1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.6. Amenable groups

(8) for any finite subset E c r, we have

1

11TEI 2: "-sll = 1;

(9) C~(r) is nuclear;^18

(10) L(r) is semidiscrete.

sEE

51

Proof. (1) =? (2): Take an invariant meanμ on £^00 (r). Being the predual
of £^00 (r), £^1 (r) is dense in £^00 (r) and thus we can find a net (μi) in Prob(r)
which converges toμ in the o-(£^00 (r),£^00 (r))-topology. Note that for each
s Er, the net (s.μi - μi) converges to zero weakly in £^1 (r) (not just weak
in £^00 (r)). Hence, for any finite subset E c r, the weak closure of the
convex subset Ef>sEE{s.μ - μ: μ E Prob(r)} contains zero. Since the weak
and norm closures coincide, by the Hahn-Banach Theorem, assertion (2)
follows.

(2) =? (3): Let a finite subset E c r and c: > 0 be given. Choose
μ E Prob(r) such that

2: 11s.μ - μ111 < c:.

sEE

Given a positive function f E £^1 (r) and r ~ 0, we define a set F(f, r) =
{t E r : f(t) > r} and let XF(f,r) be the characteristic function of this
set. For a pair of positive functions f, h E £^1 (r) and t E r, observe that
IXF(f,r)(t) - XF(h,r)(t)I = 1 if and only if r lies between the numbers f(t)
and h(t). If both f and hare bounded above by 1, it follows that

lf(t) - h(t)I = fo

1

IXF(f,r)(t) - XF(h,r)(t)ldr.

Applying this observation toμ ands.μ, we get

lls.μ-μIii= L ls.μ(t) - μ(t)I

tEr

= L fl IXF(s.μ,r)(t) - XF(μ,r)(t)ldr

tEr Jo

= fl (L IXsF(μ,r)(t) - XF(μ,r)(t)l)dr

Jo tEr

= fo

1

lsF(μ,r) .6F(μ,r)ldr.

18This is also equivalent to knowing O*(r) is nuclear. Thanks to condition (6), one direction
is immediate, but we don't yet have the tools for the other. Once we introduce The Trick (Section
3.6), it will be trivial by considering the right regular representation.