1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.6. Amenable groups 53

operator S = 111 l:::sEE As is self-adjoint. Thus, for any c: > 0, we can find a
unit vector e E £^2 (r) such that I (Se, e) I > 1-E. Letting 1e1 be the pointwise
absolute value of e, a straightforward calculation confirms
1
1-c: < l(se,e)I::::; (s1e1, 1e1) =TEI 2:(.xs1e1, !el).
sEE
Since the cardinality of Eis fixed, by taking c: sufficiently small, we deduce
that all the numbers (.A.slel, !el) must be close to 1; hence the norms 11.A.slel-
JeJ II are small, for all s E E.
(1) ::::?-(9): Let Fk c r be a sequence of F¢lner sets. For each k let Pk E
B(£^2 (r)) be the orthogonal projection onto the finite-dimensional subspace
spanned by {og : g E Fk}· Identify PkB(£^2 (r))Pk with the matrix algebra
Mpk(C) and let {ep,q}p,qEFk be the canonical matrix units of Mpk(C). One
can check that for each s E r we have ep,pAseq,q = 0 unless sq = p, and
ep,pAseq,q = ep,q if sq = p. Since Pk = :L::pEFk ep,p, we have

Pk.A.sPk = L ep,pAseq,q = L ep,s-lp·

p,qEFk pEFknsFk

Let 'Pk: C~(r) ---+ Mpk (C) be the u.c.p. map defined by x 1--+ PkxPk. Now
define a map 'l/Jk : Mpk ( C) ---+ C~ (r) by sending

1. ep,q H JFkl ApAq-i.

Evidently this map is unital; it is also completely positive, being (a scalar
multiple) of the form described in Example 1.5.13.

The 'Pk's and 'l/Jk's do the trick. Since the linear span of {.As: s Er} is

norm dense in C~(r), it suffices to check that II.As - 'l/Jk o 'Pk(.A.s)ll ---+ 0 for

all s E r. This follows from the definition of F¢lner sets together with the

foilowing computation:

'l/Jk o 'Pk ( ) As = 'l/Jk ( """" ~ ep,s-lp ) = """" ~ JFkl^1 .A. s = !Fk JFkJ n sFkl .A. s·.

pEFknsFk pEFknsFk

Hence the reduced group C* -algebra is nuclear.

(1) ::::?-(10): The maps constructed above also prove semidiscreteness of

L(r). According to Remark 2.1.3, it suffices to show that for every x E L(r)

and g,h Er,

('l/Jk 0 'Pk(x)og, oh) ---+ (xog, oh)·

If x E L(r) is given, then we can find unique scalars {as}sEr such that

(xog, oh) = as, whenever hg-^1 = s. A computation shows

"""" IFk n sFk I

'l/Jk o 'Pk(x) =~as JF. I As·

sEI' k