1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

366 12. Approximation Properties for Groups

Though the context should always be clear, one must be careful not to

mix up Acb for C* -algebras and for von Neumann algebras.

Theorem 12.3.10. Let r be a discrete group. Then

Acb(r) = Acb(C{(r)) = Acb(L(r)).

Proof. We trivially have Acb(r) 2': Acb(C{(r)) and Acb(r) 2': Acb(L(r)). To

prove the reverse inequalities at once, let a finite subset E C r and E: > 0 be

given and choose a finite-rank map B: C{(r) -----t L(r) such that llBllcb = C

and 11 - T(.X(s)*B(.X(s)))I < E: for s E E. Since 111f;llB 2 ::::; ll1f;lle2 for any 'lj;,

it suffices to show that the function <p(s) = T(.X(s)*B(.X(s))) is in £^2 (r) and

ll(mrp)JCt(r)llcb::::; C. (See Proposition D.6.) Since() has finite rank, there

exist finite sequences w1, ... , Wn E C{(r)* and x1, ... , Xn E L(r) such that

B( a) = I:~=l 1.uk (a )xk for all a E C{ (r). It follows that

n

sc(s) = T(.X(s)*B(.X(s))) = L wk(.X(s))T(.X(s)*xk)·

k=l

Since supsEr lwk(.X(s))I ::::; llwkll and 'EsEI' IT(.X(s)*xk)l^2 = llxk8ell^2 < oo for

every k, the function <p is in £^2 (r). We denote by 7r the *-homomorphism

from C{(r) into C{(r)@C{(r) given by 7r(.X(s)) = .X(s)@.X(s) for every s Er.

Let V be the isometry from £^2 (r) into £^2 (r) 0 £^2 (r) given by V 88 = (^88 0 88)
for s E r. It is not hard to check that
mrp(a) = V*(idct(r) 0 B)(a)V
for a E C{(r), and hence ll(mrp)Jq(r)llcb::::; llBllcb· 0
Proposition 12.3.11. Let A be a co-amenable subgroup of r. Then there
exist nets of u.c.p. maps
'Pi: C{(r) -----t Mn(i)(C{(A)) and 'lj;i: Mn(i)(C{(A)) -----t C{(r)
such that 'lj;i o 'Pi -----t 1 in the point-norm topology.
Proof. Fix a cross section (]": r I A -----t r and identify (r I A) x A with r via
(p, a) f----+ CT(p)a. Then we have
.Xr(s) = L esp,p 0 AA(CT(sp)-^1 sCT(p)) E lBl(£^2 (r/A) @£^2 (A)),
pEI'/A
where .Ar and AA are the left regular representations of r and A, respectively.
For a finite subset F c r I A, let
'PF: lBl( .e^2 (r /A) 0 .e^2 (A)) -----t lBl( .e^2 (F) 0 .e^2 (A))
be the compression map; observe that <pp(C{(r)) c lBl(£^2 (F)) 0 C{(A). Let
VF: .e^2 (r) -----t £^2 (r /A) 0 £^2 (F) 0 £^2 (A)