1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.3. Weak amenability 365

Corollary 12.3.7. Let I'= YIA be the wreath product ofY by A. If JYJ > 1

and A is nonamenable, then Acb (r) > 1.^12

Proof. We assume that Acb (r) = 1 and derive a contradiction. Passing to a
subgroup, we may assume that Y is cyclic. Let YA = EB A Y be the canonical
normal subgroup of r = YA ><l A. By Theorem 12.3.6, there exists a state μ
on t)Q(YA) which is both YA-invariant and Ad(A)-invariant. Let SC YA be
a system of representatives of the A-orbits. Thus, YA = { e} LJ UrESo A/ Ax
as a A-set, where s^0 = S\ { e} and Ax = { s EA: sxs-^1 = x} is the stabilizer
subgroup of x E s^0. Since Ax is finite for every x E s^0 , by averaging over
the right Ax-action, we obtain a A-equivariant u.c.p. map from .C^00 (A) into
.C^00 (A/ Ax). Collecting them together, we obtain a A-equivariant u.c.p. map
from .€^00 (A) into .€^00 (YA\ { e}). Since A is nonamenable, the Ad( A )-invariant
mean μ must be concentrated on { e}. Such a μ cannot be YA-invariant - a
contradiction. D

There exist weakly amenable groups whose Cowling-Haagerup constants
are greater than 1. Before stating the theorem, we should mention that
the definition of weak amenability extends to locally compact groups G. ·
Moreover, for a lattice r :S G, it turns out that Acb(G) = Acb(r). The
proofs of these facts are beyond the scope of this book, but we summarize
some important results below.

Theorem 12.3.8. The following statements are true:

(1) Acb(Z^2 ><l SL(2, Z^2 )) = oo;

(2) Acb(SO(l, n)) = 1 and Acb(SU(l, n)) = 1;

(3) Acb(Sp(l, n)) = 2n -1 and Acb(F 4(-20)) = 21;

( 4) if G is a simple Lie group with real rank greater than or equal to 2,

then Acb(G) = oo.

Definition 12.3.9. We say a C*-algebra A has the CBAP (completely

bounded approximation property) if there exists a net of finite-rank maps

fh: A ---t A such that ei ---t idA in the point-norm topology and sup JJeillcb :S

C. The Haagerup constant Acb(A) is the infimum of those C for which such

a net (ei) exists. We set Acb(A) = oo if A does not have the CBAP.

We say a von Neumann algebra M has the W CBAP (weak* CBAP) if

there exists a net of ultraweakly-continuous finite-rank maps ei: M ---t M

such that ei ---t idM in the point-ultraweak topology and sup JjeiJJcb '.S C.

The Haagerup constant Acb(M) is again the infimum of those C for which

such a net (ei) exists, and Acb(M) oo if M does not have the weak*

CBAP.

12 It is plausible that Acb (r) = oo.