1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.4. Another approximation property 369

for C*-algebras Ai and A2. The inequality Acb(Ai@A2)::; Acb(Ai)Acb(A2)

is trivial. To prove the opposite inequality, suppose by contradiction that

Acb(Ai@ A2) < Acb(Ai)Acb(A2), and choose 0 < Ci < Acb(Ai) such that

Acb(Ai @A2) < CiC2. It follows that there is a finite-dimensional subspace

Fi C Ai such that there is no finite-rank map I.pi on A with 1PilFi = idFi and

ll1Pillcb ::; Ci. Choose a finite-rank map <p on Ai@ A2 such that 1PIFi@F 2 =

idF 1 @F 2 and ll1Pllcb < CiC2. By the small perturbation argument, we may

assume that the range of <p is contained in E1 @E 2 for some finite-dimensional

subspaces Ei C Ai. By Lemma 12.3.16, there is ui E Fi 0 Ei such that

ltr(ui)I > CilluillAi@E~· By Lemma 12.3.14, the element u = u1 x u2 E

(F1 Q9 F2) 0 (Ei Q9 E2)i satisfies

ltr(u)I = ltr(u1)tr(u2)I

> C1llu1llA 1 @EiC2llu2llA 2 @E2

2:: C1 C2 llull (A 1 ©A2)@(E 1 ©E 2 )* ·

This is in contradiction to the fact that ll1Pllcb < C1C2, by Lemma 12.3.16.

D

It is proved in [165] that the class of groups with Cowling-Haagerup con-
stant equal to 1 is closed under free products (even allowing amalgamation
over a finite subgroup), but it is not yet known whether weak amenability
is closed under free products.

12.4. Another approximation property

Definition 12.4.1. Let A be a C* -algebra. We say that A has the OAP
(operator approximation property) if there exists a net of finite-rank contin-
uous linear maps I.pi on A which converges to idA in the stable point-norm
topology: I.pi @ idK(£2) converges to the identity map on A@ JK(.€^2 ) in the
point-norm topology.^13 We say that A has the SOAP (strong OAP) if there
exists a net of finite-rank continuous linear maps I.pi on A which converges
to idA in the strong stable point-norm topology: I.pi@ idJIB(£2) converges to the
identity map on A@ JE(.€^2 ) in the point-norm topology.

Let M be a von Neumann algebra. We say that M has the VVOAP
(weak OAP) if there exists a net of finite-rank ultraweakly-continuous linear
maps 1Pi on M which converges to idM in the stable point-ultraweak topology:
1Pi@idlIB(£2) converges to the identity map on M ®JE(.€^2 ) in the point-ultraweak
topology.

It is clear that CBAP::::}SOAP::::}OAP and that W*CBAP::::} W*OAP.

13The crucial point is that the net (\Oi) need not be uniformly bounded. The Principle of

Uniform Boundedness implies that this net cannot be a sequence (unless A has the CBAP).