1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.1. Exact groups

e

Figure 1. The Oayley

graph of 1F2 and the

boundary olF 2

s.x

f.

Figure 2. Amenability

of lF 2 acting on olF 2

173

With this geometric picture in mind, one checks that lls.μx - μs.xll :::;
2lsl/N for all s E lFr and x E 8lFr. Letting N -too, this shows that lFr acts
amenably on its ideal boundary.

The original definition. Kirchberg and Wassermann introduced exact-
ness for groups in [108]. Their definition involves short exact sequences of
r-C*-algebras - that is, an algebra A, an action a: r -t Aut(A) and an
ideal I <I A such that a 9 (I) = I, for every g E r. In this situation, I and
A/ I inherit r-algebra structures denoted by air and a, respectively, and we
refer to 0 -t I -t A -t A/ I -t 0 as a short exact sequence of I'-algebras.

Definition 5.1. 9. We say r is KW-exact if for every short exact sequence
of r-algebras, 0 -t I -t A -t A/ I -t O, the sequence

O -t I ><Irr -t A ><Ir r -t (A/ I) ><Irr -t 0

is also exact.^5

Just as for short exact sequences and maximal tensor products (see
Proposition 3.7.1), it is not too hard to show that full crossed products
always preserve exactness, due to the fact that

A><1I'

I ><1 r

contains a r-equivariant copy of A/ I. (Consider this an exercise.)

Theorem 5.1.10. A discrete group r is exact if and only if it is KW-exact.

5We can evidently identify I ><lr r, by independence of representation.
Moreover, there is a canonical surjective *-homomorphism A ><lr r ---+ (A/ I) ><lr r whose kernel
contains I ><lr r. See Exercise 4.1.4.