1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

364 12. Approximation Properties for Groups

Let A <1 r be a normal subgroup. Then, r acts on A by conjugation:

Ad(s)(a) = sas-^1 for s Er and a EA. Abusing notation, we also denote

by Ad(s) the action of s Er on £^00 (A), given by Ad(s)(f) = f o Ad(s-^1 ).

Theorem 12.3.6. Let r be a group with Acb(r) = 1 and let A <1 r be a

normal abelian subgroup. Then, there exists a A-invariant mean on £^00 (A)

which is Ad(r)-invariant.

Proof. Let 'Pi : r ----+ <C be a net of finitely supported functions such that

ll'PillB 2 :S 1 and 'Pi----+ 1 pointwise. By Theorem D.4, there are Hilbert spaces

Hi and unit vectors ei ( s), rli (t) E 1i such that 'Pi ( st-l) = (7Ji(t), ei( s)) for

every s, t Er. For each s Er, we have

limsup llei(st)-ei(t)ll :S limsup llei(st)-7Ji(t)ll +limsup llei(t) - 7/i(t)ll = o

i i i

uniformly fort EI', since limi 'Pi(s) = 1 = limi 'Pi(e). Likewise, limi ll7Ji(st)-

7Ji(t)ll = 0 uniformly fort Er. It follows that ll<,oi - 'Pio Ad(s)llB 2 ----+ O for

every s EI'.

Since A is amenable, there is a trivial character To: C{(A) ----+ <C (Theorem

2.6.8). Since the Herz-Schur multiplier m'Pi maps L(A) into <C[AJ c C{(A),

we may define linear functionals Wion L(A) by Wi =Too mcpJL(A)· Observe

that wi E L(A)* with llwill :S ll'PillB 2 :S 1. Let A be the Pontryagin dual

of A and recall that C{(A) ~ C(A) and L(A) ~ L^00 (A) via the Fourier

transform £^2 (A) ~ L^2 (A). Also, note that Ad(s) acts naturally on A and

hence on LP(A) for 1 :S p ::; oo. We denote by fi E L^1 (A) the element

corresponding to wi E L(A)*. Since llfill1 :S 1 and J fi = wi(l) = 'Pi(e)----+ 1,

we have 111 fi I - fi I I 1 ----+ 0. Let ei E £^2 (A) be the element corresponding to

lfil^112 E L^2 (A). Then, we have

li:μi.(.X( i a )ei, ei) = lim i lwi I(>.( a)) = li:μi. i Wi (>.(a)) = li:μi. i 'Pi (a) = 1

for every a E A and

li:μi. i llei - Ad(s)(ei)ll;2 = li:μi.// i lfil^1!^2 - Ad(s)(lfi1^112 )/[~2

:S li:μi.l/lfil i -Ad(s)(lfil)l/v
= li:μi. II fi - Ad( s) (fi) llv
i
:S li:μi. i ll'Pi - 'Pio Ad(s-^1 )llcb
=0
for every s EI'. It follows that et E £^1 (A) is approximately A-invariant and
approximately Ad(I')-invariant. Taking any limit point of the net (el) in
£^00 (A)*, we are done. D