1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

362 12. Approximation Properties for Groups

starts at x and eventually flows into w. It is not hard to see (cf. Figure 2)

that

n Ln/2J

'l/Jn(x, Y) := L(5wy(k)' 5w.,(n-k)) = L Xn-2m(x, Y)

k=O m=O

for any x, y ET. In particular, Xn = 'l/Jn - 'l/Jn-2 for every n 2: 2.

have Jl'l/Jnllcb :Sn+ 1 by Theorem D.4, we are done.

Since we

D

wy(k) = Wx(n - k)

y

X = Wx(O) x

Figure 2

Corollary 12.3.4. There is an increasing sequence Kn E N such that the

kernels

1

Bn(x, y) = X<Kn - (x, y) exp(--d(x, n y))

on the tree T satisfy limsupn---+oo JJBnllcb = 1.

Proof. Let 'l/Jn(x, y) = exp(-n-^1 d(x, y)). By Theorems 12.2.5 and D.11,

the kernel 'l/Jn is positive definite and hence Jl'l/Jnllcb = 1. Since Xk'l/Jn =

e-k/nXk, Theorem 12.3.3 implies that JIXk'l/Jnllcb :S 2ke-k/n for every n and

k. Therefore, for any integer K, one has

llX:::;K'l/Jnllcb :'S JJ'l/JnJlcb +II L Xk'l/Jnllcb :'S 1 + L 2ke-k/n.

k>K k>K

Thus, if Kn is chosen sufficiently large, we have llBnllcb < 1 + n-^1. D

Corollary 12.3.5. If a group r acts properly on a tree, then it is weakly

amenable with Acb(r) = 1. In particular, free groups and SL(2, Z) are weakly

amenable and their Cowling-Haagerup constants are 1.

Proof. Let T be the tree on which r acts properly. Let Bn be as in Corol-
lary 12.3.4 and set 'Pn(s) = Bn(o, s-^1 .o) for a fixed base point o ET. Then,
'Pn -+ 1 pointwise, and 'Pn is finitely supported since the action is proper.
Moreover, since 'Pn(sr^1 ) = Bn(s-^1 .o,C^1 .o), we have ll'PnllB 2 = llBnllcb-+

1. D

Here is another proof, using ideas from [162], of the corollary above,
which simultaneously proves the Haagerup property.