1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

48 2. Nuclear and Exact C* -Algebras

A. That is, EI(.\(s)) = XA(s).\(s) for s Er, where XA is the characteristic

function of A. The same result holds for universal C* -algebras.

2.6. Amenable groups

Amenable groups admit approximately 101010 different characterizations;

our goal in this section is to present a few that can be proved without

too much effort.^15

Definition 2.6.1. A group r is amenable if there exists a stateμ on R^00 (r)

which is invariant under the left translation action: for all s E r and f E

£^00 (r), μ(s.f) = μ(!).

Such a state μ is called an invariant mean.

Definition 2.6.2. For a discrete group r, we let Prob(I') be the space of

all probability measures on r:

Prob(r) = {μ E R^1 (r): μ 2:: O and Lμ(t) = l}.

tEr

Note that the left translation action of r on R^00 (I') leaves the subspace

Prob(I') invariant; hence we can also use μ 1--t s.μ to denote the canonical

action of r on Prob(r).

Definition 2.6.3. We say r has an approximate invariant mean if for any

finite subset E c r and c > 0, there existsμ E Prob(r) such that

max lls.μ-μ111 < 6.

sEE

Recall that the symmetric difference of two sets E and F, denoted El:c,.F,

is EUF\EnF.

Definition 2.6.4. We say r satisfies the F¢lner condition if for any finite

subset E c r and c > 0, there exists a finite subset F c r such that

lsF L,. Fl

~~ IFI <c,

where sF ={st: t E F}.^16 A sequence of finite sets Fn c r such that

lsFn L,. Fnl ---+ O

IFnl

for every s Er is called a F¢lner sequence.

15We'll see several more in later chapters.

(^16) Since sF i::,, F = [sF \ (sF n F)] u [F \ (sF n F)], it follows that ls~.#;FI = 2 - 2 IF
1 r;(1.
Hence the F¢lner condition is equivalent to requiring maxsEE lsf_;1(1 > 1 - e:/2, which is often
how it gets used in our context.