1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

352 12. Approximation Properties for Groups

for every x E J. We define a T-preserving u.c.p. map c.p: M ---+ M by

(c.p(a)x,fj) = (aex,ey)L2(M) (see Appendix F). Then we have

llcp(x) - xii~= llcp(x)ll~ + llxll~ - 2~(xe,ex)
:::; 2llxll~ - 2~(xe, ex)
= llxe - ex11^2
< 52
for every x E J. It follows that llcp(a) - all2 < c:llall for all a E N. Hence,
for every unitary element u E N, we have
11e-ueull~ = 2 - 2~T(c.p(u)u) < 2c.
Let ea be the circumcenter of the set {ueu: u EN unitary}. By uniqueness
of the circumcenter, we have ufou =ea for every unitary element u E N
and llfo - ell:::; (2c:)^1 l^2.
(2) :::::?- (1): Let c: > 0 be given and take J and 8 as in condition (2).
Let cp: M ---+ M be a T-preserving u.c.p. map such that llcp(x) - xll2 <
(2llxll2)-^182 for every x E J. Let (1-l, e) be the M-M-bimodule arising from
the minimal Stinespring dilation (see Appendix F). Then we have
llxe-exll^2 = 2llxll~ - 2~T(c.p(x)x*) < 82
for every x E J. It follows that there exists fo E 1-l with llfo - ell < c such
that aea = eaa for all a EN. Hence, for every a EN we have
llcp( a) - all~ :::; 211ae11^2 - 2~(ae, ea)
:::; 211ae1111ae - eall
:S 4llall2llall lie - foll
< 4c:lla11^2.
Now, for the last assertion, we assume our inclusion arises from groups,
i.e., (N c M) = (L(A) c L(r)). It is routine to check that relative property
(T) of (L(A) c L(r)) implies condition (3) of Theorem 12.1.7; hence we
only prove (3) :::::?-(2). Let c: > 0 be given and take a finite subset F c rand
/), > 0 such that (F, /),) satisfies condition (2) of Theorem 12.1.7. We will
show that >-.(F) C M and 8 = /'i,c > 0 satisfy condition (2) of the present
theorem. Let an M-M-bimodule 1-l and a unit vector e E 1-l be given such
that 11>-.(s)e - e>-.(s)ll < 8 for every s E F. Then, the unitary representation
1T of r on '}-{ given by
?T(s)( = >-.(s)(>-.(s-^1 )
satisfies ll?T(s)~ - ~II < 8 for every s E F. It follows that there exists a
?T(A)-invariant vector ~a E 1-l such that llfo - ~II :::; /'i,-^15 = c:. Since fo is
?T(A)-invariant, we have afo = ~aa for all a E L(A). D