1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

D. Positive Deflnite Functions 469

definite. In particular, for any 1-cocycle b on a group r and I > 0, the

function <p~ on r' defined by

<p~(s) = exp(-1llb(s)ll^2 ),

is positive definite.

Proof. Let k(s, t) = llb(s) - b(t)ll^2. A computation shows

exp(-k(s, t)) = exp(-llb(s) 112 ) exp(-llb(t) 112 ) exp(2~(b(s ), b(t))).

It is easy to see that (s,t) c---t exp(-llb(s)ll^2 )exp(-llb(t)ll^2 ) is a positive
definite kernel. On the other hand,

exp(2~(b(s), b(t))) =I:: (2~(b(s); b(t)) r
n:'.:'.0 n.
and it is easy to see that ~(b(s), b(t)) is positive definite and so are the
iterated Schur products (~(b(s), b(t)) r. Hence, (s, t) c---t exp(-k(s, t)) is
positive definite since it is a Schur product of positive definite kernels. D

Let b be a 1-cocycle on r. We note that <p~(e) = 1 and <p~ ~ 1 pointwise
as I~ 0. On the other hand, <p~ ~ 1 uniformly as I~ 0 if and only if bis
bounded.

Lemma D.12. Let b be a 1-cocycle of a group r and let / > 0. Let
(1r~, H~,~~) be the GNS triplet of the positive definite function <p~ on r
given in Theorem D.11. Suppose that b is unbounded on a subgroup A of r.
Then, (7f~,,H~) has no nonzero A-invariant vector.

Proof. Suppose that b is unbounded on a subgroup A and choose a sequence
(sn) in A such that llb(sn)ll ~ oo. Since llb(tsnt')ll 2: llb(sn)ll-:-(llb(t)ll +
llb(t')ll), we have ,
limsup<p~(tsnt') = limsupexp(-1llb(tsnt')ll^2 ) = 0.
n--+oo n--+oo ·

for every t, t' Er. Hence, for any ( ~ 2..:::~=l Gi1f~(ti)~ EH~; we have

limsup 1(7f~(sn)(,()I = limsup I Laiaj <p~(ti^1 sntj)I = 0.

n--+oo n--+oo i,j

By continuity, lim(7r~(sn)(, () = 0 for every ( E '}-{~ (i.e., 7f~IA is weakly
mixing). Clearly, there is no nonzero A-invariant vector. D
Exercises

Exercise D.1. Let V be a bounded subset of a Hilbert space and let

ro = inf{r 2: 0 : Vis contained in a closed ball of radius r }.