1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

50 2. Nuclear and Exact C*-Algebras

Here is the simplest example of something nonamenable.

Example 2.6.7 (Nonabelian free groups). The free group lF2 of rank two

is not amenable. Let a, b E lF 2 be the free generators and set

A+ = {all reduced words starting with a} C lF 2.

Similarly, let A- be the reduced words beginning with a-^1 and likewise

define B+ and B-. Then, for C = {1, b, b^2 , ••. } c lF2, we have

lF 2 = A+ LJ A-LJ ( B+ \ C) LJ ( B-U C)

=A+ LJ aA-

= b-^1 (B+ \ C) LJ (B-UC).

This kind of decomposition is said to be paradoxical.^17 Note that the exis-

tence of an invariant mean μ on £^00 (r) would lead to a contradiction:

1 = μ(1) = μ(xA+) + μ(xA-) + μ(xB+\C) + μ(xB-uc)

= μ(xA+) + μ(a.xA-) + μ(b-^1 ·XB+\C) + μ(xB-uc)

= μ(XA+ + a.xA-+ b-^1 -XB+\c + XB-uc)

= 2μ(1) = 2.

Since amenability passes to subgroups, it follows that all nonabelian free

groups (on any number of generators) are nonamenable.

Here is a small sample of the known characterizations of amenable

groups.

Theorem 2.6.8. Let r be a discrete group. The following are equivalent:

(1) r is amenable;

(2) r has an approximate invariant mean;

(3) r satisfies the F¢lner condition;

( 4) the trivial representation To is weakly contained in the regular rep-

resentation .A {i.e., there exist unit vectors ei E £^2 (r) such that

11.As(~i) - eill ~ 0 for alls Er);

(5) there exists a net ( r.pi) of finitely supported positive definite functions

on r such that I.pi ~ 1 pointwise;

(6) C*(r) = c~(r);

(7) C~(r) has a character {i.e., one-dimensional representation);

(^17) This paradoxical decomposition leads to the famous Banach-Tarski paradox. See Eric
Weisstein's website Mathworld (mathworld.wolfram.com) for more.