1 Kirchbergs theorem and tensorial characterizations of LLP and WEP

The words here written constitute notes for a presentation by Andreas Ns Aaserud for aseminar organized by Uffe Haagerup in Spring 2011 at the Department of Mathematics at theUniversity of Copenhagen.

The notes are based on section 13.2 of the book C-Algebras and Finite-Dimensional Ap-proximations by N. Brown and N. Ozawa (referred to below as {BO}) and on the followingarticle by G. Pisier (on which section 13.2 of {BO}, in turn, is mostly based): A simple proof ofa theorem of Kirchberg and related results on C-norms, J. of Operator Theory 35 (1996), pp.317-335 (referred to below as {P}).

The main results are Kirchbergs theorem 1.13 and its corollaries concerning WEP and LLP.

Contents

1 Kirchbergs theorem and tensorial characterizations of LLP and WEP 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Pisiers lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Pisiers theorem on tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Kirchbergs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Tensorial characterizations of LLP and WEP . . . . . . . . . . . . . . . . . . . . . . 6

1.1 Preliminaries

Definition 1.1. An operator space X is a normed space which is isometrically isomorphic to alinear subspace of B(H) for some Hilbert space H.

The following result is classical, cf. the book Operator Spaces by E. Effros and Z.-J. Ruan.

Theorem 1.2 (Arveson-Wittstock extension theorem). Let E be an operator space in a unitalC-algebra A, let H be a Hilbert space, and let T : E B(H) be a c.c. linear map. Then T extendsto a c.c. linear map T : AB(H).

Furthermore, we recall the following result from the book by Brown and Ozawa.

Theorem 1.3 (Corollary 3.5.5 of {BO}). If T : W Y and S : X Z are c.b. linear maps ofoperator spaces, then T S : W X Y Z extends uniquely to a c.b. linear map T S : W minX Y min Z . Moreover, T Scb = T cbScb.

(Here e.g. W min X denotes the closure of span{w x |w W, x X } in B(H K ), wherewe have chosen linear isometric injections W H and X K .)

We shall invoke the usual identifications Mn(A)= AMn(C) and An = ACn .Furthermore, we shall need the following two results. The second one is a generalization

of the Cauchy-Schwarz inequality (put A =C).

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Lemma 1.4 (Lemma 1.2 of {P}). Let u B(H) be unitary and let S : H H be a linear isome-try. If SuS is unitary then K = S(H) is invariant under u and u, i.e., u commutes with theorthogonal projection PK = SS onto K .Proof. For each h H , we get that h2 = uS(h)2 = PK uS(h)2+(1PK )uSh2 = h2+(1PK )uS(h)2. Thus uS(h) K for all h H , i.e., K is invariant under u. Replacing u by u,we get that K is invariant under u as well.

Lemma 1.5 (Lemma 1.3 of {P}). Let a1, . . . , an and b1, . . . ,bn be elements of a C-algebra A.Then

ai bi2 ai ai bi bi.The proof of Lemma 1.5 is an easy exercise and we leave it to the reader.We will also need the following immediate corollary of Theorem B.7 of {BO}.

Theorem 1.6 (Factorization of c.b. maps). Let A be a C-algebra, H a Hilbert space, and T : AB(H) a c.b. linear map. Then there exist a Hilbert space H, bounded operators V ,W B(H), anda -homomorphism pi : AB(H) such that

T (a)=V pi(a)Wfor all a A and V W = T cb.

1.2 Pisiers lemma

The next result is quite useful.

Lemma 1.7 (Proposition 1.7 of {P}). Let A and B be unital C-algebras and assume that {ui }iI U (A) contains the unit and generates A as a unital C-algebra. Put E = span({ui }iI ) and letT : E B be a unital linear map mapping each ui to a unitary element of B.

If T is a c.c. map, then it extends to a -homomorphism T : AB.Proof. Let B B(H) be a faithful representation of B . By the Arveson-Wittstock extensiontheorem 1.2 T : E B(H) extends to a c.c. linear map T : AB(H). Since T is a u.c.c. map, it isautomatically a u.c.p. map. (This is a well known but non-trivial fact. See e.g. V. Paulsens bookCompletely Bounded Maps and Operator Algebras.) Now, the Stinespring Theorem impliesthat there exists a Hilbert space H , a unital -homomorphism pi : A B(H), and a linearisometry S : H H such that

T (x)= Spi(x)Sfor all x A.

Next we note that for every u in {ui }iI , T (u) = T (u) = Spi(u)S is unitary. Put K = S(H)and denote the orthogonal projection onto K by PK . Then PK = SS and by Lemma 1.4 pi(u)commutes with PK . Since {ui }iI generates A, it follows that pi(A) commutes with PK . Thus

T (ab)= Spi(a)pi(b)SSS = Spi(a)SSpi(b)S = T (a)T (b).for all a,b A. Since T (a)= T (a) for all a A, we get that T is a -homomorphism. Finally,it follows that T maps into B , since T is a -homomorphism and {ui }iI generates A.

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Remark 1.8. Notice the similarity between the previous lemma and Lemma 13.2.3 of {BO}. Infact, the idea of showing that the ui belong to the multiplicative domain of T (cf. Proposition1.5.7 of {BO}), which is employed in the proof of Lemma 13.2.3 of {BO}, also could have beenused in lieu of the proof just given. Note that the same big guns, i.e., Arveson-Wittstock andStinespring, are used in either case.

Remark 1.9 (Remark 1.8 of {P}). A useful way to invoke the lemma is the following. LetA de-note the (norm-dense) unital-algebra generated by E, and u : A B a unital-homomorphism.If u|E is a c.c. map, then u extends to a -homomorphism u : AB.

1.3 Pisiers theorem on tensor products

We now turn to the following theorem, which forms an integral part of Pisiers proof of Kirch-bergs Theorem.

Theorem 1.10 (Theorem 1.1 of {P}). Let A1 and A2 be unital C-algebras and assume that{ui }iI U (A1) and {v j } jJ U (A2) contain the units and generate A1 and A2, respectively, asunital C-algebras. Put E1 = span{ui }iI and E2 = span{v j } jJ .

The following are equivalent:

(i) The inclusion map : E1E2 A1max A2 is a complete isometry when E1E2 is equippedwith the min-norm.

(ii) A1min A2 is canonically -isomorphic to A1max A2.Before we give the proof, we recall the fact that any C-norm on an algebraic tensor prod-

uct is a cross-norm, cf. Lemma 3.4.10 of {BO}, and we note the existence of a canonical sur-jective -homomorphism

F : A1max A2 A1min A2.It can be proved as follows. Consider the inclusion : A1 A2 A1min A2. Since min max on A1 A2, this map is a contraction. Extend it by continuity to a linear contractionF : A1max A2 A1min A2. This extension is clearly a -homomorphism. Since RanF is thenautomatically closed and dense in A1min A2, it follows that F is surjective.Proof of Theorem 1.10. (ii) (i): If (ii) holds, then is merely the composition E1E2 A1minA2 A1max A2. Thus (i) is clearly satisfied.

(i) (ii): Put E = E1 E2 and A = A1 min A2. Then E A and by assumption : E A1max A2 is a c.c. map. Since E = span{ui v j | i I , j J }, it follows from Pisiers lemma that extends to a -homomorphism T : A1min A2 A1max A2 and it is clear (because E1 andE2 generate A1 and A2, respectively, and T is a -homomorphism, which acts as the identityon E) that T acts as the identity on 1 A2 and A11, hence on A1 A2. Furthermore, T F isa -homomorphism of A1max A2 into A1max A2 and T F is the identity on A1 A2. ThusT F is the identity map. Since F therefore has a left-inverse, it is injective, proving (ii).

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1.4 Free groups

In the following F denotes any free group and H denotes an infinite-dimensional separableHilbert space, which we think of as `2(N). We recall the universal property of the free group.

Lemma 1.11. Suppose that {ai }iI is a set of free generators of F and let {ui }iI be a set of unitaryelements in B(K ) for some Hilbert space K . Then there exists a unique group homomorphism : FU (K ) such that (ai )= ui for all i I .

We will now consider the full group C-algebra C(F).The following result is central to Pisiers proof of Kirchbergs theorem.

Lemma 1.12 (Lemma 1.5/Remark 1.6 of {P}). Let {ui }iI consist of the unit in C(F) as wellas the unitary generators of C(F) associated to the generators of F. Let x = {xi }iI B(H) befinitely supported, i.e., xi = 0 for all i outside of some finite subset of I . Define a linear mapTx : `(I )B(H) by Tx((i )iI )=iI i xi .

The following are equivalent:

(i) Txcb < 1.(ii) ui ximin < 1.

(iii) There exist {ai }iI , {bi }iI B(H) such that ai = bi = 0 whenever xi = 0, xi = ai bi for alli I , and ai ai 1/2bi bi1/2 < 1.

Proof. We begin by showing, following the proof of Lemma 13.2.2 of {BO}, that

Txcb =

ui ximin.Since x = {xi }iI is finitely supported, we may assume that supp(x) {1, . . . ,n} for some n Nand that u1 = 1C(F). Put z =ui xi and let E be the operator space spanned by {u1, . . . ,un}.

Note thatz = (idE Tx)(w) E minB(H),

where w = i ui ei E min `n . (Here {ei }iI is the canonical basis in `n .) It follows byTheorem 1.3 that

idE Tx idE Txcb = idEcbTxcb = Txcb.Since w is a unitary element of C(F)min `(I ), as is easily checked, it follows that

zmin = (idE Tx)(w)min idE Txwmin Txcb.Let now a B(H)min`n be any contraction. Identifying B(H)min`n with B(H)n, write

a = (a1, . . . , an) with ai B(H) for all i = 1, . . . ,n. Since a =maxai, we have that ai 1for all i = 1, . . . ,n. Let, for each i , ai M2(B(H)) denote the unitary dilation(

ai (1ai ai )1/2(1ai ai )1/2 ai

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as in the proof of Theorem 13.1.3 of {BO}. Define a unital linear map : E M2(B(H)) by

(ui )= a10 ai .

It clearly maps the unitary elements u1, . . . ,un to unitary elements in M2(B(H)).Universality of the full group C-algebra (cf. Proposition 2.5.2 of {BO}) as well as of the free

group (cf. Lemma 1.11), imply first that C(Fn1) is -isomorphic to C({u1, . . . ,un}) and thenthat there exists a -homomorphism

: C({u1, . . . ,un})M2(B(H))

such that (ui )= a10 ai for all i = 1, . . . ,n. Since extends , is a c.c. map. Hence the map : E B(H) defined by (ui ) = (a0(ui ))11 = ai is also a c.c. map, being a composition ofsuch. It follows that

(idB(H)Tx)(a)min = ( idB(H))(z)min idB(H)zmin = cbzmin zmin.

We use here also the fact that idB(H) = cb, cf. Remark B.3 of {BO}. Since similarlyidB(H)Tx = Txcb, it follows that (i) (ii).

We prove now that (iii) implies (ii). Let pi : C(F)B(K ) be a faithful representation. ThenLemma 1.5 implies that

ui ximin = pi(ui )ai bi= (pi(ui )ai )(1bi ) (1ai ai )1/2(1bi bi )1/2= 1ai ai 1/21bi bi1/2= ai ai 1/2bi bi1/2 < 1.

It remains to prove that (i) implies (iii). By applying Theorem 1.6 to Tx , we obtain V ,W B(H , H) and a -homomorphism `(I ) B(H) such that Tx(a)= V pi(a)W for all a `(I )and V W = Txcb. We may assume that I is finite. Put ai = V pi(ei ) and bi = pi(ei )W ,where {ei }iI is the canonical basis in `(I ). Then ai bi =V pi(ei )W = Tx(ei )= xi and

ai ai = V pi(ei )V = V pi(ei )V V 2.Similarly, bi bi W . Since V W = Txcb, it follows that ai ai 1/2bi bi1/2 ymin. Thus y/cmax < 1 whenever c > ymin.This implies that ymax ymin, hence that ymax = ymin. Consider the inclusion map

: E1E2 A1max A2,

where E1E2 is equipped with the min-norm. We have just proved that it is isometric. Since

Mn(E1E2)= E1B(H)Mn(C)= E1B(H) A1maxB(H)= A1maxB(H)max Mn(C)=Mn(A1max A2),

where the inclusion (and equalities) denote linear (surjective) isometries whose compositionequals the nth inflation of , it follows that is a complete isometry. Thus Theorem 1.10 im-plies that A1min A2 = A1max A2, as desired.

1.6 Tensorial characterizations of LLP and WEP

Definition 1.14 (Definition 3.6.7 of {BO}). A C-algebra A B(K ) is said to have the weakexpectation property (or WEP) if there exists a u.c.p. linear map : B(K ) A with|A = idA.

We will need the following result. We skip the proof which is based on Proposition 3.6.6 of{BO} as well as The Trick, i.e., Proposition 3.6.5 of {BO}.

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Theorem 1.15 (Corollary 3.6.8 of {BO}). A C-algebra A has the WEP if and only if for everyinclusion A B of C-algebras and every C-algebra C we have a canonical inclusion AmaxC B max C .

We prove (most of) the following theorem.

Theorem 1.16 (Corollary 13.2.5 of {BO}). Let A and B be (unital) separable C-algebras.

(i) Amax B = Amin B if A has the WEP and B has the LLP.(ii) B(H)max B =B(H)min B if and only if B has the LLP.

(iii) Amax C(F)= Amin C(F) if and only if A has the WEP.Proof. We begin by proving (i) under the assumption of (ii). Since A is separable, we can finda faithful representation of A on the separable Hilbert space H . By Theorem 1.15, there existsa canonical inclusion map : Amax B B(H)max B . Furthermore, we have a canonical in-clusion map : Amin B B(H)min B (cf. Proposition 3.6.1 of {BO}). Consider the canonicalsurjections F : Amax B Amin B and G : B(H)max B B(H)min B . By (ii), G is injec-tive. Since G |AB =F |AB = idAB , it follows by continuity that G =F . Thus F isinjective, proving what we wanted.

We now prove (ii). Assume first that B has the LLP. LetU (B) = {ui }iI denote the unitarygroup in B . Let F be the free group on |I | generators {ai }iI . Define a unitary representation ofF by ai 7 ui . By universality of the full group C-algebra (cf. Proposition 2.5.2 of {BO}), thereexists a unital -homomorphism pi : C(F) B such that pi(ai ) = ui for all i I . It is clearlysurjective. Hence the map pi(a) [a] is a -isomorphism B C(F)/J , where J = kerpi.

Let z =digi be an arbitrary element of BB(H). Let E be a finite dimensional operatorsystem in B containing the di so that z E B(H). Since B has the LLP, there exists a u.c.p.,hence c.c., linear map : E C(F) such that [(pi(a))]= [a] whenever pi(a) E . Now, givene E , choose a C(F) such that e =pi(a). It follows that pi((e))=pi((pi(a)))=pi(a)= e.

Consider the -homomorphism (cf. Proposition 3.1.16 of {BO}) pi idB(H) : C(F)B(H)B max B(H). By universality of the max-norm (Proposition 3.3.7 of {BO}), it extends to a -homomorphism pi idB(H) : C(F)maxB(H) B maxB(H). In particular, pi idB(H) is a c.c.map. By Corollary 3.5.5 of {BO}, idB(H) : E B(H) C(F)min B(H) is a also a c.c. map,where E B(H) comes equipped with the min-norm. Let F : C(F)max B(H) C(F)minB(H) be the canonical -isomorphism given by Kirchbergs Theorem. Its inverse F1 is also ac.c. map. Thus

(pi idB(H))F1 ( idB(H))= idEB(H)is a c.c. map. In particular zmax zmin so that zmax = zmin. Since z B B(H) wasarbitrary, B maxB(H)=B minB(H), as desired.

Assume conversely that BmaxB(H)=BminB(H). Since B is separable, we can refine theargument above to yield a -isomorphism B C(F)/J . Consider the sequence

0B(H)min J B(H)min C(F)B(H)min B 0.

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It is exact by Corollary 3.7.3 of {BO}, since B(H)B has a unique C-norm by assumption.Hence Theorem 13.1.6 of {BO} implies that idB is locally liftable, and Corollary 13.1.4 of {BO}implies that B has the LLP.

We conclude by proving the if part of (iii).1 Indeed, since C(F) has the LP (cf. Theorem13.1.3 of {BO}), and hence the LLP, the fact that A has WEP implies by (i) that C(F)max A =C(F)min A.

1For the only if part we refer the reader to p. 456 of E. Kirchbergs paper On non-semisplit extensions, tensorproducts and exactness of group C-algebras, Invent. math. 112 (1993), pp. 449-489.

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Kirchberg's theorem and tensorial characterizations of LLP and WEPPreliminariesPisier's lemmaPisier's theorem on tensor productsFree groupsKirchberg's theoremTensorial characterizations of LLP and WEP