00, equipped with the norm (7.18)

When p = 2,82 is the classical "Hilbert-Schmidt class." The previous discussion leads to

or equivalently, by Lemma 2.7.2, (811 K,h/2 c:= (82 )oh

(completely isometrically),

where the compatibility is meant with respect to the natural inclusion 8 1 C K,.

Remark 7.13. As is well known, we have K,* c:= 8 1 isometrically. In this isometric identity, the duality can be defined by setting either

(x, y)J = tr(xy)

or

(x, y)Il = tr(xty).

In other words, this gives us two distinct isometric identifications between K,* and 8 1 . Note that the resulting two o.s. structures on 8 1 (for which K,* c:= 8 1 becomes completely isometric) depend on the choice of this duality: Each one is the opposite of the other. But since K, c:= K,op (completely isometrically, via x --+ t x ), the resulting operator spaces are completely isometric. In the preceding discussion, we chose to work with (-'.)J, but actually, since we equip 8 1 with the o.s.s. for which (K,*)OP c:= 8 11 the resulting o.s.s. on 8 1 is exactly the same as the one that we would obtain on 8 1 by requiring that the isomorphism K,* c:= 8 1 associated to (., -)Il be completely isometric (which is the convention adopted in [P2, §1]). The preceding identities allow us to equip Lp(r) with an o.s.s. More generally, the author has developed ([P2]) a theory of noncommutative vector-valued Lp-spaces, which extends the so-called Lebesgue-Bochner theory of the spaces

where 1 ::; p ::; 00, (fl, J.l) is a measure space and E is a Banach space. Recall that, in the discrete case, that is, when (fl, J.l) is just the integers (resp.

7. The Operator Hilbert Space

141

= {I, ... , n}) equipped with the counting measure, Lp(Q, p; E) is usually denoted by Cp(E) (resp. C; (E». In this new theory, E has to be equipped with an operator space structure. Then, using interpolation, the spaces

Q

can also be given a natural operator space structure. When p = 2 and E = C this reduces to the theory of the already mentioned space OH.

In the noncommutative case we replace Loo(Q,p) by an injective von Neumann algebra M C B(rt). We assume that l\f admits a faithful normal semi-finite trace T so that, as before, the predual 111* can be identified with Ll (T) in the usual way (cf., e.g., [Di3, Se, Ne, Ta3]). Then let E C B(H) be an operator space, and we set

o (T; E )def Loo = M

@min

E.

Note that, in the commutative case, this space is in general smaller than the classical space Loo(Q, It; E) in Bochner's sense but coincides with it if E or M = Loo(Q, p) is finite-dimensional. The case p = 1 has already been treated by Effros and Ruan, who introduced more generally the analog of the projective tensor product, denoted by @A. We simply set L 1 (T;

E)~fLl(T) @A E.

Then we can define (following [P2])

(7.19) Of course, when E space Lp(T).

= C, the space Lp( T; E) reduces to the previously defined

For example, consider 111 = B(C2) equipped with the usual trace (somewhat discrete) "tr". Then, if p < 00, Lp(T) is the Schatten class Sp of all compact operators such that trlTIP < 00 equipped with the norm defined in (7.18). In this particular case, the space Lp(T; E) is denoted by Sp[E]. The space Sp (resp. Sp[E]) is analogous to a discrete or atomic Lp-space. It is analogous to Cp or C; (resp. Cp(E) or e;(E» in the commutative case. These spaces satisfy all the basic properties of the classical Lebesgue-Bochner spaces Lp(Q, p; E), such as duality, interpolation, injectivity, and projectivity relative to the space of "values" E. The reader should observe that our theory yields an isometric embedding Sp

c B(H)

Introduction to Operator Space Theory

142

that is highly nonstandard! Just as in the case of OB (when p = 2), we have no explicit description of this embedding. We will only give here a brief introduction to that subject and refer the reader to the extensive text [P2] devoted entirely to this topic. To avoid any technicality, we will discuss here only the spaces S;[E]. The spaces Sp[E] behave very much the same, and in fact the embeddings ... c S;[E] c S;+1[E] c ... are completely isometric, so that Sp[E] can be viewed as the completion of the union of the spaces S;[E]. Throughout this discussion, E will be an arbitrary operator space. The normed space S;[E] can be described as follows. It is the same vector space as Mn(E) but equipped with the following norm (1 :::; p < 00): For any x in Mn(E) we set (7.20) where the infimum runs over all possible factorizations of x of the form x = a· x· b with a, bE Mn and x E Mn(E). Actually, the same definition (7.20) makes sense for Sp[E], but Sp[E] is now defined as the subspace of K ®min E formed of all the x that can be written as x = a . x· b with a, b E S2p and x in K ®min E. It is natural to denote Soo = K and Soo[E] = K ®min Fj clearly the elements ofSoo[E] can be viewed as bi-infinite matrices with entries in E. This definition is reminiscent of the obvious fact that (

L

IIYnll~ )

l/P

= inf{llclle"lIylle",,(E)}

over all factorizations of the form Yn = cnYr;. But here the noncommutativity forces us to consider separately both left and right multiplications, and thus the space S!jp makes its appearance (note that Ilells;; :::; 1 iff c can be written c = ab with Iialls"2p :::; 1 and Ilblls"21) < 1). If x E Spn[E] is a diagonal matrix, then we find simply Ilxlls;:[Ej isometrically.

= (I: Ilxiill~)l/p, so that e;(E) c S;[E]

In the case p = 1, the norm appearing in (7.20) coincides with the norm of Sf ®" E as defined in Chapter 4. Therefore we have Sf[E]* = Mn(E*) isometrically. More generally (see [P2j) we have, for any 1 < p < 00, S;[E]* = [E*] isometrically. Equivalently, we have another description of the norm of S;[E] as follows:

S;,

IIxlls;:[Ej

= sup{lIa· X· bllsf[Ej},

(7.21)

where the supremum runs over all a, b in the unit ball of S!Jv. Thus, with (7.20) and (7.21), we have two a priori different ways (dual to each other) to look at the spaces S;[E] (or Sp[E]), and this is (to a large

143

7. The Operator Hilbert Space

extent) the reason why their theory works so nicely. Note that (7.20) is projective (if E is a quotient of F, then S;[E] appears as a quotient of S;[F]) while (7.21) is injective (if E c F, then S;[E] c S;[F]). Having defined the norm in the spaces S:[E], we can now define their o.s.s. as follows: For any x in .lI1n (S:[E]) we define Ilxll n = sup{lla· x· blls;:N[EJ}, where the supremum runs over all a, b in the unit ball of S~p and where we have identified .lI1n (.lI1N (E)) with .lI1nN (E). Then it can be checked that this sequence of norms satisfies Ruan's axioms (Rt) and (R2); hence it defines an o.s.s. on St'[E] (and also on Sp[E]). With this structure, if 1 < P < 00, we have (see [P2]) completely isometrically:

Since e;(E) c S;[E] (and epee) c Sp(E]), the preceding structure induces an o.s.s. on e;(E) and epee). More generally, if (0,11) is any measure space, we can equip Lp(O, 11; E) with an o.s.s. by introducing the following norm on .lI1n (Lp(O,/I;E)): Viewing an element x in 1IIn (Lp(0,/L;E)) as one in Lp(O, JL; Mn(E)), we let (7.22)

Ilxll n = sup{lla· x· bll L,.(fl.JL:S;: [E])}, where the supremum runs over all a, b in the unit ball of

S~p.

The resulting o.s.s. on Lp(O, JL; E) is called the natural one. It coincides with the one described above using complex interpolation in (7.19). The introduction of these vector-valued (actually operator space-valued) noncommutative Lp spaces opens the way for noncom mutative analogs of a number of natural questions originating in either Banach space theory, classical analysis, or probability. To name a few of those: \Ve can study uniform convexity and the Radon-Nikodym property, maximal inequalities for martingales or the unconditionality of martingale differences (U1'lD) for operator spaces. One can also ask which Fourier (or Schur) multipliers are com.pletely bounded on L p , and this is of interest already for the most classical groups such as Z or '['. We will return to this subject in §9.5, and we illustrate it with "concrete" examples in the subsequent §§9.6-9.9. See [P2] for a full treatment. To give only a sample result, recall that, among Banach spaces, Hilbert space is characterized by the Jordan-von Neumann (parallelogram) inequality

\/x,Y

E

E

144

Introduction to Operator Space Theory·

In other words, if we denote by

T: £~(E)

---->

£~(E)

the operator defined by

then E is isometrically Hilbertian iff IIT1! ::; 1. In our theory, if E is an arbitrary operator space, the space £~(E) (denoted by E(f32E in Remark 2.7.3) is also equipped with a natural operator space structure (this is a particular case of (7.22), with (0, /-l) a two-point space). For that structure we have

Theorem 7.14. set I.

IITllcb::;

1 iff E is completely isometric to OH(I) for some

We end this chapter with a surprising estimate (improving a bound in Theorem 3.8) that uses the space OH. This is due to Eric Ricard.

Theorem 7.15. ([Ril}) Let E be any n-dimensional operator space. Then

IIIE: min(E)

--+

max(E)lIcb ::; n/2 1/ 4 •

Equivalently, a:(n)::; n/2 1/ 4 • Proof. By a well-known result on 2-summing operators (cf., e.g., [P4, p. -1

15)), IE can be factorized as E~£2~E with 1T2(U) By a well-known estimate, we have lIuIlCB(min(E),R,,) ::;

1T2(U) =

= 1 and

lIu- 1 11

= .;n.

1,

and similarly IluIICB(min(E),C,,) ::; 1;

hence

On the other hand, the identity map on £'2 (denoted below by I) defines a completely contractive inclusion I: Rn n en --+ OHn. This follows from Corollary 7.11, but it can also be verified directly as follows. Let (Ti) denote an orthonormal basis in OHn . Then, for any n-tuple (Xi) in B(£2), we have by (7.5) and (2.11.3)

7. The Operator Hilbert Space

145

hence a fortiori

which proves our claim that III: RnnCn -> OHn Ilcb ~ 1. Furthermore, we also have (by (7.9) and Exercise 3.2) III: OHn -> max(£2)llcb = III: min(t2) -> max(£2)11!~2. Hence, by Theorem 3.8

Finally, we have lin-I: max(£2) conclude

->

max(E)llcb

=

lin-III

• Corollary 7.16. ([Ril}) An'y linear mapping spaces E, F with rank ~ n satisfies

11:

E

->

F between operator

The best constant (instead of 2- 1/ 4 ) in the preceding bound is not yet known.

Remark. An important and classical fact in Banach space theory is that £2 embeds isometrically into L 1 . This embedding can be obtained by mapping an orthonormal basis to a sequence of independent complex Gaussian random variables (see §9.8). It is natural to wonder about the o.s. analog of this fact for the space OH. This question was recently solved by l\'1arius .Junge [J2]. Curiously, we cannot use Gaussian variables, either "classical" or "free" (see §§9.8 and 9.9) to embed OH completely isomorphic ally into an L)-space, either commutative or not. Nevertheless, Junge [.J2] proved that OH embeds completely isomorphically into the predual 11[* of a von Neumann algebra 11[. The completely isometric case remains open (at the time of this writing). Actually, Junge proved more generally that any quotient of RtBC embeds into l'vf* for some 11[. Since, by Exercise 7.9, OH embeds completely isometrically in a quotient of R tB C (this fact has been known for some time independently to Junge and the author), we recover the embedding of OH as a corollary. Junge's construction uses for 11/ a von Neumann algebra "of type III," that is, one that does not admit any semi-finite and faithful trace. After learning of Junge's results, we were able to show (using [PiS]) that "type III" cannot be avoided: If OB C M* completely isomorphically, then 1I[ cannot be semi finite (see [P22]).

146

Introduction to Operator Space Theory Exercises

Exercise 7.1. Let E isometrically.

= OH(I). Prove that E

~

EOP and E

~

E completely

Exercise 7.2. In a suitable realization of OH(1) there is an orthonormal basis «()i)iEI of OH(J) formed of self-adjoint operators. Exercise 7.3. For any orthonormal basis (Ti)iEI of OH(1) (with have

III =J 0)

we

~~~ III:Ti ® Til = 1.

IJI
tEJ

Exercise 7.4. Let E = OH(1). Show that Ilull = Ilulicb for any u: E ---- E, so that C B (E, E) ~ B (£2 (1» isometrically. Show that this identity is not completely isometric. Exercise 7.5. Let E have

= OH(1).

IIxIIM,,(E)

=

Show that for all x

= (Xij)

in Mn(E) we

11«(xij,Xke)lli~~2

or, more precisely,

I: (Xij' Xke)eij

1/2

® eke

i,j,k,e

min

Exercise 7.6. Let E be any operator space. Then, for any hI"'" hn in OH and any Xl, ... ,Xn in E, we have 1/2

II I:h'®X·11 OH0mlu E t

t

i.,j

min

Exercise 7.7. Let (Tlo ... , Tn) be an orthonormal basis in OHn . Compute and

IIL:

n

1

eli

® ~II

R ... 0mln OH"

.

Let u: Cn ---- OHn (resp. v: Rn ---- OHn) be any isometric map. Show that

Ilulicb = IIvllcb = nl / 4 • Exercise 7.8. Let E C B(H) be any n-dimensional operator space. For any x = (Xi)i
7. Tile Operator Hilbert Space

147

Then we define for 0 < () < 1 IlxllE(I;l)

=

Il x ll(E(O),E(1))n'

Show that there is a basis (Xi) in E with biorthogonal basis that

(~i)

in E* such

In particular, for () = 1/2, deduce from this a new proof that dcb(E, OHn) :::; y'n. Hint: Use Theorem 7.8 applied to (E, EOP)o, like in the proof of Corollary 7.9. Exercise 7.9. Show that OH is completely isometric to a quotient of a subspace of R EB C.

Chapter 8. Group C*-Algebras. Universal Algebras and Unitization for an Operator Space We first recall some classical notation from noncommutative abstract harmonic analysis on an arbitrary discrete group G (we restrict ourselves to the discrete case for simplicity). Let 7l': G --+ B(Ji) be a unitary representation on G. We denote by C;(G) the C*-algebra generated by the range of 7l'. Equivalently, C;(G) is the closed linear span of 7l'(G). In particular, this applies to the so-called universal representation of G, a notion that we now recall. Let (7l'j)jEI be a family of unitary representations of G, say,

in which every equivalence class of a cyclic unitary representation of G has an equivalent copy. Now one can define the universal representation Uc: G--+ B(Ji) of G by setting Uc

= tBjEI7l'j on Ji = tBjE1Hj .

Then the associated C* -algebra CUG (G) is simply denoted by C* (G) and is called the C*-algebra of the group G (so this is the closed linear span of {Uc(t) I t E G}). We will sometimes call it the full C* -algebra of G to distinguish it from the reduced one, which is described in the following. Let 7l' be any unitary representation of G. By a classical argument, 7l' is unitarily equivalent to a direct sum of cyclic representations; hence, for any finitely supported function x: G --+ C, we have

Equivalently,

IlL x(t)7l'(t) I ~ IILX(t)Uc(t)lI· ilL x(t)Uc(t) II = sup {IILX(t)7l'(t)II},

(8.1)

where the supremum runs over all possible unitary representations 7l': G --+ B(H) on an arbitrary Hilbert space H. More generally, for any finitely supported function x: G --+ B(Ji) we have

where the sup is the same as before. These formulas explain why C* (G) is sometimes called the "maximal" C*-algebra of G. We denote by

AC: G --+ B(£2(G))

[resp.

Pc: G --+ B(£2(G»]

the left (resp. right) regular representation of G, which means that AC(t) (resp. pc(t) is the unitary operator of left (resp. right) translation by t

8. Group C*-Algebras. Universal Algebras and Unitizatioll

149

(resp. t-I) on £2(G). Explicitly, if we denote by (OdtEC the canonical basis of £2(G), we have Ac(t)Os = Ots (resp. pc(t)os = 0st-1) for all t, s in G. Actually, we will often drop the index G and denote simply by U, A, p the representations Uc, Ac, Pc. We denote by CA(G) (resp. C;(G)) the·C*-algebra generated in B(£2(G)) by Ac (resp. pc). Equivalently, CA(G) = span{A(t) I t E G} and C;(G) = span{p(t) I t E G}. Note that A(t) and p(s) commute for all t, s in G. \Ve will study the maximal tensor product of two C* -algebras AI, A2 in more detail in Chapter 11. For the moment we will content ourselves with its definition. For any x in Al Q9 A 2, we define

Ilxll max = sup 117r(x)IIB(H)l where the supremum runs over all possible Hilbert spaces H and all possible *-homomorphisms 7r: Al Q9 A2 ---> B(H). The completion of Al Q9 A2 with respect to this norm is a C*-algebra called the maximal tensor product of Al and A2 and is denoted Al Q9max A 2. Let G I, G2 be two discrete groups. It is easy to see (exercise) that C*(GJ)

Q9max

C*(G 2) c::= C*(G I

G 2),

(8.3)

CHGJ)

Q9min

C A(G 2 ) c::= CHG 1 x G 2),

(8.4)

X

and similarly for the free product G I * G 2 (the free product of C* -algebras is defined before Theorem 5.13):

These identities can be extended to arbitrary families pair (G I , G 2 ). In particular, we have

*

iEI

C*(Gd c::= C* (

*

iEI

Gi )

(Gi)iEI

in place of the

(8.5)

.

The following very useful result is known as Fell's "absorption principle." Proposition 8.1. For a.ny unitary representation

AC

Q9

7r c::= Ac Q9 I

7r:

G

--->

B(H), we have

(unitaryequimlence).

Here I stands for the trivial representation of G in B(H) (i.e., I(t) = IH ' B(£2), we have

II L II L

a(t)Ac(t)

b(t)

Q9

Ac(t)

Q9

7r(t)11 = II L

a(t)Ac(t)ll,

Q9

7r(t) II = II L

b(t)

Q9

Ac(t)lI·

Introduction to Operator Space Theory

150

Proof. Note that AG ® 71" acts on the Hilbert space K = f2(G) ®2 H :::' f2(G; H). Let V: K -4 K be the unitary operator taking x = (X(t))tEG to (71" ( t- 1 )x( t) )tEG. A simple calculation shows that V- 1 (AG(t) ® IH)V = AG(t) ®

•

7I"(t).

The next result illustrates the usefulness of this principle.

Theorem 8.2. We have an isometric (C*-algebraic) embedding

J: C*(G)

c

C,\(G)

®max C~(G)

taking UG(t) to AG(t) ® AG(t) (t E G). Moreover, there is a (completely) contractive projection P from C~ (G) ®max C~ (G) onto the range of J.

Proof. Let x: G x G following claim:

Il Lt

X,(t,

-4

C be finitely supported. It suffices to show the

t)UG(t)ll.

::; IlL x(s, t)AG(S) ® AG(t)11

C (G)

s,t

(8.6) max

Indeed, in the converse direction we have obviously

Hence (8.6) implies at the same time that J defines an isometric *-homomorphism and that the natural ("diagonal") projection onto its range is contractive. We will now prove this claim. Let 71": G -4 B(H) be a unitary representation of G. \Ve introduce a pair of commuting representations (71"1,71"2) as follows:

7I"1(AG(t)) = AG(t) ® 7I"(t)

and

7I"2(AG(t)) = PG(t) ® I.

Note that both 71"1 and 71"2 extend to continuous representations on C,\(G). For 71"1, this follows from the preceding absorption principle. For 71"2, it follows from the fact that PG:::' AG (indeed, if W: f2(G) -4 f2(G) is the unitary taking 1St to ISt - l , then W*AG(')W = PGO)· Since 71"1 and 71"2 have commuting ranges, we have

8. Group C* -Algebras. Universal Algebras and Unitization

151

hence, restricting the left side to 8e 0 He t2(G) 02 H, we obtain (note that (Ac(s)pc(t)8 e ,8e ) = 1 if s = t and zero otherwise)

II

::; IILX(S' t)AC(S) 0 Ac(t)11

LX(t, t)7r(t)11 t

s.t

8(H)

max

Finally, taking the supremum over 7r, we obtain the announced claim (8.6). The same argument (using operator valued coefficients (x(s, t)) instead of scalar ones) shows that 1IPlicb ::; 1. • Theorem 8.3. ([BF, Jol]) Let G be a discrete grollp and H a Hilbert space. Tlle following properties of a function cp: G ---+ B(H) are equivaJent: (i) Tlle linear mapping defined on span[A(t) It E G] by M",(A(t)) extends to a c.b. map

= A(t) 0 cp(t)

M",:C~(G) ---+

B(£2(G) 02 H) witllIIM",llcb::; 1.

(ii) Tllere is a Hilbert space ii and bounded fUllctions x: G ---+ B(H, ii) and y: G ---+ B(H,ii) witll SUPtEC Ilx(t)1I ::; 1 and SUPsEG Ily(s)11 ::; 1 sllcll tlwt \;f s,t E G cp(sC I ) = y(s)*x(t). Proof. Assume (i). Then, by Theorem 1.6, there are a Hilbert space ii, a representation 7r: C~(G) ---+ B(ii), and operators V;: £2(G) 02 H ---+ ii (i = 1,2) with IIVIIII1V211 ::; 1 such that (8.7) We define x(t) E B(H, ii) and yes) E B(H, ii) by x(t)h = 7r(A(t- I )) VI (8 t 0h) and y(s)k = 7r(A(s-I))V2(8 s 0 k). Note that (A( 0) 0 cp(O))( 8t 0 h), 8s 0 k) = 1 {6=st-l} (
= (y(s)*x(t)h, k),

and we obtain (ii). Conversely, assume (ii). Define 7r: C~(G) ---+ B(£2(G)0 2ii) by 7r(x) = X0Iii' Let V;: £2(G)02H ---+ f.2(G) 02 ii be defined by VI (8 t 0h) = 8t 0 x(t)h and V2(8 s 0 k) = 8s 0 y(s)k. Then for any 0, t, s, h, k we have W2*7r(A(O))VI(8t 0 h), 8s 0 k)

= (A(O)8t, 8s )(y(s)*x(t)h, k) = (A(O) 0
152

Introduction to Operator Space Theory

hence M
Remark 8.4. Consider a complex-valued function cp: G ~ C. Then the preceding result applied with C = B(H) yields: IIM:(G) ~ C>:(G)llcb ~ 1 iff there are Hilbert space valued functions x, Y with SUPt IIx(t) II ~ 1 and sUPslIy(s)1I ~ 1 such that cp(st- 1) = (y(s),x(t)).

V s,t E G

Proposition 8.5. Let G be a discrete group and let f c G be a subgroup. Then the correspondence )..r(t) ~ )..a(t), (t E f) extends to an isometric (C*algebraic) embedding J: C>:(f) ~ C>:(G). Moreover, there is a completely contractive and completely positive projection P from C>: (G) onto the range of this embedding. Proof. Let Q = G/f and let G = UqEQ fgq be the decomposition of G into disjoint right cosets. For convenience, let us denote by 1 the equivalence class of the unit element of G. Since G ~ f x Q, we have an identification

such that )..a(t)

"It E f

= )..r(t) ® I.

This shows of course that J is an isometric embedding. Moreover, we have a natural (linear) isometric embedding V: £2(f) ~ f2(G) (note that the range of V coincides with £2(f) ® 81 in the above identification) such that

VtEf Let u(x)

)..r(t)

= V*xV. Clearly we have V t

u()..a(t))

= )..r(t)

if

tEf

E

= V* )..a(t)V. G

and

u()..a(t))

=0

if

t ¢. f.

Therefore P = Ju is a completely positive and completely contractive projection from C>:(G) onto JC>:(f). • In the case of the full C* -algebras, the analogs of the preceding results are as follows.

8. Group C* -Algebras. Universal Algebras and Unitization

153

Proposition 8.6. Let G be a discrete group. The following properties of a function cp: G ----> B(H) are equivalent: (i) Tlle linear mapping taking Ua(t) to cp(t) extends to a c.b. map Tep: C*(G) ----> B(H) witll IITeplicb ~ 1. (ii) The "multiplier" taking Ua(t) to Ua(t) 0 cp(t) extends to a c.b. map Afep: C*(G) ----> C*(G) 0 m in B(H) with IIMeplicb ~ 1. (iii) There is a unitary representa.tion 71": G ----> B(H7r) and contractions V, It': H ----> H7r SUell tlwt cp(t) = V*71"(t)W.

Vt E G

Proof. We will show (i) ~ (iii) ~ (ii) ~ (i). Assume (i). Then (iii) follows by an immediate application of the factorization theorem for c.b. maps, namely, Theorem 1.6. Assume (iii). By maximality of C*(G), the mapping taking Ua(t) to Ua(t) 071"(t) extends to a representation on C*(G); hence, since Jllep = (I0V*)[Ua(t)071"(t))(10W), we have IIMeplicb ~ 1110V*II·III0WII ~ 1, whence (ii). Assume (ii). Since the mapping Tl taking Ua(t) to 1 (the unit in q is clearly a representation on C* (G) (associated to the trivial representation on G), and since we have obviously Tep = (Tl 0 J)Mep, we obtain IITeplicb

~

•

1, whence (i).

Corollary 8.7. A function cp: G ----> C*(G) iff tllere are a unitary representation 71": G ----> B(H7r) and ~,'f/ in H7r such that cp(t) = (71"(t)~, 'f/}. AJoreover, we lwve

where the infimum runs over all possible if ~ = 'f/, Afep is completely positive.

71", ~,

'f/ for which this llOlds. Finally,

Proof. The first assertion for c.b. maps and the equality IIMeplicb = inf{II~IIII'f/II} follow from the equivalence (ii) {::} (iii) in the preceding statement, using V, W: H7r defined by VI = 'f/, WI =~. Now, if Mep is bounded, then (with the above notation) Tep: C*(G) ---->
r

1i'(t)H CHand 1I"(t) = 1i'(t)IH.

Here is a quick justification for this fact (see [Riel for a more general framework). Let {Sj I j E /} be a set of elements of G chosen so that G = UjEI Sjr is the decomposition of G into left cosets. For convenience, we assume that / contains a distinguished element, which we denote by 1, such that Sl = e (unit element of G). Let 11"0: G ~ B(H) be the trivial extension of 11" defined for all x in G by 1I"0(x) = {1I"(X) if x E ~ o otherw1se. We let fl

= €ajEI Hj , where each Hj is a copy of H. Then, for any x in G, we

define 1i'(x): fl ~ fl as the operator associated to the matrix (1I"0(S;lXSj»ij with entries in B(H). We then leave as an exercise for the reader to check that 1i' has the' announced properties. Following l\'lackey's classical terminology, 1i' is the representation "induced" by 11". Using (8.1) and the preceding extension property, it is immediate that J is an isometric embedding. Now, let 11"1: r ~ B(C) ~ C be the trivial representation (11"1 == 1) and let 1i'1: G ~ B(H) be the induced representation as above, with C C fl. Let E fl be the unit in C viewed as sitting in fl. Then it is easy to check that the function cp defined by

e

\i t E G

is nothing but the indicator function of r. Therefore, by Corollary 8.7, the multiplier Mcp defines a completely contractive and completely positive projection from C*(G) onto JC*(r). • Let F be a free group with generators (gi)iEI. The first part of the next result is based on the classical observation that a unitary representation 11": F ~ B(H) is entirely determined by its values Ui = 1I"(gi) on the generators; and if we let 11" run over all possible unitary representations, then we obtain all possible families (Ui) of unitary operators. The second part is also well known (see, e.g., [Pa5]).

8. Group C*-Algebras. Universal Algebras and Unitization

155

Lemma 8.9. Let F be a free group. Let (Ui)iEI be the family composed of the unitary generators of C*(F) (i.e., these are the unitaries corresponding to the free generators of F in the universal unitary representation of F). Let (Xi)iEI be a finitely supported family in B(H). Consider the linear map T: €oo{l) --+ B(H) defined by T«O:i)iEI) = LiEf O:iXi. Tllen we have

LUi. ® Xi iEI

= IITllcb = sup{11 LUi ®

xill

rnill },

(8.8)

C, (F)0minB(H)

where tIle sup runs over all possible families of unitaries (Ui). Actually, the latter supremum remains the same if we let it run only over all possible families of finite-dimensional unitaries (Ui). lIJoreover, if dim( H) = 00, tllen

IILUi®Xill tEl

=inf{IILYiy;111/21ILZ;Zilll/2},

(8.9)

C*(F)0min B (H)

where the infimum that runs over all possible factorizations Xi = YiZi with Yi, Zi in B(H) is actually attained. Moreover, all tllis remains true if we enlarge the family.(Ui)iEI by adding tIle unit element ofC*(F). Proof. It is easy to check by going back to the definitions that, on one hand

where the sup runs over all possible families of unitaries (ud, and on the other hand that

where the sup runs over all possible families of contractions (td. By the Russo-Dye Theorem, any contraction is a norm limit of convex combinations of unitaries, so (8.8) follows by convexity. Actually, the preceding sup obviously remains unchanged (see Exercise 2.1.1) if we let it run only over all possible families of contractions (t i ) on a finite-dimensional Hilbert space. Thus it remains unchanged when restricted to families of finite-dimensional unit aries (Ui). Now assume IITllcb = 1. By the factorization of c.b. maps we can write T(o:) = V*n(o:)W, where n: €oo{l) --+ B(H) is a representation and \vhere V, Ware in B(H, H) with 1IVIIIIWil = IITllcb. We can assume I finite and, since dim(H) = 00, H = H. Let (ei)iEI be the canonical basis of €oo(I). We set

156

Introduction to Operator Space Theory

It is then easy to check IILYiYilil/21ILZizilil/2 ::; IIVIIIIWII = IITllcb. Thus we obtain one direction of (8.9). The converse follows from the inequality

(already proved in Remark 1.13) applied to bi = Ui 0 Yi and ai Finally, the last assertion follows from the next remark.

=

1 0 Zi. •

Remark. Let {O} be a singleton disjoint from the set I and let J = I U {O}. Then, for any finitely supported family {Xj I j E J} in B(H) (H arbitrary), we have (8.10)

where the supremum runs over all possible families (Uj )jEJ of unitaries. Indeed, since

L Uj 0 x j jEJ

=

III

0 xo +

LU01~i 0Xil , iEI 1110

Xi

the right side of (8.10) is the same as the supremum of Xo + LiEI Ui 0 II over all possible families of unitaries (Ui)iEI, but (recalling U(gi) = Ui ) the latter supremum coincides with

I I0 XO+ LUi0Xill. tEl mm I

•

Remark 8.10. Let F be a free group with free generators (gi). Then, for any finitely supported sequence of scalars (ai)' for any H, and for any family (Ui) of unitary operators in B(H), we have

Indeed, this follows from the absorption principle (Proposition 8.1) applied to the function a defined by a(gi) = ai and = 0 elsewhere, and to the unique unitary representation1l" on F such that 1l"(gi) = Ui· We now return to the unital operator algebra of an operator space E, which we introduced in Chapter 6. Recall that OAu(E) is defined as the completion of the unital tensor algebra Tu(E) for the operator space structure determined by (6.3). Moreover,

8. Group C* -Algebras. Universal Algebras and Unitization

157

OA(E) is the subalgebra of OAu(E) formed by all elements with vanishing constant term.

By construction, the operator algebra OA(E) (resp. OAu(E)) is characterized by the following universal property (recall that "completely contractive" is abbreviated by c.c.): For any c.c. map a: E ---> B(H), there is a unique c.c. morphism &: OA(E) ---> B(H) (resp. unital morphism &: OA,,(E) ---> B(H)) extending a. This characterizes OA(E) (resp. OA,,(E)) lip to a completely isometric isomorphism of algebras (resp. unital algebras) among all (resp. unital) operator algebras containing E completely isometrically. The next result is a reformulation of von Neumann's classical inequality. Theorem 8.11. If E = C, then OA,,(E) can be identified completely iso-

metrically with the disc algebra A(D)

= span{zn In 2:: O}

C C(fJD).

Here D denotes the open unit disc in C. Proof. Assume E = C. Let liS denote by z E E c OAu(E) the element corresponding to the unit of C. Then any element in T" (E) can be written as P = ajz0j (aj E q, which we identify with the polynomial P(z) d . 2::0 ajzl. By (6.3) we have

2::g

where the supremum runs over all contractive T in B(H) (H arbitrary). Let peT) = aoIH + 2::~ a j Tj. We now claim

IIP(T)II :::; sup{IP(w)11 w E fJD}.

(8.11)

This is nothing but von Neumann's classical inequality. Here is a quick proof: As already noted, for any contractive T in B(H) there is a unitary V in lIh(B(H)) c::: B(H EB H) such that T = Vll . Indeed, we simply set

V = ( -(1 _

~*T)I/2

(1-TT*)1/2) T* .

Let j and p denote, respectively, the inclusion and the projection corresponding to the first coordinate in H EB H. Then T = Vll can be rewrit.t.en as T

=p

(1 0) (1 0). 0

0

V

0

0

),

Introduction to Operator Space Theory

158

and more generally (since pj = 1), for any polynomial P, we have trivially

P(T) = pP(U(O))j, where we have set

U(w) =

(~ ~) V (~ ~).

Hence IIP(T)II

= IlpP(U(O))j1l ::;

sup IIP(U(w))II, wED

but since w yields

hence

---+

P(U(w)) is obviously analytic in w, the maximum principle IIP(T)II ::; sup IIP(U(w))II; Iwl=l ::; sup{IIP(U)111 U unitary},

and, by the spectral theorem for unitary (or normal) operators, we obtain our claim: IIP(T)II ::; sup IP(z)l· zE8D

Thus we conclude that 1IPIIOAu(C)

=

IIPIIA(D),

or, equivalently, OAu(lC) ~ A(D) isometrically. Using operator coefficients instead of scalars, it is easy to check that this identification actually is completely isometric. • More generally, we have (cf. [Bol]):

Theorem 8.12. Let I be any set. Let FI be the free group with generators (gi)iEI and let Ui E C*(FI) be the unitary element associated to gi. We denote byei the canonical basis of £1(1). Then the correspondence ei ---+ Ui extends to a unital homomorphism that is a completely isometric embedding ofOAu (max(£l(I))) into C*(FI). Proof. Let E = max(£l(I)). We may as well assume by density that I is finite. Let P E K ® Tu(E). We have P = Ed Pd with Pd E K ® E®d of the form Pd = Ail ... id ® (eil ® ... ® eid)

L

with Ait ... id E K. Then, by (6.3), we have

8. Group C* -Algebras. Universal Algebras and Unitization

159

where the supremum runs over all H and over all families of contractions (Ti)iEI in B(H). By the same t.rick as in the preceding proof, the supremum remains the same if we restrict it to families of unitaries (d. [Bol]). Therefore we obtain, by the observation preceding Lemma 8.9, IIPIIK0".;nOA,,(E) =

IlL .L d

' \ I ... id

® Uil U;2 ...

'·t···l,/

which shows t.hat the correspondence pletely isometric homomorphism.

eit ei2 ... ei,/ --->

ui ,/

I

K0".;n C '(Fil

Uil Ui2

...

'

Ui,J is a com•

Corollary 8.13. Fix d 2': 1. \Vit11 the same notation as before, we lmve a completely isometric isomorphism

Proof. \Ve simply combine the preceding statement with Proposition G.G . • We will now introduce (following [Pes]) the universal C* -algebra of an operator space E, which will be denoted by C*(E). The definition stems from the next statement. Theorem 8.14. Let E be an operator space. There is a C*-algebra A and a completely isometric embedding j: E ---> A wit11 the following properties: (i) For any C*-algebra B and any completely contractive map u: E ---> B there is a representation 11": A ---> B extending u, tl18t is, such that 1I"j = u. (ii) A is the smallest C*-algebra containing j(E).

Moreover, (ii) ensures that the representation

11"

in (i) is unique.

Proof. The proof is immediate. Let I be the "collection" of all u: E with lIullcb : : : 1. Let

We define j: E

--+

--->

BlI

B by

j(x)

= EBu(x). uEI

It is easy to check that j is completely isometric. Then, if we define A to be the C*-algebra generated by j(E) in B, it is very easy to check the announceo • universal property of A.

160

Introduction to Operator Space Theory

Notation. We will denote by C*(E) the C*-algebra A appearing in the preceding statement, and we denote by C~(E) its unitization. Note that C*(E) is essentially unique. Indeed, if jl: E -> Al is another completely isometric embedding into a C* -algebra Al with the property in Theorem 8.14, then the universal property of A (resp. AI) implies the existence of a representation 11": A -> Al (resp. 11"1: Al -> A) such that 1I"j = jl (resp. 1I"dl = j). Since C* -representations are automatically contractive, we have 1111"11 ::; 1, 111I"tII ::; 1, and 11"1 = 11"-1 on Jt(E) and hence on the *-algebra generated by jl(E), which is dense in At, by assumption. This implies that 11" is an isometric isomorphism from A onto AI. • Similarly, C~ (E) is characterized as the unique unital C* -algebra C containing E completely isometrically in such a way that, for any unital C*algebra B (actually we may restrict to B = B(H) with H arbitrary), any c.c. map u: E -> B uniquely extends to a unital representation (Le. *-homomorphism) from C to B. Remark. If two operator spaces E, F are completely isometrically isomorphic, then E and F can be realized as "concrete" operator subspaces E c A and FeB of two C* -algebras A and B for which there is an isometric *-homomorphism 11": A -> B such that 1I"(E) = F. Indeed, let A = C*(E) and B = C*(F), let u: E -> F be a completely isometric isomorphism, let 11": C* (E) -> C* (F) be the ( unique) extension of u (as in Theorem 8.10), and let 0': C*(F) -> C*(E) be the (unique) extension of u- 1 • Then clearly we must have (by unicity again) 0'11" = I and 11"0' = I: hence 0' = 11"-1. • To illustrate the use of this notion, we can state a nice relationship between direct sums of operator spaces and free products. Recall that we have defined the direct sum El 871 E2 of two operator spaces (in the iI-sense) in §2.6 and the free products in Chapter 5 (before Theorem 5.13). Using formula (5.7) together with Theorem 5.1 and comparing with (5.17), the following result is easy to check (we leave the details to the reader). Theorem 8.15. Let Et, E2 be arbitrary operator spaces. canonical (C*-algebraic) isomorphisms

Then we ha've

and

Proof. It suffices to check that the free product C*-algebras C*(E1 )*C*(E2) and C~(El) * C~(E2) have the universal property characteristic, respectively,

8. Group C* -Algebra.''!. Universal Algebras and Unitization

161

of C*(E) and C~(E) when E = El EEll E2 is embedded into the free product in the natural way. • A similar result holds for N-tuples E l , ... , EN (or actually for arbitrary families) of operator spaces. Remark 8.16. It is easy to see that OA(E) (resp. OA,,(E)) can be identified with the closed (resp. unital) subalgebra generated by E in C* (E) (resp. C~(E)). Of course, C*(E) can also be identified with a C*-subalgebra of C~(E).

For any c.c. map 0": E - t B(Ha), we denote by (j : OAu(E) c.c. morphism extending 0" and by

-t

B(Ha) the

the unital representation extending 0". Note that (j can be identified with the restriction of tra to OAu(E) viewed as a subalgebra of C~(E). The following simple fact is sometimes useful. It shows the "residllal finiteness" (see [Pes]) of the universal algebras of E. Proposition 8.17. Let E be any operator space. Let

C = {O": E

-t

B(H) I dim(H) <

00,

1I001Icb ::;

I}.

Then the embedding J: C~ (E)

-t

EElaEcB(Ha)

defined by J(x) = EEl aEC7r a (X) is a completely isometric unital representation. A fortiori, when we restrict to either C*(E) (resp. OAu(E) or OA(E)), we obtain a completely isometric representation (resp. morphism). Proof. The fact that J is isometric is an easy consequence of the following elementary fact. Let Tl' ... ' Tn E B(H). Let P( {Ti , Tn) be a polynomial in Tl, ... ,Tn , Tt, ... ,T~. Then

where the supremum runs over all finite-dimensional subspaces J( c H. This implies that, for any x in the *-algebra generated by E in C~(E), we have

Ilxll ::; IIJ(x)ll, and hence that J is isometric. The completely isometric case is proved similarly using polynomials with operator coefficients instead of scalar ones. •

Introduction to Operator Space Theory

162

Let (Eo, Ed be a compatible couple of operator spaces and let Eo = (Eo, E1)o, with 0 < () < 1. We have c.c. injective maps Eo ---+ Eo + El and El ---+ Eo + El that extend to C.c. morphisms OAu(Eo) ---+ OAu(Eo + Ed and OAu(Ed ---+ OAu(Eo+Et). We claim that these morphisms are injective. Indeed, using Proposition 6.6, this can be reduced to the fact that, if a c. b. map j: E ---+ F is injective, then, for any N ~ 1, j ® ... ® j extends to an injective map from E ®h'" ®h E to F ®h'" ®h F (N times). The latter fact follows from Proposition 5.16. Using this claim, we can view (OAu(Eo),OAu(E 1)) as a compatible couple. The next result shows that the interpolation functor commutes with the functor E ---+ 0 Au (E) .

Theorem 8.18. With the above notation we have completely isometric identities and OA(Eo)

~

(OA(Eo ), OA(E1))o.

Proof. We will prove only the second identity. The proof of the first one is exactly the same. By Propositions 6.6 and 2.7.6 we have a completely isometric embedding v: Eo C (OA(Eo), OA(Et))o

obtained by interpolating between Eo

---+

OA(Eo)

and

El

---+

OA(Ed.

Let OAo = (OA(Eo),OA(Ed)o. By the universal property of OA(Eo), v extends to a c.c. morphism OA(Eo) ---+ ~Ao. To show that is completely isometric, it suffices to show by Proposition 8.17 (applied to OA(Eo)) the following claim: For any n and any c.c. map a: Eo ---+ A1n there is a c.c. OAo ---+ Mn extending a. Indeed, this claim allows us to extend morphism the embedding Eo ---+ OA(Eo) to a c.c. morphism from OAo to OA(Eo), which must be the inverse of v. To prove the preceding claim, we first assume that Eo n El is dense both in Eo and El and we make crucial use of the following identity:

v:

v

a:

liaIiCB(Ee,M,,) = li a li(CB(Eo,Mn ),CB(E lo M n

»9'

which follows from Theorem 5.22 and the discussion of the duality in §2.7 (recall that CB(E, Mn) can be identified with Cn®hE*®hRn or, equivalently, with (Rn ®h E ®h Cn)*)' Finally, the restriction that Eo n El be dense both in Eo and El can be removed a posteriori using Lemma 2.7.2. .

8. Group C*-Algebras. Universal Algebras and Unitization

163

An alternate proof, perhaps more direct (not using duality), can be given using only Propositions 5.22 and 26.14.

-

.

Let E be an operator space. We define its unitization E as the linear span of the unit and E in OA1t(E). It is easy to check that E can be characterized as the unique unital operator space containing E completely isometrically and such that any complete contraction a: E --+ B(H) admits a unique unital completely contractive extension 0:: E --+ B(H). But actually, the next result shows that this notion of unitization is essentially trivial. Proposition 8.19. TIle unitization E of an operator space E can be identified completely isometrically with tIle direct sum C EElI E, with the unit corresponding to (1,0). Proof. Let j: CEElI E --+ E be the mapping taking (A,x) to AU+x, where we have denoted by U the unit in C c OA1t(E). We claim that j is completely isometric. Let eo = (1,0) E C EEl E, let XI,"" Xn be elements of E, let ei = (0, Xi), (1 ~ i ~ n) and let ao, ... , an be in K. We have by (6.3)

Iltai0j(ei)l. t-O

=sup{llao0I+Lai0a(XitniJ,

mill

where the sup is over all complete contractions a: E hand, we have, by definition (see §2.6),

Itai 0 eill. i

t-O

= sup

{ilao 0 T + t

, __ I

IIl1l1

--+

ai 0

B(H). On the other

a(Xi)II},

where the supremum runs over all c.c. maps a: E --+ B(H) and all contractions Tin B(H). By the Russo-Dye Theorem, the sup is the same if we restrict T to be unitary, and then after multiplication by 10 T- I (on the right say) we obtain exactly the same supremum as above. Hence we conclude that

which completes the proof by (2.1.8).

•

Exercises Exercise 8.1. Prove that any unital C*-algebra (resp. separable C*-algebra) C is isomorphic to a quotient of C*(IF) for some free group IF (resp. for IF IFoo), so that C:::: C*(IF)jI for some (closed two-sided) ideal I c C*(IF).

=

164

Introduction to Operator Space Theory

Exercise 8.2. Consider an operator space El and a closed subspace E2 eEl. Let j: E2 ---; El be the inclusion map. Then j extends to completely isometric embeddings C*(E2) ---; C*(El) and OA(E2) ---; OA(Ed (and similarly in the unital case). Exercise 8.3. Let E 1 , E2 be as above. Let q: El ---; Ed E2 be the quotient map. Then the unique representation 11": C*(E1) ---; C*(EdE2) and the unique morphism u: OA(E 1 ) ---; OA(EdE2) associated to q are complete metric surjections (and similarly in the unital case). (Thus the functors E ---; OA(E) and E ---; C* (E) are "projective"; they are also "injective" by the preceding exercise.) Exercise 8.4. A discrete group G is called amenable if it admits an invariant mean, that is, a functional cp in £00 (G)+ with cp( 1) = 1 such that cp( St * f) = cp(f) for any f in £oo(G) and any tin G. It is known (and the reader should use this as an alternate definition) that G is amenable iff there is a net (fo) in the unit sphere of £2(G) such that II.\(t)fo - foll2 -,...; 0 for any tin G. Show that the following are equivalent:

(i) G is amenable. (ii) C* (G)

= C A(G).

(iii) There is a generating subset S subset E c S, we have

c

lEI =

G with e E S such thet, for any finite·

II~ .\(t)ll·

Chapter 9 Examples and Comments In Banach space theory, ever since Banach and l\Iazur's early work, several examples have played a privileged role, such as fp, Co, L p , and C(K) (with K compact). These are usually called the classical Banach spaces. In the light of more recent developments, it is tempting to extend the list to the Orlicz, Sobolev, and Hardy spaces as well as the disc algebra and the Schatten pclasses, although in some sense all these examples are derived from those of the first generation. Analogously, one could make a list of all the classical C*algebras or von Neumann algebras (see e.g. [Da2]). In the present chapter, our aim is to describe the spaces that, in our opinion, are the best candidates to appear on a list of the classical operator spaces (we have already met some examples, sllch as R,C, min(£2), max(£2), and OH).

9.1. A concrete quotient: Hankel matrices \Ve start with a natural example of a quotient operator space, namely, the space

(here Loo is equipped with its natural o.s.s., i.e., the minimal one), which can be identified with the subspace of B(£2) formed of all the Hankel matrices, that is, all the matrices (aij) in B (£2) such that aij depends only on i + j. To explain this identification, we need some specific notation and background from classical harmonic analysis. \Ve denote by Lp the space Lp on the torus 11' equipped with the normalized Haar measure m. \Ve denote by H p the su bspace of Lp formed of all functions cp with Fourier transform CP vanishing on the negative integers and by H~ c Hp the subspace of those cp such that cp(O) = o. l\Iore generally, given a Banach space X, we denote by Lp(X), Hp(X) and H~(X) the analogous spaces of X-valued functions on 11' (1 ::; p < 00). To any cp in L oo , one classically associates the Hankel matrix (aij) with entries aij = cp( -i - j) \Ii, j ~ O.

It is easy to see (identifying £2 to H 2 ) that this matrix defines a bounded linear operator u
166

Introduction to Operator Space Theory

is an isometry from L:xAMn)/H!(Aln) into Mn(B(£2)) = B(£2(£2))' Hence, in the language of operator spaces, we can reformulate the "vectorial" Nehari Theorem as follows: Theorem 9.1.1. The correspondence 1jJ embedding of L(XJ/H! into B(£2)'

->

1-l[1jJ] is a completely isometric

This is a special case of Theorem 9.1.2, which is proved below. Thus, Hankel matrices provide us with a very natural and concrete "realization" of the quotient Loo/ H! as an operator space (the existence of which is guaranteed a priori by Ruan's Theorem). It would be interesting to have an analogous description in the general case for the quotient E1 / E2 of two operator spaces with E2 c E 1. Even in the particular case when E1 = L(XJ and E2 is a translation invariant subspace, this does not seem to be known (see, however, Exercise 8.2). We will prove in the following a generalization to Hankel matrices with entries in a von Neumann algebra. For that purpose, we need more notation. Let M c B(H) be a von Neumann algebra on a Hilbert space H. For simplicity, we assume H separable. We say that a function rp: 11' -> AI is a-measurable if for any x, y in H the function (rp(·)x,y) is measurable. We will denote by Loo(A1) the tensor product (von Neumann sense) L(XJ(1I')0M. Equivalently, L(XJ(AI) is formed of all equivalence classes of bounded a-measurable functions rp: 11' -> AI equipped with the norm Ilrpll(XJ = ess sup IIrp(t)IIJI,f· tEl'

Note that, for such a function rp, the Fourier transform ip: Z defined by the formula:

v x,y E H

(ip(n)x, y) =

->

Al can be

J

(rp(t)x, y)e-intdm(t).

We denote by H!CM) the subspace formed of all rp in L(XJ(M) such that ip(n)=OVn~O.

For any rp in L(XJ (M) let

be the operator of multiplication by the (operator-valued) function rp (taking x E L 2 (H) to the function t -> rp(t)x(t)). Clearly

(9.1.1) We will work with the subspaces H 2 (H)

Hg(H).

c

L2(H) and Hg(H).l = L 2(H)

e

9. Examples and Comments

167

The Hankel operator H[
The associated matrix is given by V i,j

~

0

(H[ifJ);j

=


Note that IIH[ifJII ~ II if II "". Here again H[
Theorem 9.1.2. The correspondence

Proposition 9.1.3. Consider if in Loo(M).

(i) Let Tep: £2(Z; H)

~

£2(Z; H) be tl1e operator defined by the matrix

(Tep)ij

=
(i,j E Z).

Then IITepil = II
Tben IITepil = IIifiloo. (Note that Tep may be viewed as the restriction of Tep to £2(Z+; H).) (iii) Consider a function b: Z ~ M. Let T(b): £2(Z+; H) ~ £2(Z; H) be the operator defined by the matrix T(b)ij = b(i - j), (i E Z, j E Z+). Then T(b) is bounded iff there is a (necessarily unique)

II
{If

(
Introduction to Operator Space Theory

168

x

where the supremum runs over all x, y in the unit ball of L 2 (H) such that and fj are finitely supported. In particular, we may assume x = E;'-N x(j)e ijt with x(j) E H. We then have

(Al",x, y) =

L

($(i - j)x(j), fj(i)) =

L

($(i - j)x(j - N), fj(i - N));

i,jEZ

hence which establishes (ii). (iii) Assume first that there is cP in Loo(1\J) with $ = b. Then Ilr(b)11 = IIr",lI, and hence in this case (iii) follows from (ii). In general, for any 0 < r < 1, let br : Z -> M be defined by br(n) = r1n1b(n). Assume that r(b) is bounded. Then clearly sUPn IIb(n)1I < 00, which implies that the series CPr = EnEZ br(n)e int is absolutely convergent. Hence by the first part we have IICPriloo = IIr(br)1I ~ IIr(b)lI. Let cP be a weak-* cluster point • in Loo (M) of {CPr} where r -> 1. It is easy to see that $( n) = b( n). Our main result will follow from the following beautiful lemma proved by several authors, including Parrott [ParI].

Lemma 9.1.4. Consider a decomposition K

K. We view an operator on K as a matrix

= Kl EB K2 of a Hilbert space

(~ ~)

with operator entries.

Then

and this infimum is attained. Moreover, if Kl = K2 = H and if b, c, d are all in a von Neumann algebra M C B(H), thef!. this infimum is attained 01] an element a of M. Proof. It suffices to show that, if the right side is that

II (~ ~) II ~

1. So assume the right side is

~

~

1, then there is a so 1. Equivalently, we

have both IIb*b + d*dll ~ 1 and llce* + dd*1I ~ 1 (see Remark 1.13), which implies b*b ~ 1 - d*d and cc* ~ 1 - dd*. The latter ensure that we can find operators V, W with I!VII ~ 1, IIWII ~ 1 such that b = V(1 - d*d)1/2 and c = (1- dd*)1/2W. We then set a = -Vd*W. Let U

(1 - dd*)1/2) d = ( -(1 _ d*d)1/2 d* .

It is easy to check that U is unitary and, moreover,

9. Examples and Comments hence we conclude

I (~ ~) I : :

169

1. If b, c, d all belong to AI, it is easy to see

that, in the above argument, we can find V, W in l1f so that a = - V d* lV also lies in M. • Theorem 9.1.5. Let (a(n)k:::o be a sequence in M. Let r = (a(i + j))i,j'20 be the associated Hankel matrix. Then IIfIlB(l'2(H» ::::: 1 iff tllere is cP in Loo(M) with IIcplloo ::::: 1 such that \fn~O

a(n)

= ij5( -n).

Proof. If a( n) = ij5( -n) for all n ~ 0, then r is the matrix associated to 1t[cp]; hence Ilfll ::::: Ilcplloo and the "if" part is obvious. Conversely, assume Ilfll ::::: 1. We claim that we can find operators a(-I), a( -2), a( -3), ... in M such that the function b: Z ---+ AI defined by b(n) = a( -n) satisfies IIr(b)11 ::::: 1. We can first find a( -1) in M so that the matrix a( -1) ( a(O) a(l)

a(O) a(l) ...

a(l) ... ...

... ) . .. . ..

has norm ::::: 1. This is possible: Indeed, this matrix can be written as ( a( ~1)

with

II(~ ~)II

=

II(~ ~)II

= Ilfll;

~)

he~cetheexistenceofa(-l) follows

from the preceding lemma, since we assume lin ::::: 1. Repeating the same argument, we can find a( -2) in l1f so that a(-2) ( a(-l) a(O)

a(-l) a(O) ...

a(O) ... ...

... ) . .. . ..

has norm < 1. Continuing exactly in the same way, we find a( -3), a( -4), and so on. This gives us an extension of a to the whole of Z such that, if we set b(n) = a( -n), we have Ilr(b)1I ::::: 1. Then. by Proposition 9.1.3(iii), there is cp in Loo(M) with IIcplloo ::::: 1 such that b = 'P. Hence a(n) = 'P( -n) for all n~O.

•

Proof of Theorem 9.1.2. By Theorem 9.1.5, for any 1/J in Loo(M) we have

111i[1/J]1I = inf{llcplioo 11t[cp] = 1i[1/J]} = inf{lI1/J + hll oo I hE H!(M)}.

170

Introduction to Operator Space Theory

This shows that 1/J ---+ 1i[1/J] defines an isometric embedding of Loo(A1)/ H!(M) into B(H2(H), H8(H).L). Replacing M by Mn(M) (n ;::: 1), we obtain that this embedding is actually completely isometric. • Corollary 9.1.6. Let Aft C B(HI ) and M2 C B(H2) be two von Neumann algebras. Let A = (Aij) and B = (Bij) be Hankel matrices wit11 entries, respectively, in MI and M 2. Assume that there is a bounded linear map v: MI ---+ M2 with IIvll ::; 1 such that V(Aij) = Bij for all i,j. Then

Proof. By the preceding result, for any bounded Hankel matrix (Aij) with entries in MI there is cp in Loo(Md with 11'1'1100 = II (Ai}) II such that Aij = $( -i - j). We claim that

II(v($( -i - j)))11 ::; 11($( -i - j))II· The following, slightly incorrect, argument is easier to understand on a first reading: Since Ilv: MI ---+ M211 ::; 1, we have IIJ®v: Loo(Md ---+ L oo (M2)1I ::; 1; hence, passing to the quotient v defines a contraction from Loo(Jtld/ H!(Md to L oo (M2 )/H!(M2 ), and the above claim now follows from the preceding result. The incorrect point in the preceding argument lies in the fact that, since cp(t) is only a-measurable, the a-measurability oft ---+ v(cp(t)) is questionable when v is not weak-* continuous, so that J ® v is not a well-defined map from LooCMd to LooCM2)' However, this is easy to repair: Fix 0 < r < 1. Then CPr(t) = LnEZ rlnl$(n)e int is strongly measurable, so the preceding objections do not apply and we have t

---+

Letting r

i 1, we obtain the announced result lI(v(Aij))11 ::; II(Aij)lI.

•

Remark. Actually Lemma 9.1.4 itself reflects a "concrete" realization of a quotient operator space, namely, fl.f2/ S, where S C Af2 is the one-dimensional subspace spanned by ell. Indeed, Lemma 9.1.4 implies that the mapping

Af2

---+

. (a b) (0 b) tB (0 0)

Af2 tBM2 takmg

c

d

to

0

d

c

d

.

defines (after passmg

to the quotient modulo its kernel) a completely isometric embedding of AI2 /S into M2 tB M 2 • By a rearrangement of the basis, this also shows that the mapping

9. Examples and Comments

171

defines a completely isometric embedding of Ah/Cel2 into 1Il2 EB A12. In that

= (~ ~) to the set of strictly upper triangular matrices (Le., such that a = c = d = 0). case, infb

II (~ ~) II

appears as the distance of the matrix x

More generally, for any n 2 1, let Tn C AJn be the subspace of upper triangular matrices (a;j), that is, such that a;j = 0 for all j < i. Let Pk be the orthogonal projection onto span(el, ... ,ek). Then, for any matrix x = (aij), we have inf

Ilx - y/l

yET"

=

sup /1(1 - Pk)xPkll·

l~k
l\Joreover, the mapping

x

--+

E9

(1 - Pk )xPk

l~k
defines a completely isometric embedding of lIln lTn into the (n - I)-fold direct sum Mn EB ... EB Un. This can be proved by repeated applications of Lemma 9.1.3. There is an analogous result for more general "nest algebras" (for which Tn is a prototypical case), called "Arveson's distance formula." See either [DaI] or [Pow] for more on all this. Remark 9.1.7. Let us denote simply by C the subspace formed in Loo by the continuous functions, and let A = Hoo n C be the subspace of C formed by the boundary values of analytic functions. By the vectorial version of Hartman's Theorem (cf. [NiD, the restriction to the continuous functions of the correspondence appearing in Theorem 9.1.1 is a complete isometry between the quotient operator space CIA and the space of all compact Hankel operators on £2' In particular, this implies that the operator space CIA is exact in the sense of §I7 below. On the other hand (this remark is due to Junge), the results of [HPI, Theorem 2.2] suggest that the space Lool Hoo might be exact, but this does not seem clear at the moment. Until recently, we knew of no example of a quotient of a commutative C*algebra (by a closed subspace) that is not exact as an operator space, but an example of this kind has just been exhibited by N. Ozawa [OzI]. An operator space that is (completely isometrically) a quotient of a minimal operator space is called a Q-space; equivalently, t.hese are subspaces of quotients of commutative C* -algebras. The class of Q-spaces seems to be quite interesting to study. See [Ri2] for an investigation of an analog of the Haagerup tensor product for Q-spaces.

172

Introduction to Operator Space Theory

9.2. Homogeneous operator spaces

We say that an operator space is homogeneous (resp . .A-homogeneous) if, for all u: E ~ E, we have Ilulicb

= lIuli (resp. Ilulicb ::;: .Allull)·

In this case, every surjective isometry on E is a complete isometry. For example, it is easy to check that, for any Banach space E, the spaces min(E) and max(E) (defined in Chapter 3) are homogeneous. Actually, the most interesting case seems to be the Hilbertian one: We will say that E is Hilbertian if it is isometric (as a normed space) to a Hilbert space. When E is only .A-isomorphic to a Hilbert space for some .A > 1, we will say that E is .A-Hilbertian. The spaces Rand C (introduced in (0.1) and (0.2» as well as min(t'2), max(t'2), and OH are examples of homogeneous Hilbertian operator spaces. The homogeneous Hilbertian operator spaces reproduce the same operator space structure inside themselves; when a space E is a mixture of two distinct structures, for instance, E = REEl C, then it is not .A-homogeneous for any .A. More precisely, we have: Proposition 9.2.1. Let E c B(H) be a Hilbertian operator space. Then E is homogeneous iff, for any unitary operator U: E ~ E (here "unitary" refers to the Hilbert space structure of E), we have IlUlicb ::;: 1. In tlIat case, whenever two subspaces E1 c E and E2 C E and llave the same Hilbertian dimension, they are completely isometric. Proof. The first assertion follows from the Russo-Dye Theorem: The unit ball of B(E) is the closed convex hull of its unitaries ([Ped, p. 4]). Then the second assertion follows: Indeed, let u: E1 -~ E2 be an isometric isomorphism. After comparing with the orthogonal projection onto E 1 , we can extend u to u: E ~ E such that Ilull = 1 and, if E is homogeneous, we have Ilulicb = lIull, whence Ilulicb ::;: 1. Similarly (reversing the roles) lIu- 11lcb ::;: 1. Thus we conclude that u is completely isometric. • Proposition 9.2.2. Every .A-homogeneous operator space is completely isomorphic to a homogeneous (i.e., l-homogeneous) one E with dcb(E, E) ::;: .A. Proof. Let (3 be the family of all the operators u: E ~ E with Ilull ::;: 1. Let Bu = B(H) for all u in (3. Consider the direct sum (f7uE~Bu (equipped with the operator space structure defined in §2.6) and ~et J : E ~ EEluE~Bu be the isometric map defined by J(x) = (U(X»UE~· Let E be the range of J. Then J is an isometry from E onto E. Moreover, IlJllcb :::; .A and 11J;~1I1cb ::;: 1. Finally, by its very definition, the space

E is homogeneous (i.e., I-homogeneous).

•

9. Examples and Comments

173

Remark. The consideration of a few simple examples like R, C, or OH gives the impression that a homogeneous Hilbertian operator space might be detenllined by its two-dimensional subspaces. But it is not so: Using [r\'IP], C. Zhang [Z2J exhibited for each integer N a pair of homogeneous Hilbertian (infinite-dimensional) operator spaces that have the same (Le. completely isometric) N-dimensional subspaces bllt which are not even completely isomorphic. The examples that we will review in this chapter demonstrate the rich diversity of the homogeneous Hilbertian operator spaces appearing in the literature. Nevertheless, it seems reasonable to expect that a elassification or a parametrization of this class of operator spaces will be available soon.

9.3. Fermions. Antisymmetric Fock space. Spin systems Let I be any set. Consider a family of operators in B(H) satisfying the following relations, called the canonical anticommutation relations (in short, CAR): (CAR) {

+ Vj*Vi

'Vi,j E I

ViVj*

'Vi,jE1

ViVj+VjVi=O.

=

6;)1

Here, of course, we have set 6 jj = 1 if i = j and 6;j = 0 if i i- j. The closed span in B(H) of the family [Vi liE 1J gives us a very interesting example of an operator space. We will denote it by (I). We set = (N) and n = <1>( {I, 2, .. , n}). Let us first justify the existence of such families. Let 1i = £2(I) equipped with an orthonormal basis (ei)iEI, let Ho = te, and let Hn = 1i An (antisymmetric Hilbertian tensor product) for n ~ 1. \Ve define the antisymmetric Fock space associated to 1i as follows:

For each h in 1i, it is classical to denote by c(h) (resp. a(h» the so-called operator of "creation (resp. annihilation) of particle" defined by 'Vx E H

c(h)x = h /\ x

(resp. a(h) = c(h)*).

Then we have the relations 'Vh, k E 1i

a(h)a(k)* +a(k)*a(h)

=< h, k > I

et

a(h)a(k) +a(k)a(h)

= o.

If we let Vi = a(ei) (or if we set Vi = c(e;) for all i), we obtain a family with the announced properties. We refer the reader to [Gu2, Gu3, EvL, BRJ for more information.

The next result is classical.

174

Introduction to Operator Space Theory

Theorem 9.3.1. The operator space (I) is (isometrically) Hilbertian and homogeneous. Up to a complete isometry, it only depends on the cardinality of I (and not on the particular family (Vi)iEf chosen to define it). Proof. Let (Oi)iEf be a finitely supported family of scalars and let T = LiEf 0i Vi· By the CAR we have

Then Hence whence (9.3.1) Therefore, the space (1) = span[Vi liE I] is isometric to £2(1). Let H' be another Hilbert space and let {Wi liE I} be a system in B(H') satisfying the CAR. Let A (resp. B) be the C* -algebra with unit generated by {Vi liE I} (resp,{Wi liE I}). By a classical argument (see [BR, p. 15J), one shows that there is a C* -algebraic isomorphism 7r: A -+ B such that 7r(Vi) = Wi for all i. A fortiori (since, by Proposition 1.5, 7r is completely isometric), the spaces span[Vi] and span[Wi ] are completely isometric. This justifies the last assertion in Theorem 9.3.1. Finally, let us show that (I) = span[Vi] ~ £2(I) is homogeneous: According to Proposition 9.2.1, it suffices to show that every unitary U: (I) -+ (I) is completely contractive. But, as shown by a simple computation, if U is unitary, then the operators Wi = U(V;) still satisfy the CAR; hence the mapping U extends to an isometric representation from A onto B. A fortiori, U is a complete isometry and 1lUlicb = 1. Remark. We should mention that some variants of the space (I) appear in the theory of Clifford algebras. On the other hand, a space very similar to (I) also appears in the study of certain domains of holomorphy, called Cartan factors of type IV (cf. [HaJ). In the latter theory, the term "spin system" is used to designate a family of Hermitian and unitary operators {Ui liE I} in B(H) such that 'Vi

i= j

UiUj + UjUi = O.

For instance, the classical Pauli matrices

form a spin system.

9. Examples and Comments

175

There is (essentially) a one-to-one correspondence between spin systems and systems satisfying the CAR. Indeed, if (Uj)jEI and (Uj')jEI are two systems such that their disjoint union forms a spin system, then the system (ltj)jEI, defined by ltj = (Uj + iUj')/2, satisfies the CAR. In the converse direction, if {Vi 1 i E 1} is a family satisfying the CAR, then t.he operators Ui = Vi + ~* form a spin system. The subspace of B(H) generated by a spin system (Ui);EI clearly is isomorphic to £2(1). To be more explicit, we must distinguish the real case from the complex one. We have

V(o:;) E 1R(I)

V(O:i)

E C(1)

(2: 10:;12)1/2 =

(2: IO:iI 2)1/2 ~

II 2: O:iU;.II,

II 2: O:iUili

~ 21/2(2:

IO:iI 2)1/2.

More precisely (see [Ha]), we have

Remarks. It is known that, if 1 = {l, ... , n}, t.he unital C* -algebra generated by <1>(1) = <1>n is isomorphic to the algebra 11[2" of 2n x 2n matrices (one can obtain an entirely explicit realization of <1>n by using tensor products of the Pauli matrices). This remark shows that <1>(1) is an "exact" operator space with constant = 1 in the sense of §l7. It is possible to show that, if 1 is infinite, for any completely isometric embedding <1>(1) c B(H), there is no c.b. projection from B(H) onto <1>(1) (but there is a bounded one). Except for these few facts, there is very little available informat.ion on the operator space structure of <1>(1). It would be interesting to compute its dual <1>(1)* as well as the iterated tensor products <1>(1) ®min ... ®min <1>(1) (n times) in the style of [HP2] (note, however, that <1>(1) can be easily distinguished from all the examples reviewed in this chapter; see Chapter 10). The field of fermionic analysis is a very active one at the moment. See, for example, [M] for the probabilistic viewpoint and also [BoSl, BoS2, CL, BCL, CK, PX]. It is only natural to expect that there will be further points of contacts with operator space theory.

9.4. The Cuntz algebra On Let S1, ... , Sn E B(H) be isometries such that

The C*-algebra generated by St ... , Sn is called the Cuntz algebm (cf. [Cu]) and is denoted by On. One can show that it does not depend on the particular

176

Introduction to Operator Space Theory

choice of the sequence {Si} but only on n. It is tempting to study the operator space En = span[sl' .. , sn], but this is disappointing: A very simple calculation shows that for all al, ... ,an in B(H), we have II Lai 0 sill = II Lai'ailll/2. It follows that En is completely isometric to Cn. Similarly, span[si, .. , s~] is completely isometric to Rn. Of course, similar remarks hold for the C*algebra 0 00 generated by a sequence of isometries (Si) such that L~ SiSi' ::; I (this time the equality is not required for the unicity of 0 00 ), Another very important family of isometries is provided by the creation operators on the full Fock space. To define this we need to introduce some notation. Let H be a Hilbert space. We denote by :F(H) (or simply by :F) the full Fock space associated to H, that is to say, we set 'Ho = C, 'Hn = H~!m (Hilbertian tensor product) and finally

:F = EBn~o 'Hn· We consider from now on 'Hn as a subspace of :F. For every h E :F, we denote by C(h): :F ~:F the operator defined by:

C(h)x=h0x. More precisely, if x = >'1 E 'Ho = Cl, we have C(h)x = >'h and if x = Xl 0 X2 ... 0 Xn E 'H n , we have C(h)x = h 0 Xl 0 X2 ... 0 x n . We will denote by n the unit element in 'Ho = Cl. Now assume H = C~ with its canonical basis (ei). We denote :Fn = :F(C~) and we set (i=l, ... ,n). Then {C i } is a family of isometries on :Fn with orthogonal ranges, and we have n

LCiC: = I - Po, I

where Po is the orthogonal projection onto cn. Now let {xd be any n-tuple in B(H) (H arbitrary). We will denote by £ {Xi} the closed linear span of I and all the products of the form XiI Xi2 ... Xip XiI' .. Xiq (p 2:: 0, q 2:: 0). In particular, taking q = 0, we see that

span[I,xi] C £{Xi}' The space £{Xi} can also be viewed as the span of all products P{Xi}Q{Xi}, where P{Xi } and Q{Xi } are polynomials in noncommutative variables {Xi}' Note that our polynomials are allowed to have a "constant term" equal to a multiple of the unit in the free unital semi-group generated by {X~, ... , X n }. In a series of papers ([Pul-3]) G. Popescu stud;ed various extensions of von Neumann's inequality for n-tuples of operators; in particular, he proved the following.

9. Examples and Comments

177

Theorem 9.4.1. ((Pu3i) Assume IIL~ x~x;11 S 1. For any finite family Pk(Xl, ... ,Xn ), Qk(Xl, ... ,xn ) of polynomials in noncommuting variables {Xl, ... , Xn} we have

More generally, tile linear mapping

defined by .

is completely contractive. This shows that {£ 1, ... , £n} is in some sense "maximal" among all the n-tuples {xd with IILxixil1 S 1.

Remark. In [Pu3], Popescu also observed that for "analyt.ic" polynomials, the Cuntz isometries {sd can be substitut.ed t.o {£d in t.he preceding st.at.ement.. Namely, for any polynomial P we have

Moreover, the correspondence £i -> Si extends to a complet.ely isometric unit.al homomorphism on the (nonself-adjoint) algebra generated by {£i}'

Corollary 9.4.2. We llave completely isometrically

and similarly with {Si} substituted to {£i}. Proof. Let us denote by eo the unit in C. \Ve will show that the correspondence u: C EIh C n ~ span[l, el , ... , en] defined by u(eo) = land u(eil) = (i is completely isometric. Note that, by Remark 1.13, the restriction of u t.o en is completely contractive. Hence, by the definition of 611, we have lIullcb S 1. To show that Ilu-llicb S 1, it suffices to establish the following claim: For any family of matrices {ao, al, ... ,an} in !lIN we have (9.4.1 )

Introduction to Operator Space Theory

178

where the norm is the norm in MN(IC EB1 Cn) and MN(B(Fn)). By Theorem 8.19, the left side of (9.4.1) is equal to the supremum of

where v: C n ---- B(H) runs over all possible maps with IIvll cb ::; 1. But, if we let Xi = v(eil), we clearly have (see Remark 1.13) IIExixill1/2 ::; Ilvllcb ::; 1. Hence we find

Ilao ® eo + L and since

1IFlicb =

ai ® ei111 = sup lIao ® I

+ La; ® Xi I '

1, we obtain

Thus we conclude lIu -111 cb ::; 1 and the last assertion concerning {Si} follows • from the preceding remark. Remark. It would be interesting to study more generally the relation between . an operator space and the C* -algebra that it generates. For some information in this direction, see [KiV, Zl, KiW, Pll].

9.5. The operator space structure of the classical Lp-spaces In this section, we describe the operator space structure of the usual (Le. commutative) Lp-spaces. We will see that these spaces can be equipped with a specific structure, which we call their natural structure. We refer to [P2] for more details. Let (0, A, JL) be a measure space. We will denote by Lp(JL) the associated Lp-space of complex-valued functions. Given a complex Banach space X, we will denote by Lp(JL; X) the Lp-space of X-valued measurable functions in Bochner's sense. It is well known that

with () = lip. A priori, this formula is an isometric identity that is only valid in the category of Banach spaces, but the complex interpolation of operator spaces will allow us to extend this formula to the operator space setting. The case p = 00. First, if A is a C*-algebra, it possesses a privileged operator space structure associated to any realization of A as a C*-subalgebra of B(H). Indeed, by Proposition 1.5, iftwo C*-algebras are isomorphic (as C*-algebras), then they are completely isometric, and hence they are identical as operator spaces. We will say that this particular structure is the natural operator space structure on a C* -algebra.

9. Examples and Comments

179

In particular, if p = 00, we have a natural operator space structure on the space Loo(Jl) or on the space C(T) of all continuous functions on a compact set T. It is easy to check that this natural structure is determined by the isometric identity C(T) Q9min K = C(T; K), where the space on the right-hand side is the space of K-valued continuous functions equipped with its usual norm. This space coincides with the injectiye tensor product (in Grothendieck's sense) of C(T) and K. In other words, the natural operator space structure on E = £'XJ(/L) or E = C(T) makes it identical to the space min(E), as defined in Chapter 3. The case p = 1. If p = 1, again the choice is clear: The natural structure is defined as the Olle induced on Ll (/L) by the dual space Loo(/L)* equipped with its dual operator space structure. Explicitly, this means that the norm of lIIn (L 1 (Jl)) is by definition the norm induced by CB(Loo(Jl),lIfn ). (Note that lIfn (L I (Jl)) can be identified with the a(Loo(p), Ll (It ))-continuous linear maps from Loo(Jl) to lIfn .) For the resulting operator space structure Oll L 1 (Jl), we have L 1 (Jl)* = Loo(/L) completely isometrically. Proposition 9.5.1. The operator space obtained by equipping Ll (/1) with its natural o.s.s. (as defined above) coincides with max(L l (p)). Proof. Indeed, let B = L 1(/1) and let E = max(B). By (2..5.1) (or Corollary 5.11), the inclusion E ---> E** is a complete isometry, and, by Exercise .3.2, we have E** = (max(B))** = ma..x(B**) = (min(B*))*. Therefore the o.s.s. induced by max(B**) on B coincides with that ofmax(B). In tIle present case, we have max(B**) = (min(B*))* = Loo(/l)*, so we conclude that tIle opera,tor space structure of max(L l (Jl)) is tIle same as tlJat induced by Loo (Jl)*, where L oo (/l) is equipped with its natural (i.e. minimal) operator space structllre. • In particular, in the case n = N, we have a natural o.s.s. on fl. It is not hard to verify that this natural o.s.s. on (1 also coincides wit.h the one obtained by considering £1 as the dual of c{). vVe can describe the associated norm II Ilmin on K Q9 fl in the following manner: Let. (en) be the canonical basis of C1. For any finite sequence (an) in K (or in B (£2)) we have (9.5.1) where the infimum runs over all possible decomposit.ions an = bncn in K (resp. B(f 2 ))· We will give two more descriptions of the same structure, in (9.5.3) for p = 1 and in (9.6.1). Analogously, we can describe the natural o.s.s. of Ll (f.l) as follows. Let f E K ®min L 1 (Jl). We may consider f as a K-valued function on n. We

Introduction to Operator Space Theory

180 then have

IlfllK:®mlnLl(/L) = inf

{III

I III

g(t}g(t}*dt.t(t} 1/2

I

h(t}*h(t}dt.t(t} 1/2} ,

where the infimum runs over all possible decompositions of f (measurable) K-valued functions.

(9.5.2) as a product of

The formula (9.5.1) (resp. (9.5.2}) is the quantum version of the fact that the unit ball of £1 (resp. L 1 (t.t)} coincides with the set of all products of two elements in the unit ball of £2 (resp. L 2 (t.t)). These two formulas (9.5.1) and (9.5.2) can be deduced from the fundamental factorization Theorem 1.6.

The case 1 < p < 00. For the general case, we use interpolation. Consider the operator space structure on Lp(t.t} corresponding to (L oo (t.t},L 1 (t.t}}o as defined in §2.7. By definition, the norm on Mn(Lp(t.t)} is the one induced by the space (Mn (L oo (t.t}},Mn (L 1 (t.t}}o with () = lip. Once again, we will say that this is the natural operator space structure on Lp(t.t). We can describe more explicitly this structure using the Schatten classes Sp already introduced (see (7.19}). Indeed, let f E K @ Lp(t.t) (viewed as a K:-valued measurable function). Then we have (9.5.3) where the supremum runs over all a, b in the unit ball of S2p. Since Lp(t.t} has been equipped with an o.s.s., we may now unambiguously discuss completely bounded maps u: Lp(t.t} ---4 Lp(t.t}. These turn out to be easy to describe. Proposition 9.5.2. A linear map u: Lp(t.t}

---4 Lp(t.t} is completely bounded (on Lp(t.t) equipped with its natural o.s.s.} iff the mapping u@Is" is bounded on Lp(t.t; Sp}. Moreover, we have

It has been known for a long time that the Hilbert transform is bounded on the (vector-valued) Lp-space of Sp-valued functions for any 1 < p < 00 (see also in [Bou1] the analogous result for martingale transforms). This implies that, if 1 < p < 00, the Hilbert transform is completely bounded on Lp (on the torus or on JR.). The same is valid for the Riesz transforms and their Gaussian analogs on the n-dimensional torus or on JR.n; see, for example, the discussion in [P15]. It is then tempting to compare boundedness and complete boundedness for maps on Lp. By Proposition 9.5.2, it is easy to check that, in the cases· p = 1,2 or 00, every bounded map on Lp is automatically completely bounded.

9. Examples and Comments

181

However, if 1 < p 1= 2 < 00, there are examples of Fourier multipliers on Lp of the torus that are not c.h. Such examples (implicit in [P4, pp. 110--113]) can be obtained easily by observing that the norm in 8 p is invariant under all transformations (aij) ----> (E;aijE'j) for any choice of signs E; = ±1, E'j = ±1, but that there exist, if 1 < p 1= 2 < 00, mappings that are unbounded on 8 p and of the form (aij) ----> (E;ja;j) with Eij = ±1 for i :::; j and (say) Eij = 0 otherwise. Fix (E;j) such that the latter mapping is unbounded on 8p- Then let u: Lp ----> Lp be the Fourier multiplier defined as follows: If n is of the form n = 3 k + 31 with k :::; l, then u{e;nt) = Ekleint and u{e int ) = 0 otherwise. By a classical variant of the Khintchine inequalities (cf., e.g., [LR, p. 65]), it is known that u is bounded on L p , but by the choice of (cij) (and by the invariance property ofthe 8 p-norm recalled above) u is unbounded on Lp{8p), and hence it is not c.b. Similarly, it follows from [HP1] that there are Fourier multipliers bounded on Hi (of the torus) but not c.b. The preceding definitions can be extended in the case of noncommutative Lp-spaces. Consider, for instance, the Schatten class 8 p • It is well known that, if () = lip, we have an isometric identity (9.5.4) Here, it would be more appropriate to denote by 8= the space K! The space K can be equipped with its natural o.s.s. as a C* -algebra and 8 1 with the dual o.s.s. associated to the identity 8 1 = K*, relative to the duality defined for x E 8 1 , Y E K by (x, y) = tr(t xy). Then, the formulas (2.7.3) and (9.5.4) allow us to define an o.s. structure (again called natural) on 8p- By Theorem 7.10, the space 8 2 is then completely isometric to OH(fir x fir) or, equivalently, to OH (see Remark 7.13 and the discussion preceding it). As we already mentioned, given an operator space E c B(H), exactly the same idea leads to the definition of a natural o.s.s. on the E-valued Lp-space that remains valid in the noncommutative case. When p = 1, we define 8 1 [E] as the "projective operator space tensor product" of 8 1 with E, introduced in §2.8 and denoted by 8 1 ®" E. By [ER8], we have (9.5.5) 8dE*] ~ (K ®min E)* completely isometrically. When 1 < p < 8 p [E] = (K

00

®min

we define

E, 8t[E])

°,

(9.5.6)

where () = lip· Then the following result holds (cf. [P2, Lemma 1.7])

Proposition 9.5.3. Fix p with 1 :::; p < 00. Let u: E ---+ F be a linear map between two operator spaces. Then u is c.b. (resp. completely isometric) iff

182

Introduction to Operator Spa,ce Theory

the map Isp ®u defines a bounded (resp. an isometric) linear map from Sp[E] to Sp[F]. Moreover, we have

Ilulicb = IIIsp ® UIlS,,[Ej--+S,,[Fj'

:0

:1 '

Furthermore (see [P2, Corollary 1.4), if 1 :::; Po, P1 :::; 00 and = 1;;:,0 + then for any compatible couple of operator spaces (Eo, Ed we have completely isometrically (9.5.7) Spo[(Eo,E1 )0] = (Spo[EO],Spl[E 1])o. We refer the reader to [P2] for more details.

9.6. The C"-algebra of the free group with n generators Let Fn (resp. Foo) be the free group with n (resp. countably many) generators, and let {gll g2, ". }) be the generators. Let 11": F 00 -4 B (H) be a unitary representation of the free group. We will see that, in several instances, the operator space E(1I") spanned in B(H) by {1I"(gi) Ii = 1,2,,,.} has interesting properties. We will first illustrate this with the universal representation U: Foo -4 B(1t), which generates the full C" -algebra C" (F00), as introduced in Chapter 8. The other classical choice for 11" is the left regular representation A , which generates the "reduced" C"-algebra C~(Foo). This case is discussed in the next section, §9.7. We let E[j = span[U(gi) I i = 1,2, '" n] and Eu

= span[U(gi) Ii:::: 1].

By Lemma 8.9, for any finite sequence (ai) in B(£2)' we have (9.6.1)

where the supremum runs over all sequences (Ui) of unitary operators in B (H) and over all possible Hilbert spaces H. Actually, the supremum remains unchanged if we restrict ourselves to H = £2 or to H finite-dimensional with arbitrary dimension. But then formula (2.11.2) shows that, if we denote by (en the dual basis to the canonical basis of Co (equipped with its natural 0.S.8.), we also have

II L ai ® U(gi)lImin = II Eai ® e;IIB(i2)®mlnCo' Therefore, the mapping u: Co - 4 Eu that takes ei to U(gi) is a complete isometry. Hence, we have proved:

9. Examples and Comments

183

Theorem 9.6.1. The operator space Eu spanned by the generators in C* (Feo) is completely isometric to fl equipped with its natural operator space structure (or, equivalently, its o.s.s. as the dual of co). Similarly, Efj is completely isometric to

fr.

The formula (9.6.1) can be viewed as the "quantum" analog ofthe classical formula

\I(..\i) E

C(I)

11(..\i)II£1 = L I..\il = sup{1 L..\i zill

Zi

E C,

Iz;1 =

I}.

(9.6.2) The space Eu gives us a concrete realization of the space fl as an operator space. More generally, for any measure space (0, It), one can describe t.he natural operator space structure of Ll(O,lt) (induced by Leo(O"t)*) as follows. For all f in L 1 (0, It) ® B(f2)' we have

IIfIILd!1'JL)0I11iIlB(~2) = sup

{II!

f(w) ® g(w)dJL(w)

I

B(f2)0I11iIl B (f2)

},

(9.6.3)

where the supremum runs over all functions 9 in the unit ball of the space of Leo-functions with values in B(f2)'

9.7. Reduced C*-algebra of the free group with n generators

Let G be a discrete group. We have defined in Chapter 8 the left regular representation..\: G -+ B(f2(G)). Recall that ..\(t) is the unitary operator of left translation by t on f2(G). We denote by C~(G) the C*-algebra generated in B(f2(G)) by {..\(t) I t E G} or, equivalently, CHG) = span{..\(t) I t E G}. Clearly, we have a C* -algebra morphism Q: C*(G)

-+ C~(G)

that takes U (t) to ..\( t). By elementary properties of C* -algebras, it is onto and we have C~(G) ~ C*(G)j ker(Q). In general, ker(Q) =I- {O}, but one can show that C~(G) = C*(G) (Le., ker(Q) = {O}) iff G is amenable. The free groups are typical examples of nonamenable groups. The fact that the algebras C~ (G) and C* (G) are distinct in this case is manifestly visible on the generators. Indeed, if we let

E>: =

span[..\(gi)

Ii

= 1, .. , n]

184

Introduction to Operator Space Theory

we can see that E>. is a very different space from its analog in the full C*algebra, namely, the space Eu studied in §9.6. Indeed, as Banach spaces, we have Eu ~ €l and E>. ~ €2. The first isomorphism is elementary (see Theorem 9.6.1), while the second one is due to Leinert [Le]. Using Haagerup's ideas from [H2], one can describe the operator space structure of E>. as follows (see [HP2] for more details). Consider the space B(€2) EB B(€2)' equipped with the norm II(x EB y)1I = max{llxll,llyll}. In the subspace REB C c B(€2) EB B(€2)' we consider the vectors 8i (i = 1,2, ... ) defined by setting

We will denote by RnC the closed subspace spanned in REBC by the sequence

{8d· This notation is compatible with the notion of "intersection" defined in §2.7, provided we view the pair (R, C) as a compatible pair using the transposition mapping x - 4 tx as a way to embed R into C. This means that we let X = C, we use x -4 tx to inject R into X, and we use the identity map of C to inject C into X. Similarly, we will denote by Rn n C n the subspace of R n C spanned by {8 i I i = 1,2, ... , n}. It is easy to verify that, for any Hilbert space H and for any finite sequence (ai) in B(H), we have

Then, we can state (see [HP2]):

Theorem 9.7.1. The space E>. is completely isomorphic to R n C. l\fore precisely, for any finite sequence (ai) in B(H), we have (9.7.1) so that tIle mapping u: R n C -4 E>. defined by u(8i ) = .>.(gd is a complete isomorphism satisfying lIulicb = 2 and Ilu-11l cb = 1. Moreover, the map P: C~(Foo) -4 E>., defined by P.>.(t) = .>.(t) ift is a generator and P.>.(t) = 0 otherwise, is a c.b. projection from C~(Foo) onto E>. with norm IlPlicb : : ; 2. (Similar results hold for E'): and Rn n C n .)

ct

Proof. Let G = Foo. Let c G (resp. Ci- C G) be the subset formed by all the reduced words that start by gi (resp. gil). Note: Except for the empty word e, every element of G can be written as a reduced word in the generators admitting a well-defined "first" and "last" letter (where we read from left to right). Let p/ (resp. Pi-) be the orthogonal projection on €2(G) .

9. Examples and Comments with range span[8t check that

It

E

ct]

(resp. span[(8 t

E

Ci-D. Then it is easy to

+ )..(g;)(1 - Pi-) + p/ )..(g;)(1 - P;-), + p/ )..(gi),

)..(gi) = )..(gi)Pi= )..(gi)Pi-

= )..(gi)Pjso that )..(gi)

It

185

= Xi + y; with (note xixi = Pi- and

y;y;

= p/)

Therefore, for any finite sequence (ai) in B(H) we have by (1.11) (note a; 0 X; = (ai 0 1)(1 0 x;) and similarly for ai 0 y;) Ill: ai 0 )..(gi) II

~

Ill: ai 0 Xiii

~ IIl:a aill i

i 2 /

+ Ill: ai 0

Yill

+ IIl:a: ai ll i / 2

~ 211l:a; 08ill· The converse follows from a more general inequality valid for any discrete group G: For any finitely supported function a: G -+ B(H) we have max {11l:a(t)*a(t)ll

i 2 / ,

IIl:a(t)a(t)*ll

i 2 / }

~ IIl:a(t) 0

)..(t)llmill· (9.7.2)

To check this, let T = 2::a(t) 0 )..(t). For any h in BH we have T(h 0 8e ) 2:: a(t)h 0 8t . Hence IIT(h 0 8e )11 = (2::t Ila(t)hI1 2 ) 1/2, and hence Ill: a(t)*a(t)ll i / 2

= sup (l: Ila(t)hI1 2) 1/2 hEB/I

=

~ IITII.

Similarly, since T* = 2::a(t- 1 )* 0)..(t), we find

IIl:a(t)a(t)*ll

i 2 /

~ IIT*II = IITII,

and we obtain (9.7.2). If we now return to the case G = IF",,, we find that (9.7.2) implies the left side of (9.7.1). Moreover, if P is as in Theorem 9.7.1, we have

(I0 P) (l: a(t) 0 )"(t») = l : a(gn) 0 )..(gn). Hence, if T

= 2:: a(t) 0 )..(t), we have

11(10 P)(T) I

~ 2 max {liE a(gn)*a(gn)11 1/ 2 , liE a(gn)a(gn}* 111/2} ::; 2 max

{liE a(t)*a(t) 111/2 , liE a(t)a(t)* 111/2} ,

186

Introduction to Operator Space Theory

and by (9.7.2) this is:::; 211TII. Thus we conclude, as announced, that 1IPlicb :::;

2.

•

We will now describe the operator space generated by the free unitary generators {.\(gi) I i = 1,2, ... ,} in the noncommutative Lp-space (1 :::; p < 00) of the free group Foo. Let us denote by r the standard trace on CHFoo) defined by r()..(f» = f(e). Let Lp(r) denote for 1 :::; p < 00 the associated noncommutative Lp-space. The space Ll (r) is the predual of the von Neumann algebra generated by )..(Foo), which we will denote by Loo(r). When 1 < p < 00 and () = lip, the space Lp(r) can be identified with the complex interpolation space (C! (F00), Ll (r»o or, equivalently, with (Loo (r), Ll (r»o . Using §2.7, we may view Lp(r) as an operator space. Let us denote by Ep the closed subspace of Lp( r) generated by the free generators {.\(gi) I i = 1,2, ... } (note that Eoo = EA)' We may view Ep as an operator space with the o.s.s. induced by Lp(r). Clearly, the orthogonal projection P from L2(r) to E2 is completely contractive (since, by Theorem 7.10, L2(r) can be identified with OH(I) for a suitable set I). On the other hand, by Theorem 9.7.1, that same projection P is completely bounded from C!(Foo) onto EA' Actually, the proof of Theorem 9.7.1 shows that P extends to a weak-* continuous projection from Loo(r) onto EA' By transposition, P also defines a c.b. projection from Ll(r) onto E 1 . Therefore, by interpolation, P defines a completely bounded projection from Lp(r) onto Ep for any 1 < p < 00. Moreover, by Proposition 2.7.6, the existence of this simultaneous c.b. projection ensures that the space Ep can be identified completely isomorphic ally with (EA' Edo with () = lip. In addition, El ~ (EA)*' By Theorem 9.7.1, we have EA ~ R n C and by duality El ~ R + C; hence we have (completely isomorphically) Ep ~ (RnC, R+C)o. We will compute the latter space more explicitly in Theorem 9.8.7, but let us state what we just proved. Corollary 9.7.2. Let Lp(r) denote the noncommutative Lp-space of the free group and let Ep be the closed subspace generated by tIle free generators {.\(gi) I i = 1,2, ... }. Then we have, completely isomorphically (with () =

lip), Ep

~

(RnC,R+C)o,

(9.7.3)

where, as before, we use the transposition mapping as the continuous injection from R to C, which allows us to view (R, C) as a compatible couple. Moreover, the orthogonal projection from L2(r) onto E2 defines a c.b. projection from Lp(r) onto Ep for all 1 :::; p :::; 00. Moreover, the equivalence constants in (9.7.3) as well as liP: Lp(r) ---+ Epllcb remain bounded when p runs aver the whole range 1 :::; p:::; 00. We will see later (Theorem 10.4) that the space RnC is very different from the Fermionic space ~ described in §9.3. Nevertheless, the following simple statement holds.

9. Examples and Comments

187

Proposition 9.7.3. Let (V;) be a sequence satisfying tlle CAR.. For any finite sequence (ai) in B(H), we llave

II Lai QSH5i li min ::; hll Lai 0 V;llmin, so that the mapping v: q,

--->

RnC defined by v(V;) = 8i has norm IIvllcb ::;

/2.

Proof. It suffices to show the same result for q,n' \Ve may assume that VI, ... , Vn are in a von Neumann algebra equipped with a normalized trace r (for example, ]1.[2" equipped with its usual normalized trace). Then let T =

L ai 0 Vi. The CAR and the trace property imply r(\tj* V;) = r(V; \tj*) = O. Hence, if we set T = Lai 0 V;, we obtain II Laiaill = 211(I 0 r)(T*T) II < 2I1TII~in' and similarly II Laiaill ::; 21IT11~in' We conclude that

•

(9.7.4)

\Ve will now generalize Theorem 9.7.1: Instead of the generators {9d we consider all the reduced words of length d, d being a fixed integer. Recall that (after all possible cancellations have been made) any element t of a free group can be viewed as a reduced word in the generators and their inverses. We denote by It I the length (i.e., the number of letters) of this reduced word. For instance, for the empty word, we have lei = 0, and also 19i1 = Igi 1 1= 1, 19i9; 11 = 2 Vi #- j, and so on. Let G be a free group, freely generated by {9j lie f}. Let Wd = {t E G Iitl = d}. We will study the operator space spanned inside C~ (G) by {,X( t) I t E Wd}. \\'hen d > 1, the analog of RnC is a bit more difficult to describe, because there are now (d + 1) ways (and not only two) to write an element t in H'd as a reduced product L = (3-y with 0 ::; 1(31 ::; d and 1(31 + l-yl = d. Indeed, we can take for (3 the word formed by the first j letters of t and for -y the one formed of its last d - j letters. Actually, we will look at the d + 1 ways to decompose t as t = (3-y-l instead of t = (3-y, but this does not really matter here. This idea leads to a decomposition that we now describe precisely. Let H be any Hilbert space. Let a = {a(t) I t E G} be a finitely supported family in B (H). For any fixed 0 ::; j ::; d, we consider the matrix {a ~~] I (3 E Wj, -y E Wd-j} defined by [d,j]

a{3,'"'(

= a ((3-y -1)1 {1{3,",(-II=d}'

Note that here 1(3-y- 1 1 = d expresses that there is no cancellation in the product (3-y-l. We denote by Iiall [d,j] the norm of this matrix as an operator acting from l2(Wd _ j ; H) to l2(Wj ; H). The following extension of Theorem 9.7.1 is due to Buchholz [Buel] and Haagerup (unpublished).

Introduction to Operator Space Theory

188

Theorem 9.7.4. Let G be a free group and let d 2: 1. With the above notation we have (9.7.5) !v[oreover, the projection Pd: C~(G) ---+ span[A(t) I t E WdJ defined by PdA(t) = A(t)l{tE W d} is c.b. and satisfies IIPdl1 :s: d + 1 and IIPdli cb :s: 2d.

Proof. Let T =

2:

a(t) 0 A(t). We claim that there is a decomposition

tEWd

T = To

+ Tl + ... + Td

with each Tj of the form

"~

Tj =

[d,jj

a{3,"'(

0

u{3v"'(,

1{3I=j,l"Yl=d-j

where

u{3, v"'(

E B(f2(G))

are such that

As we will see, this implies

and then the triangle inequality yields the right side of (9.7.5). We now proceed to check this claim. Note that if we denote by es,t(s, t E G) the matrix units in B(f 2 (G)), we have T

=

L

a(be- 1 )1{lbc- 1 1=d} 0 eb,c·

b,cEG

,X

Note that Ibe- 11= d iff we can write (uniquely) b = (3x (reduced) and e= (reduced) with 1(3,-11 = 1(31 + 1,1 = d. Note that if 1(31 = j, after reduction the word be- 1 consists of the first j letters of b followed by the last (d - j) letters of e- 1 • Moreover, the equalities l(3xl = 1(31 + Ixl and h'xl = III + Ixl are a convenient way to express that the words (3x and 'YX are reduced. A simple calculation then shows that, if 1(31 = j, we have Ibl = j + Ixl and lei = d - j + Ixl; hence j = (Ibl- lei + d)/2. Thus the set of all pairs (b, e) in G x G such that Ibe- 11= d can be decomposed as a disjoint union Co U C 1 U··· U Cd, where Cj

= {(b, e) Ilbe-il = d, Ibl - lei + d = 2j}.

9. Examples and Comments Let Tj

Lg T

Tj =

j

189

L(b,c)ECj a(bc- 1) 0 eb,c' By the preceding discussion we have T

=

=

and moreover

L

aUh,-l)l{lth,-ll=d} 0

liJl=j,hl=d-j

Tj

L

e!3x,")'x1{liJxl=I!3I+lxl}1 {I")'xl=hHlxl}'

xEG

=

a(,B')'-1)1{iJ")'-11=d} 0UiJ V ")"

L 1!3I=j,hl=d-j

Moreover, a simple calculation shows L

UiJU~

II3I=j

= L L eiJx,iJx1{liJxl=liJl+lxl} = 1!31=j x

Hence IlL UiJU~11 ::; 1 and similarly of the above claim.

L

eb,b·

b:lbl~j

IlL v~v")'l1 ::; 1. This completes the proof

Now, an elementary verification shows that, if 9 is any fixed element in G (for instance 9 = e),

and hence

IITjll = IITj 0eg,gll::; La~~)010eiJ,")' = Ilall[d.j). 13,")'

Then, by the triangle inequality, we have d

IITII ::; L

lIall[d,j) ::; (d

+ 1) max{lIall[d,j) 10::; j

::; d},

o

which establishes the right side of (9.7.5). To establish the left side, we will first study the projection Pd. Let a: G ~ B(H) be a finitely supported function. Let Y =

L a(t) 0 A(t) tEG

and

T

= (I 0

Pd)(Y)

=

L

a(t) 0 A(t).

Introduction to Operator Space Theory

190

Then, with the same notation as before (recallllTjll :::; Ilalhd,j]), we will show that for any 0 < j < d we have Ilall[d,j] :::; 211Y11 and that, if either j = 0 or j = d, the factor 2 can be removed. Let H t = H ® 8t . We have an orthogonal decomposition H ®2 f2(G) = EeHt, tEG

relative to which Y is represented by a matrix (Y(s, t)) with coefficients in B (H) defined by Y(s,t) = a(sC 1). Now fix j with 0 :::; j :::; d. If we restrict the above matrix to act from EB1tl=d-j Ht to EB1tl=j H t , we find

(9.7.6) 1,BI=j,bl=d-j Now assume first that a is supported on Wd so that Y = T and the sum in (9.7.6) can be restricted to 1.8),-11 = d. Then, the last estimate yields Ilall[d,j] :::; IIYII = IITII, whence the left side of (9.7.5). \Ve will now estimate the norm of Pd' We no longer assume that the coefficients (a(t)) are supported on W d , but we assume that they are scalar. Then it is easy to check that lIall[d,j] :::;

(L la[d,j](t3),-1W)1/2 :::; (L la(t)12)1/2 :::; ,B,,,(

tEG

IlL

a(t)A(t)11

= IIYII·

tEG

Thus, we obtain IITII :::; L:~ IITjl1 :::; L:~ Ilall[d,j] :::; (d + 1)11Y11 and hence IIPdll :::; d + 1. We will now estimate the c.b. norm of Pd' We return to the general case of a finitely supported function a: G -+ B(H). Note that if either j = 0 or j = d, (9.7.6) implies lIall[d,j] :::; IIYII. Now assume 0 < j < d. For any tin G of positive length we denote by get) the last letter of t (equal to a generator or the inverse of one), and we define x(t) = 8g (t), so that (assuming again lsi> 0) (x(s), x(t)) = 0 iff the product st- 1 is reduced (in othe~ words, iff Ist- 11 = lsi + Itl) and otherwise (x(s),x(t)) = 1. By Exercise 1.5 and the triangle inequality, we deduce from (9.7.6)

L li3I=j,bl=d-j

a(t3),-1)(t3),-1)®e,B,,,(1-(x(t3),xb))) :::;2I1YII,

9. Examples and Comments

191

or, equivalently, (9.7.7) This gives us

and hence

IIPdlicb

~

•

2d.

Remark 9.7.5. Fix an integer D 2: 1. Let a: G -+ B(H) be a finitely supported function with support in lVo U ... U WD . Let ad: G -+ B(H) be the function equal to a on l,vd and equal to zero elsewhere, so that

L

aCt)

Q9

A(t) =

Itl:SD

L

ad(t)

Q9

A(t).

O:Sd:SD

We then have

2- 1 max max {IIadll[d 'J} < O
-

_1-

.J

-

"a(t)

~ Itl:SD

~ 2D(D

Q9

A(t)

+ 1) O:Sd:SD max ma..x {IIadll[djJ}' O:Sj:Sd •

(9.7.8)

Indeed, the left side is an immediate consequence of (9.7.7) while the right side follows from the triangle inequality and (9.7.5).

9.8. Operator space generated in the usual Lp-space by Gaussian random variables or by the Rademacher functions Let (n, A, P) be a probability space. We will say that a real-valued Gaussian random variable (in short r.v.) is standard if E"( = 0 and E"(2 = 1. We will say that a complex-valued Gaussian r.v. 1 is Gaussian standard if we can write 1 = 2- 1/ 2 h' + h") with ,,(', "(" real-valued, independent, standard Gaussian r.v.'s. Let hn In = 1,2,oo.} (resp. 1n In = 1,2,oo.}) be a sequence of real(resp. complex-) valued independent standard Gaussian r.v.'s on (n, A, P). As is well known, for any finitely supported sequence of real (resp. complex) scalars (ank.!he r.v. S = L: ani (resp. L: a;1j) has the same distribution as the variable S = (L: laiI 2 )1/2"(1 (resp. (L: laiI 2 )1/2 1t). In particular, we have for any finitely supported sequence of complex scalars (9.8.1)

Introduction to Operator Space Theory

192

Let Qp be the subspace of Lp (O, A, P) generated by FYn I n = 1,2, ... }. Then, as a Banach space, Qp is isometric to £2 for all 1 ~ p < 00. (To simplify, we will discuss mostly the complex case in the sequel, although the real case is entirely similar provided we restrict ourselves to lR-linear transformations.) Moreover, for any isometry U : Qp ~ Qp the sequence {U{1i) I i = 1,2, .. } has the same distribution as the sequence {1i I i = 1,2, .. }. If we equip Qp with the o.s.s. induced by Lp{O, A, P), it follows (see Proposition 9.5.2) that U is a complete isometry from Qp to Qpo Therefore, by Proposition 9.2.1, we have Proposition 9.8.1. For any 1 Hilbertian operator space.

~

p <

00,

the space Qp is a homogeneous

Let {en I n = 1,2, ... } be a sequence of independent, identically distributed (in short LLd.) r.v.'s on (O, A, P) with ±1 values and such that P{en = +1} = P{en = -I} = 1/2. The reader who so wishes can replace {en I n = 1,2, ... } by the classical Rademacher functions (rn) on the Lebesgue interval; this does' not make any difference in the sequel. Let Rp be the subspace generated in Lp{O, A, P) by the sequence (en). The analog of (9.8.1) for the variables (en) (or, equivalently, for the Rademacher functions) is given by the classical Khintchine inequalities (cf., e.g., [LTl, p. 66] or [DJT p. 10]), which say that, for 1 ~ p < 00, there are positive constants Ap and Bp such that, for any scalar sequence of coefficients (an) in £2, we have (9.8.2) (Note that we have trivially Bp = 1 if p ~ 2 and Ap = 1 if p :::: 2.) This implies that, as a Banach space, Rp is isomorphic to £2 for any 1 ~ P < 00. A fortiori, Rp and Qp are isomorphic Banach spaces if 1 ~ p < 00. We now wish to describe the operator space structure induced by Lp on Qp (resp. Rp). By (9.5.3), this can be reduced to the knowledge of the norm

Il L 'YnXnll L,,(fl,P;Sp) II

(resp. LenxnIiL,,(fl,p;S,,» when (xn) is an arbitrary finite sequence of elements of Sp. In other words, to describe the o.s.s. of Qp (resp. Rp) up to complete isomorphism, it suffices to produce two-sided inequalities analogous to (9.8.1) and (9.8.2) but with coefficients in Sp instead of scalar ones. The noncommutative versions of Khintchine's inequalities proved in [LuP] and [LPP] are exactly what is needed here. The case 1 < p < 00 is a remarkable result due to F. Lust-Piquard ([LuP]). The case p = 1 comes from the later paper [LPP], which also contains an alternate proof of the other cases.

193

9. Examples and Comments Theorem 9.8.2. (i) Assume 2 ~ p < 00. Tllen tllere is a constant tllat, for any tinite sequence (xn) in Sp, we llave

B~

sucll

max{II(l:x~xnf/21Is,., 11(l:xnx~)1/2tJ

(9.8.3)

~ IIl:cnXnll L,.(O,P:s,.)

~B~ max {I (l: x~xn ) 1/211 s,.' I (l: XnX~) 1/211 sJ . (ii) Assume 1 ~ p ~ 2. Tllen tllere is a positive constant for any tinite sequence (xn) in Sp, we llave

A~

stlcll tl1at,

(9.8.4) wllere we llave set

AJoreover, similar inequalities are valid witll a real or complex Gaussian U.d. sequence h'n) or (1n) in tIle place of (cn). Finally, tIle same inequalities are valid wIlen Sp is replaced by allY nOllcommutative Lp space associated to a semi-tinite faitIlfulnormal trace 011 a von Neumann algebra.

Remark. (Observed independently by l\Jarius Junge). We claim that there is a numerical constant C such that, for all 2 ~ p < 00,

Since this is only implicitly contained in [LPP] and it might be of independent interest, we will give the details explicitly. Let us denote by PI: L2 -+ R2 the orthogonal projection. Recall that the K-convexity constant of a Banach space X is defined as follows:

By a standard averaging technique, one easily verifies that

When X K(Lp(Sp»

=

Sp, since fp(Sp) embeds isometrically into Sp, we have

In [LPP] it is proved that the constants A~ are uniformly bounded when 1 :::; p :::; 2. By duality, it follows from the preceding = K(Sp).

194

Introduction to Operator Space Theory

estimate of 'IPI ®JXIILp(X)-+Lp(X) for X = Sp that there is a numerical constant C' such that, for all 2 ~ p < 00, B~ ~ C' K(Lp(Sp)) = C' K(Sp). By [MaP, Remark 2.1OJ, the latter constant is dominated by the type 2 constant, and by [T J2J the type 2 constant of Sp is equal to the best constant in the classical (scalar) Khintchine inequalities Bp (at least when p is an even integer) that is of order pl/2 when p -+ 00. Thus we obtain our claim that there is a numerical constant C such that, for all 2 ~ p < 00,

Remark. By Proposition 9.5.2, we can deduce from the known general results on Gaussian and Rademacher series in Banach spaces (cf. [MaP, Corollaire 1.3]) that the spaces 9p and 'Rp are completely isomorphic for any 1 ~ p < 00. Of course, in the case p = 2, 92 and 'R2 are completely isometric to 0 H since the space L 2 (n, A, P) itself is completely isometric to OH(J), where the cardinal J is its Hilbertian dimension. Let us now identify 9p and 'Rp as operator spaces. We start with the case p = 1, which is particularly interesting. Consider the direct sum R EElI C (as defined in §2.6) and its subspace ~ c R EElI C defined by ~

=

({x, _tx) , x E R}.

In accordance with the definitions in §2.7, we will denote by R+C the quotient operator space (R EElI C)!~. Since R EElI C is equipped with a natural o.s.s. (see §2.6), the space R + C itself is thus equipped with a natural o.s.s. as a quotient space (see §2.4). It is easy to see (cf. §2.7) that (R n C)* = R + C

completely isometrically.

(9.8.5)

In particular, R + C is isomorphic to £2 as a Banach space. We will denote by (O'i) the natural basis of R + C that is biorthogonal to the basis (8i ) of ~l. ~ R n C. Equivalently, if we denote by q: R EElI C -+ R + C the canonical surjection, then we have O'n = q(el n EEl enl). Similarly, we will denote by Rn + C n the quotient operator space (Rn EEll Cn)! ~n' with ~n = ({x, _tx) , x ERn}. We also can identify Rn + C n with the subspace spanned in R + C by (0'1. ... , O'n). Moreover, we have (Rn n C n )* = Rn + Cn completely isometrically. Now we can reformulate the main result of [LPP) in the language of operator spaces as follows:

Theorem 9.8.3. The space 'R 1 (resp. 91) is completely isomorphic to R+C . via the isomorphism that takes Ci (resp. ::Yi) to O'i·

"'·"It

Proof. It is easy to verify that the norm appearing in Theorem 9.8.2 is the dual norm to the natural norm of the space K®min (RnC). Equivalently,

9. Examples and Comments

195

in the notation of §9.5 (using (9.5.5» it coincides with the norm in the space SI [(R n C)*] = SI [R + C]. Therefore, the linear mapping that takes Ci to O'j defines an isomorphism from SdRd onto SdR + C]. Thus, the announced result for Rl follows from Proposition 9.5.3 applied with p = 1. The case of 91 is analogous. • By combining (9.8.5) with Theorem 9.7.1 we obtain a surprising connection between the standard Gaussian (or ±l-valued) independent random variables and the generators of C~(Foo):

Corollary 9.8.4. We lmve (E)..)* ~ 91 ~ Rl completely isomorphically. More precisely, let us denote by ("\*(9j)) the system in (E>.)* that is biortllOgona1 to (..\(9i». Then the mapping u : Rl ---> (E>.)* (resp. u: 91 ---> (E>.)*) defined by U(Ci) = "\*(9i) (resp. U(1i) = "\*(9j») is a complete isomorpbism. IVlore generally, for all 1 ::; p ::; 00, let us denote by R[p] (resp. C[p]) the operator space generated by the sequence {e1j I j = 1,2, ... } (resp. {ei1 I i = 1,2, ... }) in the operator space Sp equipped with its natural o.s.s. defined by interpolation (see (9.5.4». Here again we set Soo = JC. Note that R[oo] (resp. C[oo]) obviously coincides with the row (resp. column) space R (resp. C). A moment of thought (recall Exercise 2.3.5) shows that R[l] ~ R* and C[l] ~ C* completely isometrically; therefore we may identify R[l] with C on one hand, and C[l] with R on the other. IVloreover, we obviously have a natural projection simultaneously completely contractive from Sl to R[l] and Soo to R[oo] (and similarly for columns). This implies, by Proposition 2.7.6, that if we make the couple (R, C) into a compatible one (as we did in Chapter 7) by using transposition to inject R into C, then we have completely isometric identifications (with () = l/p)

R[p]

=

(R, C)o

C[P]

= (C, R)o.

Now, if we use (9.5.7), we find

(9.8.6) and Sp[C[P]] = (Soo[C], SdR])o. If we view Soo[R] (resp. SdC]) as a space of sequences of elements of Soo (resp. S1), then the norm in Soo [R] (resp. S1[C]) is

I

1/21Is=

I

1/211sJ

easily seen to be (Xj) ---> (L XjXi) (resp. (Xj) ---> (L XjX;) Therefore, since we have simultaneously contractive projections (see the discussion before Proposition 2.7.6) onto the corresponding subspaces of Soo [Soo] and SdStl, we find that the norm in (Soo[R], SIlC])o coincides with (Xj) ---> p • In other words, for any sequence (Xj) in Sp we have, by (9.8.6),

II(Lxjxj)1/21Is

(9.8.7)

Introduction to Operator Space Theory

196 Similarly, we have

IILXi 0 eilIISp[C[pJ] = II (LX;Xi) 1/211s"

(9.8.8)

We denote by R[p]nC[p] the subspace of R[P]EBC[p] formed of all couples of the form (x,t x). On the other hand, we denote by R[P] + C[p] the operator space that is the quotient of R[P] 671 C[P] modulo the subspace formed of all couples of the form (x, _tx). Then, by (9.8.7) and (9.8.8), the norm appearing on the left in (9.8.3) is equivalent to the natural norm ofthe space Sp[C[P]] n Sp[R[p]] or, equivalently, Sp[R[P] n C[P]]. Similarly, the norm III in Theorem 9.8.2 (case p ~ 2) is equivalent to the natural norm of the space Sp[C[P]] + Sp[R[P]] or, equivalently, Sp[R[P] + C[P]]. This allows us to state:

",P

Theorem 9.S.5. Let 1 < p < 00. The space 9p (or the space 'Rp)) is completely isomorphic to R[P] + C[P] if p ~ 2 and to R[P] n C[P] if p 2: 2. Proof. First observe that the natural norm in the space Sp['Rp] is equal to the norm induced by Lp(O, A, P; Sp), by the isometric case in proposition 9.5.2. Then, by (9.8.3) and the preceding discussion, the latter norm is equivalent to the natural norm of either the space Sp[R[p] n C[P]] if p 2: 2 or the space Sp[R[P] + C[P]] if p ~ 2, whence the announced complete isomorphisms by • Proposition 9.5.3. Remark. Note that Theorem 9.8.3 is nothing but the natural extension of Theorem 9.8.5 to the case p = 1. Remark 9.S.6. In the Banach space setting, it is well known that the orthogonal projection P2: L2 --4 92 (resp. Q2: L2 --4 'R 2) extends to a bounded linear projection Pp: Lp --4 9p (resp. Qp: Lp --4 'Rp), provided 1 < p < 00, and this fails if p = 1 or p = 00. (Warning: It is customary in harmonic analysis to consider that Pp and P2 are the "same" operator, since they coincide on simple functions.) In the operator space setting, the situation is analogous: For any 1 < p < 00, Pp (resp. Qp) is a c.b. projection from Lp onto 9p (resp. onto 'Rp). This can be seen easily using Proposition 9.5.2 and the fact that, when 1 < p < 00, Sp is a K-convex Banach space in the sense of [P16]. (See also [TJ1, p. 86] or [DJT, p. 258].) This can also be viewed as a corollary of Theorem 9.8.4, since the latter result implies that (Qp)* ~ 9p' and ('Rp)* ~ 'Rp' (completely isomorphically) when 1 < p,p' < 00 with ~ + -;, = 1. Indeed, the complete boundedness of the natural mapping

(9p )* =

Lp' /G;

--4

9p '

is clearly equivalent to the complete boundedness of Pp, and similarly for Qp. Using the last assertion in Theorem 9.8.2, we obtain another striking isomorphism.

9. Examples and Comments

197

Theorem 9.S.7. Let 1 :s; p < 00. Let Ep be as in Corollary 9.7.2. Then the correspondence Ci -+ )..(gi) (resp. 1i -+ )..(gi») is a complete isomorphism between the spaces Rp (resp. Qp) and Ep. Proof. It suffices to prove this for p ;:::: 2. Let (Xi) be a finitely supported sequence of elements of Spo Let

and

II(Xi)ll,\ = IILxi ® )..(g;) II S,,[E,.j . By Proposition 9.5.3, it suffices to show that these two norms are equivalent for each 2 :s; p < 00. By Proposition 9.5.3 again (isometric case), the norm II . lie coincides with the norm induced by Sp[Lp(n, A, P)], which, by "Fubini's Theorem" (cf. [P2, (5.6)]), coincides with that of Lp(n, A, P; Sp), or, equivalently, with the norm in the middle of (9.8.3). Similarly, the norm II 11,\ coincides with the norm induced by the space Sp[Lp(r)]. The latter space is identical (by Fubini again; cf. [P2, (5.6)]) to the noncommutative Lp-space associated to B(€2)®M equipped with the trace tr ® r, which we will denote simply by Lp(tr ® r). Let (~;) be any choice of signs ~i = ±l. By a well-known result, the linear map that takes )..(g;) to ~i)..(gi) (i = 1,2, ... ) extends to an isometric C* -representation from Al to Al and (by transposition) also defines a complete isometry from Af* to M*. By interpolation, the same map defines a complete isometry on Lp(r) for all 1 < p < 00. In particular, this implies that

and, therefore, after averaging over

II (Xi) 11,\ =

~i =

±1, we have

J

II (CiXi)lI,\dP.

n

(Note: A slightly weaker result, namely, the equivalence of the last two expressions, follows from Corollary 9.7.2, and this suffices for the present argument.) If we set Yi = Xi ® )..(gi) E Sp ® Lp(r), this yields

II (Xi) 11,\

=

J

IILC;Yill L,.(tr

<8Ir)

dP;

hence, by (9.8.3),

II(Xi)ILx :s; B~ max

{II (LY;Yi) 1/211

L,,(tr <8Ir)

'

I (LYiY;) 1/211

L,.(tr <8Ir)

}

Introduction to Operator Space Theory

198 But since

Yiyi = XiX; Q9 1 and yiYi = XiXi Q9 1, we obtain finally

lI(xi)IIA

~ B~ max {II (L X;Xif/21Is" ,II (Lxix:) 1/2 I sp } ~ B~II(xi)IIA.

In other words, we conclude, that the norms

II·IIA and II· liE

are equivalent .•

Curiously, the preceding result combined with the earlier Corollary 9.7.2 implies a result of independent interest on the couple (R, C), as follows.

Corollary 9.S.S. For any 1 < p < isomorpllic identities

00

and ()

=

lip we have completely

(R n C, R + C)o ~ RlP] n ClP]

if p ? 2

(RnC,R+C)o~RlP]+ClP]

ifp~2.

Proof. Indeed, by Corollary 9.7.2, the left side can be identified with E p , and, by Theorem 9.8.5, the right side can be identified with np. Thus the result follows from Theorem 9.8.7. • Remark. By a different argument (see [P2, p. 109-110]), one can show that the equivalence constants appearing in the preceding corollary remain bounded uniformly when p runs over the whole interval) 1, 00(. Thus, together with (9.7.3), this implies completely isomorphic identities Ep ~ RlP] n ClP] Ep ~ R[p]

+ C[p]

if p ? 2 if p

~

2,

with equivalence constants independent of 1 < p < 00. 'When p is an even integer, the best possible constants are computed in the remarkable paper [Buc2].

Remark 9.S.9. Let k ? 1 be a fixed integer. Let (e;)n~1. (e;)n~1. ... ' (e~)n~1 be independent copies of the original sequence (en) as above, on a suitable probability space (0, A, P). Let us denote by n~ the subspace of Lp(O, A, P) spanned by the functions of the form e~k (nl ? 1, n2 ? 1, ... ). Then, modulo a simple reformulation, the results of the paper [HP2] describe the operator space structure of the space nt for any k = 1,2, ... and its dual. (The Gaussian case is similar by general arguments.) The paper [HP2j also describes the space EA Q9min ••• Q9~in EA (k times) and proves that nt is completely isomorphic to (EA Q9min •.• Q9 m in EA) *. Here, of course, the isomorphism constants depend on k.

e;l e;2 ...

9. Examples and Comments

199

Concerning n~ for 1 < P < 00, it is easy to iterate the inequalities appearing in Theorem 9.8.2 to obtain (after successive integrations) two-sided inequalities describing the operator space structure of n~. To describe these iterated inequalities, assume for simplicity that k = 2. Let (Xij) be a matrix with entries in Sp, with only finitely many of them nonzero. Then both x = (Xij) and the transposed matrix t x = (x ji) can be viewed as elements of Sp on the Hilbert space £2 ED £2 ED .. " and we denote the corresponding norms simply by lix II s,' and Iltxlls". Then, after iteration, (9.8.3) becomes, when 2 ~ p < 00,

}

(9.8.9)

s,'

i,j

';(B;l' max

L,,(O.P;S,,)

{IIXIIS"

lI'xlis.,

Moreover, the orthogonal projection induces a c.b. projection from Lp(0., P) onto n~, for any k = 1,2, ... , so that (9.8.9) can be dualized to treat the case 1 < p < 2. We do not spell out the corresponding inequality. Again, when 2 ~ p < 00, these inequalities can be interpreted as describing n~ as completely isomorphic to the intersection of four operator spaces, as follows. First recall that, for any Hilbert space H, we denote by He (resp. Hr) the operator space obtained by equipping H with the O.S.s. of the column (resp. row) Hilbert space. Let us define Hr[P] = (Hr' Heh/ p and Hc[p] = (He, Hrh/ p ~ H,.[p'] for 1 < p < 00. (Also set, by convention, Hr[oo] = H r , He [00] = He,

Hr[l] = He, Hc[l] = Hr.) Then (9.8.9) can be interpreted as saying that n~ is completely isomorphic to the intersection SpnS~pn(S2)e[P]n(S2)r[P]. Thus we can extend essentially all the preceding discussion of np to the spaces n~. In particular, here is what becomes of Corollary 9.8.8 in the case k = 2: Let () = lip, 1 < p < 00, and let us denote K by Soo. Then the interpolation space

is completely isomorphic to the intersection

200

Introduction to Operator Space Tl1eory

and to the sum

In the case p = 1, the results of [HP2] show that n~ is completely isomorphic to the sum (S2)r+(S2)c+Sl +Srp • The case of a general k > 2 can be handled similarly, and we obtain for p 2: 2 (resp. p :::; 2) the intersection (resp. the sum) of a family of 2k operator spaces. We leave the details to the reader (see [HP2] for the cases p = 1 and p = 00). Remark. By a well-known symmetrization procedure, one can deduce from the Khintchine inequalities that, for any sequence (Zn)n::~:l of independent mean zero random variables in Lp (1 :::; p < 00), we have (for any n)

Note that the partial sums Sn L~ Zi form a very special class of martingales. The preceding inequalities were extended to the case of general martingales by Burkholder, Davis, and Gundy (see [Bur]). For a noncommutative version of the Burkholder-Gundy inequalities, with an application to Clifford martingales and stochastic integrals, see [PX].

9.9. Semi-circular systems in Voiculescu's sense In his recent and very beautiful theory of "free probability," Voiculescu discovered a "free" analog of Gaussian random variables; see [VDN] (see also [HiP]). This discovery gives a new insight into a remarkable limit theorem for random matrices, due to Wigner (1955). In Wigner's result, a particular probability distribution plays a crucial role, namely, the probability measure on lR. (actually supported by [-2,2]) defined as follows: /.lw(dt) = 1[-2,2J

J4=t2 dt/27r.

We will call it the standard Wigner distribution. We have

In classical probability theory, Gaussian random variables playa prominent role. They usually can be discussed in the framework attached to a family (-Yi)iEI (resp. (7i)iEI) of independent identically distributed (LLd. in short) real- (resp. complex-) valued Gaussian variables with mean zero and

9. Examples and Comments

201

L 2 -norm equal to 1. When (say) I = {I, 2, ... ,n} the distribution of CYi)iEl (resp. CYi)iEl) is invariant under the orthogonal (resp. unitary) group O(n) (resp. U(n)).

In Voiculescll's theory, stochastic independence of random variables is replaced by freeness of C* -random variables. We will review the basic definitions below. After that, we will introduce a free family (lFi)iEl of C* -random variables, each distributed according to the standard \Vigner distribution. Tlwse are called free semi-circular variables. The family (Wi)iEI is the free analog of CYdiEI in classical probability; it satisfies a similar distributional invariance under the orthogonal group. But actually, since we work mostly with complex coefficients, we will also introduce a free family (lV;)iEI that is the free analog of CYi)iEI; their "joint distribution" satisfies an analogous unitary invariance. Such variables are called free circular variables. \Ve now start reviewing the precise definitions of the basic concepts of "free probability," following [VDNJ. Definitions. A C* -probability space is a unital C* -algebra A equipped ,dtll a state 'P (a state is a positive linear form of norm 1). lYe will say that an element x of A is a C*-random variable (in short, C*-r.v.). Ifx is self-adjoint, we will say tl1at it is a real C* -r. By definition, the distribution of a real C* -r. v. x is tlle probability measure ILx on JR such that

'T.

It follows that, for any continuous function f: JR 'P(f(x)) =

J

---->

f(t)/Lx(dt),

JR, we haye (9.9.1)

Indeed, we can approximate f by a sequence of polynomials uniformly on every compact subset. Hence, ill particular, for all 0 < p < 00, (9.9.2) l\,Ioreover, if 'P is "faithful" on the C* -algebra Ax generated by x (meaning that 'P(y) = 0 for y ~ 0 implies y = 0), then the support of /Lx is exactly the spectrum of x, denoted by a(x). Therefore, we can record here the following fact: Let (A, 'P) and (B, 1/J) be two C* -probability spaces with 'P and 1/' faithful. Let x E A and y E B be two real C* -r. v. with the same distribution, that is, such that /Lx = /Ly. Then we necessarily have IIxll = lIyll (where IIxll is the norm. in A and lIyll the norm. in B).

202

Introduction to Operator Space Theory

This property is immediate, since

Ilxil = SUp{IAII AE a(x)}.

(9.9.3)

It can also be obtained by letting p tend to infinity in (9.9.2). Note that it suffices that

=

J~tr(a(w))dP(w).

Assume moreover that a(w) = a(w)* almost surely. Let (Al(W), ... , An(W)) be the eigenvalues of the matrix a(w). Then the distribution J-ta of the real C*-r.v. a is nothing but

JL 1

J-ta

=

~

n

8)";(w)dP(w).

1

Definitions. Let (An,
More generally, we can define the joint distribution of a family x = (Xi)iEI of real C*-r.v.'s, but it is no longer a measure: We consider the set of all polynomials in a family of noncommuting variables (Xi)iEI. First we define

then we extend F linearly to a linear form on P(J). We will say that F is the "joint distribution" of the family x = (Xi)iEI. If we give ourselves for each n such a family (Xf)iEI with distribution Fn, we say that (Xf)iEI converges in distribution to (Xi)iEI if F n converges pointwise to F.

9. Examples and Comments

203

Let (A,cp) be a C*-probability space and let (Ai)iEI be a family of subalgebras of A. We say that (Ai)iEI is free if CP(ala2 ... an) = 0 every time we have aj E Ai.i,il -I- i2 -1- ... -I- in and cp(aj) = 0 Vj. Let (Xi)iEI be a family of C*-r.v.'s in A. Let Ai be the unital algebra (resp. C* -algebra) generated by Xi inside A. We say that the family (Xi)iEI is free (resp. *-free) if (Ai)iEI is free. The preceding definitions are restricted to C*-r.v.'s, which correspond to bounded r.v.'s in the commutative case, but, by convention, we will use the same definition whenever we are dealing with a sequence in a noncommutative Ll-space. So, from now on, convergence in distribution or weak convergence means the convergence of all moments just like in the preceding definitions (of course, for this to make sense, we implicitly assume that all moments exist, like in the Gaussian case, for instance). When the limit distribution is determined by its moments (in particular, if it is compactly supported, or in the Gaussian case) this coincides with the usual notion of convergence in distribution. We can now reformulate Wigner's Theorem in Voiclliescu's language. Fix n ~ 1. We introduce the random (real symmetric) n x n matrix

an

=

(9ii h:S;ij:S;n

with entries defined as follows: {9ij 1 i ::; j} is a collection of independent Gaussian real-valued r.v.'s with distribution N(O, l/n) (Le., E(9ij) = 0 and EI9ijl2 = l/n) and 9ij = 9ji Vi > j. We assume these (classical sense) random variables defined on a sufficiently rich probability space (for instance, the Lebesgue interval). Let An = Loo(O, A, P; lIIn) and let CPn be the state defined on An by setting Vx E An

CPn(X)

=

J~tr(x(w))dP(w).

Then, Voiculescu's reformulation of Wigner's Theorem is Theorem 9.9.1. If we consider an as a real C*-r.v. relath;e to (An' CPn), tllen we have tlle weak convergence of probability measures: I-lG"

-4

I-lw

when n

-4

00.

More generally, Voiculescu showed: Theorem 9.9.2. Let (ai )iEI be a family of independent copies (in the llsual sense) of the random variable an. Then, when n - 4 00, the family (ai)iEI converges in distribution to a free family (Wi)iEI of real C*-r.v.'s eac11 with the same distribution equal to I-lw.

We will say that a real C*-r.v. X is semi-circular if there exists A > 0 such that the distribution of AX is equal to I-lw. If A = 1, and if X admits exactly

Introduction to Operator Space Theory

204

J.lw for its distribution, then we will say that x is semi-circular standard. We then have cp(x) = 0, cp(X2) = 1. (We should warn the reader that our standard normalization differs from that of [VDN].)

Actually we even have an almost sure result as follows:

Theorem 9.9.3. Let (Gf )iEI be as above. Then, for almost all w in 0, we have sup IIGf(w)IIMn <

00

n

for each i and, moreover, the distribution of (Gf(W))iEI (on Mn equipped with its normalized trace) tends to that of (Wi)iEI when n --+ 00.

Proof. This is based on a (so-called) "concentration of measure" argument. Actually, the most basic form of Sobolev's inequality in the Gaussian setting will suffice for our purposes. Let {gm} be any collection of independent standard (Le., N(0,1)) real-valued Gaussian random variables. Then it is well known (cf., e.g., [Chel) that, for any polynomial F = F(gbg2," .), we have (9.9.4) where

Now let P and let

= P({Xd)

be a polynomial in noncommutative variables {Xi},

As we already mentioned, we have almost surely lim sup IIGfllM" n-+oo

and also for any p <

< 00

00

limsuplEIIGfll~n n-+oo

<

00.

•

Using this, it is rather easy to deduce from (9.9.4) that there is a constant Cp depending only on P such that (9.9.5) Indeed, it suffices to majorize IIV' FII2 accordingly when P is a monomial, that is a finite product Xit X i2 ••• Xi", and in that case the preceding observations

9. Examples and Comments

205

and the specific normalizations ofTn and G? yield (9.9.5). Then (9.9.5) implies for any E > 0 P(lFn -lEFnl > E) < 00,

L

and hence for almost all w lim sup IFn -lEFnl = O. n ....... oo

On the other hand, by Voiculescu's central limit theorem (i.e., Theorem 9.9.2) we know that lEFn ---> r(P({Wd))· Hence, we conclude that for almost all w we have

Tn(P({G?(w)}))

--->

(9.9.6)

T(P({Wd))·

Finally, taking a suitable countable intersection, we find a subset n' c n with p(n') = 1 such that, for any win n', (9.9.6) is satisfied by all polynomials P with (say) rational coefficients and moreover sUPn IIG?(w) II < 00. This yields the announced result. •

Remark. In the preceding proof the passage from convergence in distribution to the almost sure one is based on a very general principle called the concentration of measure phenomenon. which has many important applications in geometry and analysis (see [l'dS]). As pointed out by Voiculescu in [V02, pp. 216-217], the random matrix situation (either in the Gaussian or unitary case, for instance) is usually so "extremely concentrated" that results such as the preceding one are then easy to derive using this principle. Note in passing that we chose to use only the elementary Sobolev bound (9.9.4), but more refined estimates are actually available (see, e.g., [1\1S] or [P8, pp. 4448]). For other methods to obtain similar almost sure results see [HiP, T]. In Voiculescu's theory, the analog of an independent family of standard real Gaussian variables is a free family of standard semi-circular C* -r.v. 'so Such a family can be realized on the full Fock space, as follows. Let H = £2(1). Recall we denote by F(H)(or simply by F) the full Fock space associated to H; that is to say, we set 'Ho =
We consider from now on 'Hn as a subspace of F. For eVf'ry h by €(h): F ---> F the operator defined by: €(h)x

=

h@x.

E

F, we denote

Introduction to Operator Space Theory

206

More precisely, if x = A1 E ?to = C1, we have £(h)x = Ah, and if x = 0 X2'" 0 Xn E ?tn, we have £(h)x = h 0 Xl 0 X2'" 0 X n • We will denote by n the unit element in ?to = C1. The C* -algebra B(:F) is equipped with the state c.p defined by cp(T) =< Tn, n > . Xl

Let (ei)iEI be an orthonormal basis of H. The pair (B(:F),cp) is an example of a C*-probability space. Moreover, cp is tracial on the C*-algebra generated by the operators £(ei) + £(ei)* (i E I), that is, we have cp(xy) = cp(yx) for all x, y in this subalgebra. (Note however that cp(£(h)*£(h)) = (h, h) and cp(€(h)f(h)*) = 0, so that cp is not tracial on the whole of B(:F)!) In this subalgebra, let

Then the family (Wi)iEI is an example of a free family of standard semicircular C* -r.v.'s, or, in short, a standard semi-circular free family. This family enjoys properties very much analogous to those of a standard independent Gaussian family (gdiEI. Indeed, for every family (O:i)iEI E lR(I) with E O:~ = 1 the real C* -r.v. S = EiEI O:i Wi admits /-lw as its distribution. This is analogous to the rotational invariance of the usual Gaussian distributions. More explicitly, this means that, for every continuous function f: lR ---+ lR, we have

cp(f(S» =

!

f(t)/-l\V(dt).

In particular, by the fact preceding (9.9.3), for all finitely supported families of real scalars we have

II LO:iWili = 2(Lo:~)1/2. iEI Thus, the operator space lR-linearly generated by (Wi)iEI is isometric to a real Hilbert space. We now pass to the complex case. Let (Zi)iEI be a family of (not necessarily self-adjoint) C*-r.v.'s. We can then consider the distribution F of the family of real C*-r.v.'s obtained by forming the disjoint union of the family of real parts and that of imaginary parts of (Zi)iEI. We will say that F is the joint *-distribution of the family (Zi)iEI. Of course, if the family is reduced to one variable Z, we will say that F is the *-distribution of Z. Note that the data of the *-distribution of (Zi)iEI are equivalent to those of all possible moments of the form

cp(Xit X i2 where Xi

= either Zi

••

.Xin ),

or Z; and where il, i2 •...• in are arbitrary in I.

9. Examples and Comments

207

We now come to the analog of complex Gaussian random variables. Let (W', W") be a standard semi-circular free family (with two elements). We set W = ~(W' + iW"). Every C*-r.v. having the same *-distribution as TV (resp. as AlV for some A > 0) will be called "standard circular" (resp. "circular"). Suppose we are given a (partitioned) orthonormal basis {ei liE I} u {Ii liE I} of H. Then, one can show that Wi = £(ei) + £Ui)* is a *-free family of standard circular C* -r.v.'s (in short, a standard circular *-free family). Now, let (TVi )iEI be any *-free family formed of standard circular variables. Then, for any finitely supported family (O:i)iEI of complex scalars with L IO:il 2 = 1, the variable S = LiEI O:i Wi has the same *-distribution as W. As above, we have (9.9.7) II LO:iWil1 = 2(L IO:iI 2)1/2 (one can verify that IIWII

= 2).

Let VI be the operator space spanned by this family {lVi 1 i E I}. By (9.9.7), VI is isometrically Hilbertian and (Tl'i)iEI is an orthonormal basis. l\Ioreover (see [VDN, p. 56]) for any isometric transformation U: VI -+ VI the family (U(Wi»iEI has the same *-distribution as (Wi)iEI. In particular, by the next lemma, this implies that the operator space VI is homogeneous.

Lemma 9.9.4. Let (A, if) and (B, f/!) be two C*-probability spaces with if and'l/J faitllful. Let (Zi)iEI and (Yi)iEI be two families ofC*-r.v. 's ill A and in B, respectively, admitting the same joint *-distribution. Let A z (resp. By) be the C*-algebra generated by (Zi)iEI (resp. (Y;)iEI), and let Ez C Az (resp. E y C By) be tIle operator space spanned by the families. Then the linear mapping U defined by U(Zi) = Yi extends to a complete isometry from E z onto E y and actually to an isometric representation from Az onto By. Proof. Without restricting the generality, we may replace the family (Zi )iEI by the disjoint union of the families (Z;)iEI and (Z;)iEI , and similarly for the family (Yi)iEI. Then let P = LO:iti2 ... i~.Zil ... Zh. be a polynomial with complex coefficients in the noncommutative variables (Zi )iE I. We set

Then, since (Zi) and (Y;) have the same joint *-distribution, P* P and 7r(P)*7r(P) have the same distribution; hence, by the fact stated before (9.9.3), since if and 'l/J are faithful, IIPII = 117r(P)II. In particular, 7r extends to an isometric representation from Az onto By. A fortiori (cf. Proposition 1.5), t.he restriction U of 7r to Ez is completely isometric. •

208

Introduction to Operator Space Theory

Actually, it is very easy to identify the operator space VI (up to complete isomorphism), as the next result shows (see [HP2] for some refinements).

Theorem 9.9.5. The operator space VI gellerated by a standard circular *free family (Wi)iEI is Hilbertian and homogeneous. Similarly, the closed span of a free semi-circular family {(Wi)iEI} is 2-Hilbertian and 2-homogeneous. Moreover, if (say) I = N, each of these spaces is completely isomorphic to R n C or, equivalently, to E>.. Proof. We already saw that VI is Hilbertian and homogeneous. Let (ai)iEI be a finitely supported family in B(H). The identity Wi together with L £( ei)£( ei)* ::; I yields

=

£( ei) + £( ei)*

II I>i ® Willmin ::; II I>i ® £(ei)IImin + II I>i ® £(ei)*IImin ::; II L ai a ill 1/ 2 + II L ai ai'II 1/ 2 whence (9.9.8) Conversely, it is easy to check that ip(Wt W j ) = ip(Wj Wt) 0 if i :I j and = 1 otherwise. Hence, letting T = L ai ® Wi, we have II L ai a; II = 11(1 ® ip)(T*T)II ::; IITII~in' and similarly we have II Laiai'li ::; IITII~in' It follows that (9.9.9) The inequalities (9.9.8) and (9.9.9) imply that span[Wi liE I] is 2-Hilbertian and 2-homogeneous. For simplicity, we assume I = N in the rest of the proof. By (9.7.3) and Theorem 9.7.1, the last two inequalities imply that the closed span of (Wi)iEI is completely isomorphic to E>. or, equivalently, to R n C. Finally, as the variables lVj = (lV; + iW;')2- 1 / 2 appear as a sequence of "blocks" (normalized in £2) on a standard semi-circular system, the same inequalities (9.9.8) and (9.9.9) remain valid if we replace (W;);EJ by (W; )iEI. Therefore, we conclude that VI itself is completely isomorphic to E>. or to R n C. This· last point can also be deduced from the concrete realization Wi = £(ei)+£(Ji)* already mentioned for a standard circular *-free system. •

Remark 9.9.6. Let M be the von Neumann algebra generated by a free semicircular family (Wi)iEI. We assume I = N for simplicity. Recall a classical notation: For any x in M, we define Xip E M .. by Xip(Y) = ip(Yx) for all y in M. Thus we obtain a continuous injection M - M .. that allows us to

9. Examples and Comments

209

consider the int.erpolation spaces (1I1,lIf*)o for 0 < () < 1. Let us denot.e for simplicity L'X!(
T(x) =

L8

i


Let V: R n C -+ 1If be the mapping defined by V(8 i ) = Wi. By (9.9.8), the composition VT: M -+ M satisfies IWTllcb ~ IWlicb ~ 2. Moreover, VT is the adjoint of an operator on l\{* and VT "coincides" with P on the *-algebra generated by (Wi)iEI. Therefore, VT is a completely bounded projection from III onto Woo, which naturally extends P. By transposition, we obtain a c.h. projection from LI (
Theorem 9.9.7. For simplicity, let I = N. Let (Wi)iEI (resp. (Wi)iEI) be a standard free semi-circular (reps. *-free circular) family. For 1 ~ P ~ 00, let Wp (resp. Wp) be the closed span of (Wi)iEI (resp. (Wi)iEI) in Lp(
Introduction to Operator Space Tl1eory

210

Note that there is a Fermionic variant in which a free semi-circular family appears as the limit or suitably normalized matrices with entries satisfying the CAR. On the other hand, the reader will find in [Sk] a description of the applications of Voiculescu's theory to von Neumann algebras.

9.10. Embeddings of von Neumann algebras into ultraproducts In this section, we discuss ultraproducts in the von Neumann sense. This is a slightly different notion from the usual one considered in §2.8. We will prove in detail that the von Neumann algebra of a free group embeds into a (von Neumann sense) ultraproduct of finite-dimensional matrix algebras. We will also describe several important related open questions for which operator space theory might be useful. Let {M(n) I n ~ I} be a sequence of von Neumann algebras equipped with normal faithful traces {Tn I n ~ I} with Tn(1) = 1. Let B = foo({l\f(n) I n ~ I}), and let U be a free ultrafilter on N. We define a functional f E B* by setting for all t = (tn)"l~l in B

Clearly fu is a tracial state on B. Let Hn be the Hilbert space associated to the GNS construction for (M(n), Tn). Note that Hn can be viewed as the noncommutative version of the L 2-space over a probability space. (Indeed, if M (n) is commutative, Tn can be identified with a probability on its spectrum and Hn ::= L2(Tn». Let ~n be the (cyclic) unit vector of Hn corresponding to the identity of M(n). We then have 'V

Xn

E

M(n)

Let Hu = llHn/U. We will denote by (h,;,) the equivalence class in Hu of a bounded sequence (h n ) with hn E Hn for all n. Clearly, M(n) acts by left and right multiplication on Hn. Therefore, passing to the ultraproduct we obtain representations

L:

B

-+

B(Hu)

and

R:

BOP

-+

B(Hu)

defined by

But actually, this Hilbert space Hu is ''too large." We need to reduce it and to consider the restriction of Land R to a smaller space H u , which we now describe. The space Hu can be defined as the Hilbert space associated to the tracial state fu in the GNS construction applied to B. We denote by L: B

-+

B(Hu)

and

R: BOP

-+

B(Hu)

9. Examples and Comments

211

the representations of B corresponding to left and right multiplication by an element of B. More precisely, let p(x) = limu Tn(X~xn)1/2. Clearly p is a Hilbertian semi-norm on B. Let

Iu = ker(p). Then Iu is a closed two-sided ideal, and Hu is defined as the completion of B / Iu equipped with the Hilbertian norm associated to p.

= (tn)n in B we denote by. i the equivalence . class of tin Bllu. Then Land R are defined by L(x)t = ti and R(t)t = &. Clearly, since lu is For any t

tracial, these are contractive representations of B on Hu. Note that we have a natural isometric embedding (9.10.1)

Hu'---->Hu,

which takes t to t. Clearly Land R are nothing but the restrictions of Land R to the invariant subspace Hu. The next result seems to go back to Mac Duff's early work (see [Sa]). Note that the kernel Iu is not weak-* closed in B, so the fact that the quotient B IIu is a von Neumann algebra is a priori somewhat surprising. We call B IIu the (von Neumann) ultraproduct of the family (M(n), Tn) with respect to U. Note that a priori, this is a quotient of the (Banach space) ultraproduct described in §2.8. Theorem 9.10.1. The kernels of Land R coincide with the set'

After passing to tlle quotient, Land R define isometric representations Lu: Bllu

~

B(Hu)

and Ru: BOPllu

~

B(Hu)

with commuting ranges. Moreover, lu defines a faithful trace such that, if q: B ~ B I Iu denotes the quotient map, we have ru(q(t))

=

lu(t)

[Lu(Bllu)l' = Ru(BOPllu)

and

TU

on B I Iu

"It E B.

Finally, tlle commutants satisfy [Ru(BOPllu )]' = Lu(Bllu).

In particular, the ranges Lu (B I Iu) and Ru (BOP I Iu) are von Neumallll subalgebras of B(Hu ), and tlley are factors (i.e., their center is reduced to the scalars) if all the M (n) are themselves factors. Proof. Let In be the unit of M(n) and let fu(t)

=

(L(t)~,~)

=

~ =

(In)n

(R(t)~,~).

E

B. We have

212

Introduction to Operator Space Theory

If L(t) = 0, then L(t*t) = 0, which, by the preceding line, implies fu(t*t) = OJ hence t E Iu. Conversely, if t E I u , then x*t*tx E Iu for any x in Band

hence (Cauchy-Schwarz) fu(x*t*tx) = 0, which means tx = 0 for all x in B or, equivalently, L(t) = O. A similar argument applies for R, so we obtain that ker( L) = ker( R) = Iu. Then, after passing to the quotient by Iu, Land R define the isometric representations Lu and Ru with the same respective ranges. Therefore, Lu and Ru still have commuting ranges. Finally, let T E B(Hu) be an operator commuting with Lu(B/lu), that is, T E Lu(B/lu)'. We will show that T must be in the range of Ru. Let

/3 = T(~)

E

Hu·

We claim that there is b = (b n ) in B such that /3

T

= b and that

= R(b) = Ru(b).

Indeed, we have for any t = (tn) in B TL(t)~

= L(t)T~ = L(t)/3j

(9.10.2)

hence (9.10.3) We now use the embedding (9.12..1). Let (/3n) with /3n E Hn and sup II/3nIIH.. < 00 be a representative of /3 in Hu. Then (9.10.3) implies for any tin B (9.10.4) Let /3n = hnvn be the polar decomposition of /3n in Hn (see the next remark) with hn E H n , hn ~ 0, Vn partial isometry in .M(n), and hn = (/3n/3~)1/2. Fix c > O. Let Pn be the spectral projection of h n associated to )IITII + c, 00(. Note that /3n/3~Pn = h;Pn ~ (IITII + c)2Pn. A priori (Pn) defines an element of flu, but actually, since /3 E Hu, it is rather easy to show that (Pn) also corresponds to an element in Hu. We leave this point to the reader (see the next remark for some clarification). Hence (9.10.4) implies (with tn = Pn)

This forces limu Tn(Pn) = 0, and hence limu Tn(/3n/3~Pn) = 0., Therefore, if we set finally bn = (1 - Pn)hnvn, we find IIbnll :::; 11(1 Pn)hnll :::; IITII + c and lI/3n - bnllkn :::; IIPnhnvnllkn :::; Tn(/3n/3~Pn)j hence limu II/3n - bnllHn = O. Let b = (bn). Note that bE B with IIbllB :::; IITII + c. Then, going back to (9.10.2), we obtain finally

TL(t)e = L(t)/3 = L(t)b = (tnbn) = Ru(b)L(t)e.

9. Exa.mples a.nd Comments

213

This shows that T = Ru(b), which completes the proof that Lu(Bj1u)' = Ru(BOPj1u). The same argument clearly yields Ru(BOPj1u)' = Lu(Bj1u). and hence Lu (B j Iu)" = Lu (B j Iu ), which proves (von Neumann's bicomnl\ltant theorem) that Lu(Bj1u) is a von Neumann algebra. •

Remark. In the above, we invoked the polar decomposition in H n , viewing Hn as the noncommutative L 2 -space relative to Tn and using its structure as a bimodule over M(n). But actually, we will apply the preceding result only in the case when A/(n) is a matrix algebra. In that case, all the points left. to the reader in the preceding proof are especially easy to verify and the polar decomposition is then done in M(n) itself. Remark 9.10.2. By construction, Hu appears as the closure in Hu of the subspace of all elements of the form (b;,) with sUPn IlbnIlAI(n) < 00. Alternatively, Hu c flu can also be described as the subspace corresponding to the "uniformly square integrable" sequences. More precisely, let fi = ((3n )" be an element of flu, with sup" II.BnIIH.. < 00. Then fi belongs to Hu iff

or iff e~~ limu Tn(.B~.Bn1{i3~i3n>e})

= 0,

where we have denoted (abusively) by l{be} the spectral projection of the Hermitian operator h for the interval )c, 00(. A group G is called residually finite if there exists a collection of finite groups (G;) and homomorphisms '-P;: G --+ G; separating the points of G: that is, for any finite subset S c G there is an i for which the restriction of '-P; to S is injective. Without loss of generality, we may assume that G i = Gjr;, where each C eGis a normal subgroup with finite index and '-Pi is the canonical quotient map. Thus, G is residually finite iff it admits a family of normal subgroups with finite index (r i ), directed by (downward) inclusion and such that niEI G i = {e}.

Corollary 9.10.2. ([Wal]) Let G be any countable residually finite discrete group. Then V N( G) embeds into an ultraproduct of the form B j Iu as above with all the algebras AI (n) finite-dimensional. Proof. Let r 11 be a decreasing sequence of normal subgroups of finite index with intersection reduced to the unit e, and let G n = G jr n' Let '-Pn: G --+ G n denote the quotient morphism and let JI[(n) = VN(G n ) equipped with its normalized trace T 11' We consider B = £00 ( {1If (n) In;::: I}) and its quotient Bj Iu as above. For any tin G, we denote by yet) the equivalence class modulo Iu of (),G .. (t))n~l E B.

Introduction to Operator Space Theory

214

Let

7

denote the normalized trace on VN(G). We then have

IW 7 {.XG .. (t)) = 7(AG(t)).

"It E G

n

This implies that the family {y(t) I t E G} has the same *-distribution with respect to 7U as the family {AG(t) I t E G} with respect to 7. Therefore, by Lemma 9.9.4, the von Neumann algebras that they generate are isomorphic, • via the isomorphism taking AG(t) to y(t). The following fact is classical. Lemma 9.10.3. Free groups are residually finite. Proof. Let G = Fl. Let {gi liE J} be the (free) generators. Let C c G be a finite subset. It suffices to produce a (group) homomorphism h: G --+ r into a finite group r such that, for any e in C, we have h(e) '" er if e '" e, where er denotes the unit in rand e the unit in G. We may assume that C c G', where G' is the subgroup generated by a finite subset {gi liE J} of the generators. Let k = max{lel leE C} (recall lei is the length of e). We then set S

= {t E G' Iitl :::;

k}.

We will take for r the (finite) group of all permutations of the (finite) set S. For any i in J, we introduce

Then clearly Si C Sand giSi C S. Hence (since ISil = IgiSil and S is finite) there is a permutation O'i: S --+ S such that O'i(S) = giS for any S in Si' Thus if s,t E S and if git = s (or, equivalently, t = g;ls), we have O'i(t) = S (or, equivalently, t = O';I(S)). Thus it is easy to check that if a reduced word t = g:11 g:; ... g:::: (m :::; k ei = ±1) lies in S (note that, by definition of S, e and all the subwords of t also lie in S), we have

Therefore, if we define h: G --+ r as the unique homomorphism such that h(gi) = O'i Vi E J and O'(gi) = er Vi 1. J, we find, for t as before, h(t) = 0':110':; .. . 0':::: and h(t)(e) = t in particular, we have h(t) '" er whenever t E S and t '" e. Since C C S, we obtain the announced result. • Consequently:

9. Examples and Comments

215

Theorem 9.10.4. ([lVal]) Tile \'011 NeunuI11ll algebra of the free groups lF n or lFoo embeds into a von Neumann uitraprodlzct of matrix a.lgebraB. We wish to give a second proof of the preceding result using Voiculescu's "matrix model" for free semi-circular systems. This approach is well known to specialists. See also [T] for more along this line. Theorem 9.10.5. Let !If be the von Neumann algebra generated by a free semi-circular system (VVi)iEI (as in Theorem 9.9.2) witll I coulltable. Then !If embeds illto a VOll Neumanll ultraproduct of matrix algebras. Proof. Let Tn (resp. T) be the normalized trace 011 AIn (resp. on AI). Let B = £00 ( {lIIn I n 2:: I}). Let U be any free ultrafilter on N and let Iu be as before. By Theorem 9.9.3 (using the same notation), we know that, for almost all w, (Gi(w)k::l is in B. I\,Ioreover, if we denote by Gi(w) the equivalence class modulo Iu associated to (Gi(W))n;?:l, then the family {Gi(w) liE I} has the same distribution with respect. to (BjIu,Tu) as the system {Wi liE I} with respect to T. Thus, by Lemma 9.9.4, the von Neumann algebra.,> they generate must be isomorphic. • In [Col], Alain Connes observes in pa.'>sing the above Theorem 9.10.4 and casually asks whether the same embedding is valid for any III factor. This has become one of the main open problems in von Neumann algebra theory. Actually, since, by desintegration, any finite von Neumann algebra (on a separable Hilbert space) can be seen as the "direct integral" of a field of factors, one may reformulate Connes's question as follows: Problem. Let (!If, T) be a von Neumann algebra (on a separable Hilbert space) equipped with a faithful, norma.l, and normalized trace T. Is tllere a.lways a (trace presenTing) embedding of Af (as a von Neumanll suba.lgebra) in an ultraproduct of matrix algebras?

9.11. Dvoretzky's Theorem

It is natural to wonder whether there is an analog for operator spaces of the famous Dvoretzky Theorem on the spherical sections of convex bodies. This fundamental result can be formulated as follows: Every infinite-dimensional Banach space X contains, for each c > 0, a sequence of subspaces En C X (n = 1,2, ... ) such that En is (1 + c)-isomorphic to e~ for every n. Vitali Milman (see [MS] and [FLM]) gave a remarkable proof based on Paul Levy's isoperimetric inequality on the unit sphere of lRn. It is then tempting to try and find in every (infinite-dimensional) operator space X a sequence of subspaces (En) uniformly completely isomorphic to OHn . Unfortunately,

216

Introduction to Operator Space Tlleory

this is absurd: For a counterexample it suffices to consider a homogeneous Hilbertian X different from OH (for instance, R or C). Then, by Proposition 9.2.1, every n-dimensional subspace En C X is completely isometric to the ndimensional version of X (for instance, Rn if X = R). Therefore, one cannot find inside X anything else but "copies" of X. The best that one can hope to find in an arbitrary (infinite-dimensional) operator space seems to be a sequence of almost Hilbertian and almost homogeneous subspaces (En) (here "almost" means up to €) with dim En = n for aU n. In this form, it is rather easy to adapt Milman's ideas (the concentration of measure phenomenon; see [MS]) to the quantum situation. There is a slight difficulty due to the noncompactness of the unit ball of the quantum scalar coefficients, that is, the unit ball of IC (see the discussion in §2.11). To circumvent this difficulty, we choose the language of ultraproducts, which is convenient in the present situation. See §2.8 for precise definitions. With the ultraproduct terminology, we can reformulate Dvoretzky's Theorem as follows: Theorem 9.11.1. Let (Xi)iEI be a family of infinite-dimensional Banacll spaces (or merely sucll tllat limu dim(Xi ) = 00). Tllen tlleir ultraproduct lliEIXi/U contains a subspace isometric to £2. The "quantum" version of this statement is then (cf. [P9]): Theorem 9.11.2. Let (Xi);E! be a family of infinite-dimensional operator spaces (or merely sucll tllat lirnu dim(X;) = 00). Tllen tlleir ultraproduct lliEI Xi/U contains a subspace completely isometric to an infinite-dimensional Hilbertian llomogeneous operator space. This statement explains perhaps why Hilbertian homogeneous operator spaces appear so often in "nature."

Chapter 10. Comparisons In the previous chapters, we have met the following examples of homogeneous (or A-homogeneous) Hilbertian operator spaces:

OH,

<1>,

R,

C,

R n C,

R

+ C,

min(€2),

max(€2)'

We will now show that all the spaces in that list are mutually completely nonisomorphic. More precisely, let E, F be any two spaces in the preceding list, and let En, Fn be their n-dimensional version. We will show that dcb(En , Fn) --+ 00, and we will give an asymptotic estimate of dcb(En' Fn) when n --+ 00. Our aim is to illustrate what can be done with the ideas of operator space theory, by elementary tools. In the very early days of Banach space theory, methods were discovered to distinguish, up to isomorphism, the so-called classical Banach spaces, such as the spaces €P or LP (see [Ba, Chapter 12]) and also to estimate the Banach-Mazur distances d(€~, €~) with 1 :=:; p =I- q :=:; 00. Our aim in this chapter is to initiate an analogous study for operator spaces. Many estimates in this chapter come from C. Zhang's PhD thesis ([Z3]) or can be easily derived by his methods. (The only exception may be the fact that and max(R 2) are not isomorphic, which was only established in [JP].) The following result, from [Z3], will be very convenient throughout this chapter (see also [ZI, Z2]). Proposition 10.1. Let E, F be two Hilbertian homogeneolls operator spaces. Assume for simplicity that E and F are isometric to €2. Let (en) (resp. (in)) be an orthonormal basis of E (resp. F). Let En = span[eJ, ... , en] (resp. Fn = span[ft, ... , In])·

(i) Let Un: En

--+ Fn (resp. u: E --+ F) be the (isometric) linear map defined by un(ei) = Ii (resp. u(ei) = Ii). Then we llal'e

(10.1) and dcb(E, F) = Ilullcbllu-11lcb = sup Ilunllcbllu~lllcb.

(10.2)

n

(ii) We also have (10.3)

Proof. Using the polar decomposition of an n x n matrix, it is easy to see that dcb(En, Fn) = inf{lIvllcbllv-1Ilcb}, where the infimum runs over all maps v: En --+ Fn of the form v(ei) = Adi with Ai > O. Let a be a permutation of [1,2, .. , n]. Let Uu (resp. Vu ) be the unitary transformation defined on En by

Introduction to Operator Space Theory

218

Uu(ei) write:

= eU(i) (resp. Vu(fi) = !u(i»). Since En, Fn are homogeneous, we can

Since ~ L:u Vu-l V Uu = (l/n)(L: Ai)Un , we deduce Ilunllcb(l/n)(L: Ai) ::; IIvllcb. Similarly, exchanging the roles of En and Fn , we find Ilu;;lllcb(l/n) (L: Ail) ::; IIv- 11Icb. By Cauchy-Schwarz, we have 1 ::; n- 1 (L: Ai)n- 1 (L: Ai 1), whence Taking the infimum over all possible v, we find

Since the inverse inequality is obvious, we obtain (10.1). Let w: E ---4 F be an arbitrary isomorphism. By Proposition 9.2.1, w(En) is completely isometric to Fn. Therefore we have dcb(En' Fn) ::; deb (En , w(En)) ::; Ilwllebllw-11Ieb, whence deb(En , Fn) ::; deb(E, F). This implies (10.2)' sup Ilunllebllu;;llleb ::; dcb(E, F). n

Note then that lIulicb (resp. lIu- 1I1eb) is the nondecreasing limit of lIunlleb (resp. lIu;;ll1eb) when n ---4 00. From the obvious inequality dcb(E, F) ::; lIullcbllu-11Ieb, we deduce the converse of (10.2)' and obtain (10.2). To show (ii), observe that if u: E~ ---4 En denotes the map defined by the canonical isometry from E~ onto En, and if v: OHn ---4 En is an arbitrary isometry, we have u = vtv hence lIulleb = Ilvll~b by (7.9). • Remark. In [Z2], Zhang made the interesting observation that, for any normed space E,

deb(min(E), max(E» = II min (E)

---4

inax(E)lIeb,

or, equivalently, deb(min(E),max(E» is equal to Paulsen's constant a(E) described in Chapter 3. The argument is quite simple: Let iE: min(E) ---4 max(E) be the identity. Then, for any complete isomorphism u: min(E) ---4 max(E) we have iE = (iEU- 1 )u, but since max(E) is homogeneous, we have lIiEU-Illeb = IliEU-111 = lIu- 111; hence

IliEllcb::; which implies

lIiEU-Illcbllulieb :$

lIu-Ililiulieb :$ Ilu-Illebllulleb,

IliElieb :$ deb(min(E),max(E».

10. Comparisons

219

•

The converse is obvious.

Convention. Throughout the sequel, we will write an ~ bn if there exist positive constants C1 and C2 such that C1 an :S bn :S C2an for all n. Notation. Let En and Fn be two (isometrically) Hilbertian operator spaces of dimension n. Assume Fn homogeneous. If u: En --+ Fn and v: En --+ Fn are two isometries; then Ilulicb = Ilvllcb. Indeed, uv- 1: Fn --+ Fn and vu- 1: Fn --+ Fn are isometric; hence by the homogeneity of Fn we have Iluv- 1 1lcb = Ilvu- 1 1lcb = 1, which obviously implies lIulicb :S Iluv-1l1cbllvllcb = Ilvllcb and conversely IIvllcb :S Ilulicb. We will simply denote by

the c.b. norm of any isometry from En into Fn. By the triangle inequality, this c.b. norm is :S

11.,

whence (10.4)

Therefore we can rewrite (10.1) as follows: (10.1)' We start by estimating the distance dcb(OHn , En) when E is anyone of the spaces in the above list. Recall that we know (Corollary 7.7) that, if En is an arbitrary n-dimensional operator space, we have dcb(OHn , En) :S yn. Proposition 10.2. The following estimates hold: (10.5)

dcb(OHn , min(e~)) = dcb(OHn , max(e~)) ~ yn, deb (OHn , n) = dcb(OHn , ~) ~ dcb(OHn , Rn n C n ) = dcb(OHn , Rn

Vii,

+ Cn) =

(10.6) (10.7)

n 1/ 4 •

(10.8)

Proof. \Ve have

IIOHn

--+

Rnllcb =

lit eli Q9 elil1112 1

IDltl

= n

1/ 4 ,

(10.9)'

Introduction to Operator Space Theory

220

and similarly with C n instead of Rn; whence (10.5). We could also deduce this from (10.3) and the identity dcb(Cn , Rn) = n already mentioned above. By Theorem 3.8

II min(e~) ~ max(e~) Ilcb := n,

(10.10)

whence (10.6) by (10.3) and (10.1)'. Proof. To show (10.7), let {Vi} be as in §9.3. Note that IIQHn ~ nllcb = II 2:: Vi ® Vill:fi~' Then (with the same notation as in the proof of Proposition' 9.7.3) II 2:: Vi ® Villmin ~ 2::7 T(ViY:*) = n/2, and hence IIOHn ~ nllcb ~ (n/2)1/2. Conversely, we have obviously (by the triangle inequality) 112:: Vi ® -

1/2 Vi II min

~

y'n, and hence (10.11)

By Proposition 9.7.3 and by (2.11.4), we have IIn ~ OHnllcb ~ J2. This establishes (10.7), taking into account (10.1), and Theorem 9.3.1. Finally, to check (10.8), we observe

In the converse direction, the inequality (2.11.4) implies IIRnnCn ~ OHnllcb = 1. Then by (10.1) we have (10.8). • Note in passing that, since IIRn n C n ~ OHnllcb 9.7.3) IIn ~ Rn n Cnllcb := 1,

=

1 and (see Proposition (10.12)

we have (taking (10.4) into account) (10.13) and hence lIn

~ ~lIcb :=

1. By (10.11) and (10.3) we therefore have II~ ~ nllcb := n.

Moreover, since we have trivially IIOHn ~ min(e~)lIcb =

(10.14)

II max(e~)

~

OHnllcb = 1, (10.6) implies

II min(e~)

~ OHnllcb

= IIOHn ~ max(~)lIcb := Vn.

(10.15)

By Proposition 1O.1(ii) we immediately deduce from the preceding statement:

10. Comparisons

221

Corollary 10.3. The following estimates hold:

In the next statement we compare

cf>n

to the other spaces in the list.

Theorem 10.4. The following estimates hold: (10.16)

deb(cf>n,Rn nCn)::o.;n,

(10.17)

+ Cn ) ::0 .;n, deb(cf>n' min(£~)) ::0 n, debe cf>n, max( £~)) ::0 .;n.

( 10.18)

debe cf>n, Rn

Proof. It is easy to check that that

(10.10) (10.20)

II L \!;*Vill ::0 II L Vi\!;* 11::0 n

(in fact

= 11), so (10.21)

Conversely, we have (10.12); hence a fortiori (recalling (10.4)) (10.22) whence (10.16). To check (10.17), let c = IIRn n C n

-+

cf>nllcb.

vVe have clearly

Conversely, we can show (cf. [HP2, Remark 1.2]) that there exists a decomposition Vi = a; + b; with II L aia; 111/2 S; c and II L bib; 1 1 / 2 S; c. This yields (with the notation in the proof of Proposition 22.3, using r(b;bi) = r(bib;))

Hence finally c 2': vn(2v2)-1. Thus we have c = By (10.12) and (10.1)' we then obtain (10.17).

IIRn nCn

-+

cf>nllcb::O

vn·

To check (10.18), note that by (10.21) and (10.22) we have (a fortiori)

whence (10.18).

222

Introduction to Operator Space Theory

To check (10.19), note that (trivially) can write by (10.14)

II~n ~

min(£2')llcb = 1. Then we

n ~ II~~ ~ ~nllcb ::; II~~ ~ min(£2')llcb II min(£2') ~ ~nllcb

=11 min(£2') ~ ~nllcb,

whence (by (10.4»

II min(£2') ~ ~nllcb ~ n.

Thus we obtain (10.19).

To check (10.20), note that (trivially) II max(£2') (10.13) and (10.15»

~ ~nllcb =

1. Then (by

whence dcb(~n, max(£2'» ::; C..;n with C independant of n. In the converse direction, the results of [JP] show that max(£2» is not exact, contrary to~. The corresponding estimate from [JP] (see Exercise 19.2 below) is d s dmax(£2')) ~ 4- 1 ..;n, whence dcb(~n' max(£2')) ~ 4- 1 ..;n. Thus we obtain (10.20). • It is now easy to complete these results. First we can compare Rand C to the other spaces in the list (recall dcb(R n , Cn) = n): Theorem 10.5. The following estimates hold:

dcb(Rn ,min(£2'»

= dcb(Cn ,min(£2'» = dcb(Rn ,max(£2'» = dcb(Cn, max(£2'»

= ..;n.

Proof. By (1.5) we have IIRn ~ Cnllcb = IICn ~ Rnllcb = ..;n, from which it immediately follows that IIRn ~ Rn n Cn Ilcb = IICn ~ Rn n C n Ilcb = ..;n. On the other hand, we can verify easily (using (1.11) and (3.1»

II min(£2') ~ Rnllcb =

II min(£2') ~ Cnllcb =

¥n.

(10.23)

It is then an easy task to prove the missing estimates using the duality. Let us now compare Rn n Cn and Rn + Cn to the remaining spaces.

•

10. Comparisons

223

Theorem 10.6. The following estimates llOld:

Proof. From (10.23) we deduce that

II min(f2) --+ Rn n Cnllcb A fortiori, we have

IIRn

II min(f~)

+ Cn --+ max(f~)llcb = n ~II min(f~) --+

--+

--+

Rn

+ Cnllcb

Vii·

= ~

vn, whence, by duality,

vn. Therefore, we can write by (10.10)

max(£~)llcb ~

II min(f2)

--+

Rn

+ CnllcbllRn + Cn

max(f2 )llcb ~

Viii I min(f2) --+ Rn + Cnllcb,

and we obtain

II min(£2) --+

Rn

+ Cnllcb ~ Vii·

Since we have trivially IIRnnCn --+ min(f~)llcb = IIRn+Cn --+ min(e~)llcb = 1, we obtain the estimates on the first line and those on the second one follow ~~ili~

•

We can now state, as a recapitulation: Theorem 10.7. Tile operator spaces

OH,

~,

R,

C,

R n C,

R

are mutually completely nonisomorphic.

+ C,

min(f 2),

max(£2)

PART II OPERATOR SPACES AND C*-TENSOR PRODUCTS

Chapter 11. CO-Norms on Tensor Products. Decomposable Maps. Nuclearity Let AI, A2 be two C· -algebras. Their algebraic tensor product Al ® A2 is an involutive algebra for the natural operations defined by

and

Then a norm II lion Al ® A2 is called a C·-norm if it satisfies Ilxll = Ilx·ll,

IIxYIl :::; Ilxllllyll

and

Ilx·xll

= IIxl1 2

for any x, y in Al ® A 2. It can be shown that we then automatically have (11.0) This subject was initiated in the 1950s by Turumaru in Japan. Later work by Takesaki [Tal] and Guichardet [Gu1] lead to the following striking result. Theorem 11.1. There is a minimal CO-norm II Ilmin and a maximal one II Ilmax, so that any CO-norm II ·11 on Al ® A2 must satisfy Ilxlimin :::; Ilxll :::; IIxll max . ~Ve denote by Al ®min A2 (resp. Al ®max A 2) tlle completion of Al 129 A2 for the norm II Ilmin (resp. II Ilmax).

The maximal C* -norm is easy to describe. We simply write

where the supremum runs over all possible Hilbert spaces H and all possible *-homomorphisms 7r: AIQ9A2 --+ B(H). It is eaRY to see that, for any such 7r. there is a pair of (necessarily contractive) *-homomorphisms 7ri: Ai --+ B(H) (i = 1,2) with commuting ranges such that

Conversely, any such pair 7ri: Ai --+ B(H) (i = 1,2) of *-homomorphismR with commuting ranges determines uniquely a *-homomorphism 7r: Al 129 A2 --+ B(H) by setting 7r(al ® a2) = 7rI(aI)7r2(a2). Thus, we can write for any x = Eal ®a% in Al ®A 2

where the supremum runs over all possible such pairs.

Introduction to Operator Space Theory

228

Then the inequality II II:::; II IImax follows by considering the GelfandNeumark embedding of the completion of (AI ® A 2 , II II) into B(H) for some H. All this goes back to [Gu1]. The lower bound II IImin:::; II II is due to Takesaki [Tal] and is much more delicate. For a proof, see either [Ta3] or [KaR]. The minimal norm can be described as follows: Embed Al and A2 as C*subalgebras of B(H1) and B(H2)' respectively. Then, for any x = E a}®a~ in Al ® A 2, Ilxllmin coincides with the norm induced by the space B(HI ®2 H 2), that is, we have an embedding (Le. an isometric *-homomorphism) of the completion, denoted by Al ®min A 2, into B(Ht ®2 H 2). In other words, the minimal tensor product of operator spaces, considered in §2.1, when restricted to two C* -algebras coincides with the minimal C*tensor product. Let (B I ,B2 ) be another pair of C*-algebras and consider c.b. maps Ui: Ai --+ Bi (i = 1,2). Then (as emphasized already in §2.1) Ut ®U2 defines a c.b. map from At ®min A2 to Bl ®min B2 with lIut ® u211cb = Il u tllcbll u2l1cb. In sharp contrast, the analogous property does not hold for the max-tensor products. However, it does hold if we moreover assume that Ut and U2 are completely positive, and then (see e.g. [Ta3, p. 218], [Pal, p. 164], or [Wa2, p. 11]) the resulting map Ul ®U2 is also completely positive (on the max-tensor prod uct), and we have

This follows from the next two statements.

Theorem 11.2. Let cp: At ® A2 --+ C be a linear form and let u",: Al be the corresponding linear map. The following are equivalent:

--+

A2

(i) cp extends to a positive linear form in the unit ball of (AI ®max A2)*' (ii) u",: At

--+

.42

is a contractive c.p. map, that is, lIu",1I :::; 1, and

L(U",(Xij), Yij) ;::: 0 "In "Ix i,j

E

Mn(At}+ Vy

E

Mn (A 2)+.

Proof. Assume (i). Recall (11.0). Then clearly Ilu",1I :::; 1. Moreover (by the GNS construction) there are a representation 71': Al '8lmaxA2 --+ B(H) and in the unit ball of H such that cp(.) = (7I'(·)e, e). We may assume that 71' = 71't . 71'2 as above. Let X,Y be as in (ii). Let Z = yl/2, so that Yij = Lk ZkiZkj (note Zik = zki)' We claim that the matrix (7I'1(Xij)7I'2(Yij)) is positive. Indeed, for each fixed k the matrix

e

11. C*-Norms on Tensor Products. Decomposable ~l\faps. Nuc1earity

229

is positive and, since 7r1, 7r2 have commuting ranges, we have 7r1(Xij)7r2(Yij) = Lk 7r2(Zkd*7rl(Xij)7r2(Zkj)'

This proves our claim. Let [ ~ EEl··· EEl~. Then we have

E

L(U
H E9 ... EEl H (n times) be defined by [ =

= ([ 7r I(Xij) 7r2(Yij)][,[) ~

0,

which shows that (i) =} (ii). Conversely, assume (ii). Consider t = L aj 0 bj in Al 0 A 2· Then t*t = Li,j aiaj 0 bibj, and hence by (ii)
)

~ 0,

which shows that