12.3. Weak amenability 361and 0 is not contained in the ultraweakly closed convex hull of { wdw* : w E
B unitary}. By Theorem F.12 and Lemma F.17, we are done. D
Exercises
Exercise 12.2.1. Let ri be groups with a common subgroup A and let
r = ri *A r2 be the amalgamated free product. Let <pi be a unital positive
definite function on ri which is bi-A-invariant: <pi(asb) = cpi(s) for every
s E ri and a, b E A. Prove that there is a unital positive definite function
( <p1 * <p2) on r such that
(cp1 * <p2)(as1 · · · Snb) = <pi(1)(s1) · · · <pi(n)(sn)
for a, b EA and Sj E ri(j) \A with i(j) # i(j + 1).
Exercise 12.2.2. Let ri be groups with the Haagerup property and let
A c ri be a finite common subgroup. Prove that the amalgamated free
product ri *A r2 has the Haagerup property.12.3. Weak amenability
Before proceeding, you may wish to review Appendix D on Schur and Herz-
Schur multipliers.
Definition 12.3.1. A discrete group r is said to be weakly amenable if
there exists a net (<pi) of finitely supported functions on r such that <pi ~ 1
pointwise and limsup llcpillB 2 '.S 0.^10
The Cowling-Haagerup constant Acb(r) is the infimum of all C for which
such a net (cpi) exists. We set Acb(r) = oo if r is not weakly amenable.
Example 12.3.2. Amenable groups are weakly amenable. The class of
weakly amenable groups is closed under taking subgroups and Cartesian
products. If r = LJ ri is a directed union, then Acb(r) =sup Acb(ri)·
Let T be a tree (identified with the vertex set) and d be the graph
metric. Let Xn be the characteristic function on {(x,y) E T^2 : d(x,y) = n}
and X-S.K = 'I:;~=O Xn· For simplicity, we denote by llBllcb the ( cb-)norm of
the corresponding Schur multiplier me.
Theorem 12.3.3. We have llxnllcb :S 2n for every n 2: 1.
Proof. Fix a geodesic ray w in T; that is, w is any isometric function from
.N into T.11 For every x E T, there exists a unique geodesic ray Wx which
lORecall that the Herz-Schur norm llcpllB 2 is ::; C if and only ifthere exist families of vectors
(~s)sEr and (7Jt)tEr in a Hilbert space 7-l such that cp(sr^1 ) = (7/t, ~s) for every s, t E r and
SUPs,tEr llMll7Jtll '5. 0..
llif there is no geodesic ray, just attach one to T. This does not affect anything because the
restriction to a rectangular subset does not increase the Schur multiplier norm.