1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.1. Kazhdan's property (T) 351

Unfortunately, providing examples to which this theorem applies is not

so easy -though there are plenty! (See [201].)

Property (T) for von Neumann algebras.

Definition 12.1.16. Let N C M be an inclusion of finite von Neumann

algebras and r be a faithful normal tracial state^6 on M. We say that an

inclusion N C M has relative property (T) if for any c: > 0, there exist a

finite subset ~ c M and o > 0 with the following property: If <p: M --t M

is a r-preserving u.c.p. map such that ll<p(x)-xll2 < <5 for every x E ~'then

ll<p(a) - all2 < c:llall for all a EN.

We say M has property (T) if the identity inclusion M c M has relative

property (T).

Remark 12.1.17. The condition that <p be unital and r-preserving can be

relaxed: If <p: M --t Mis a c.p. map such that <p(l) :S 1 and r o <p :Sr, then

the map <p : M --t M defined by

<p(x) = <p(x) + (~ = ~: ~~i)) (1 - <p(l))

is a u.c.p. map such that r o <p = <p. We also note that r o <p :S r implies

ll<p(a)ll2 = r(<p(a)<p(a))^112 :S r(<p(aa))^112 :S r(a*a)^112 = llall2

for every a EM, by Proposition 1.5.7. In particular <pis normal.

Theorem 12.1.18. Let N C M be finite van Neumann algebras and r be
a faithful normal tracial state on M. The following are equivalent:
(1) the inclusion N C M has relative property (T);
(2) for any c: > 0, there exist a finite subset~ CM and o > 0 with the
following property: If 'H is an M-M-bimodule and~ E 'H is a unit
vector such that (a~,~) = r(a) = (~a,~) for all a E M and such
that llx~ - ~xii < <5 for every x E ~' then there exists fo E 'H with
llfo -~II < c: such that afo = foa for all a EN.
Moreover, if (N c M) = (L(A) c L(I')) for an inclusion Ac r of groups,
then the above conditions are equivalent to
(3) the inclusion A c r has relative property (T).

Proof. (1) ==?- (2): Let c: > 0 be given and take ~ and o as in Defini-
tion 12.1.16. Let 'H be an M-M-bimodule and ~ E 'H be a unit vector
such that (a~,~) = r(a) = (~a,~) for all a E M and llx~ - ~xii < <5

6It turns out that thi~ definition does not depend on the choice of faithful normal tracial
state -but this isn't obvious. See [144] for a proof, and much more.