342 12. Approximation Properties for Groups( 4) ~ (5) is a tautology.
(4) =?-(2): We again prove the contrapositive. Let E1 c E2 c · · · c r
be an increasing sequence of finite subsets with U En = r. If condition (2)
does not hold, then for every n there exist a unitary representation (7rn, 1-ln)
and a vector en E 1-ln such that
4-n11en - Pnenll > sup lien - 1l'n(s)enll =:On,
sEEn
where Pn is the orthogonal projection from 1-ln onto the subspace of all
A-invariant vectors. Define a map a: r ----t $1-ln bya(s) = (en -:~(s)en) E EBHn.
2 n n n=l
The infinite series in the formula is convergent and a is a 1-cocycle of r with
coefficients in ( E9 7r n, $1-ln). By (the proof of) Lemma 12 .1. 5, for every n,
there exists sn EA such that llen-7rn(sn)enll 2: llen-Pnenll· It follows that
II a( sn) II 2: 2n and a is unbounded on A.
(2) =?- (3): Let (E, "') be as in condition (2) and tp be any positive
definite function on r with tp ( e) = 1. Then, denoting by ( 7r <p, 1-l<p, e<p) the
GNS triplet, we have
sup 11-<,o(t)I ::::: sup 117r<p(t)e<p - e<pll lle<pll
tEA tEA
=sup 117r<p(t)PJ_e<p - PJ_e<pll
tEA
::::: 2"'-1max117l' sEE <p( s )e<p - e<p II= 2"'-l max(2~(1-<,o(s)))^112.
sEE
This clearly implies condition (3).
(3) =?-(1): Let (7r, 7-l) be a unitary representation of r which contains
almost r-invariant unit vectors (en)· Then, the sequence (tpn) of the positive
definite functions on r defined by Son(s) = (7r(s)en,en), converges pointwise
to 1. By assumption, convergence is uniform on A, which implies that en
is (A, 1)-invariant for sufficiently large n. By Lemma 12.1.5, there exists a
nonzero A-invariant vector.
(2) =?- (6): See the proof of Corollary 6.4.7. Note that if (E, "') is a
Kazhdan pair for r, S is any generating subset of r and n is chosen such
that E c (SU s-^1 )n, then (S, "'/n) is also a Kazhdan pair. D
Our next lemma gives a very useful spectral characterization of property
(T). But first, note that the uniform convexity^3 of a Hilbert space 1-l implies(^3) A Banach space X is uniformly convex if for each c: > 0, there exists 8 > O with the following
property: For all x, y EX with llxll, llYll ::::; 1 and llx -Yll ~ c:, we have that II "'tY II ::::; 1 -8.