1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.3. Weak amenability 363

Proof. For simplicity, we assume that T is uniformly locally finite. As in

the proof of Theorem 12.3.3, we fix a geodesic ray win the tree T and, for

every x E T, denote by Wx the unique geodesic ray which starts at x and

eventually flows into w. For every z E lI.lJ = {z E CC : I z I < 1}, we define a

function (z: T-+ £^2 (T) by

00

(z(x) = ~Lzk8wx(k)'

k=O

where J1 - z^2 denotes the principal branch of the square root. The series
above converges absolutely in z and uniformly in x:

II( (x)ll^2 = 11-z^2 I f lzl^2 k = ll - z

2

I

z k=O 1 - I z 12.

In particular, the function lI.lJ 3 z 1-+ (z E £^00 (T, £^2 (T)) is holomorphic.
Hence the function z 1-+ Bz, defined by

Bz(x, y) = ((z(y), (z(x)),

is also a holomorphic function into the Banach space of Schur multipliers.
A calculation similar to that in the proof of Theorem 12.3.3 yields

00

Bz(x, y) = (1 - z^2 ) L zk+^1 8wx(k),wy(l)

k,l=O

oo n

= (1-z^2 ) :L :L zn8wx(k),wy(n-k)

n=Ok=O

00

= (l _ z2) L zd(x,y)+2m

m=O

= zd(x,y)_

Now, suppose that a group r acts properly on T and define r.pz E B2(r) by
r.pz(s) = zd(s.o,o), for a fixed base point o E T. It follows that the function
ID 3 z 1-+ rpz E B2(r) is holomorphic. Since (r = (r for all r E ~' 'Pr is
positive definite for 0 < r < 1. Moreover, 'Pr -+ 1 as r -+ 1 and hence it
suffices to show that 'Pr belongs to the Banach subspace F C B2 (r) spanned
by finitely supported functions. Since T is uniformly locally finite and the
action of r is proper, there exists C > 0 such that l{s E r : d(s.o, o) :::;:
n}I :::::: en for every n. It follows that l{Jz E £^1 (r) c F for lzl < c-^1. This
implies that lI.lJ 3 z 1-+ r.pz + F E B2 (r) / F is a holomorphic function which
is zero for lzl < c-^1. By uniqueness of holomorphic extensions, 'Pz E F for
all z E ll.ll. D