5.1. Exact groups 171(2) the left translation action of r on the Stone-Cech compactification
,er is amenable (i.e., for any finite subset E c r and E > O, there
exist a finite subset F c r and a function m: r -+ Prob(F) c
Prob(r) such thatsup{jjs.mt - mstll : s EE, t Er}::::; c);
(3) r acts amenably on some compact topological space.
Moreover, if r is countable and exact, then r acts amenably on a compact
metrizable space.Proof. First observe that amenability of the canonical action on ,er is equiv-
alent to the parenthetical remark in condition (2), by Lemma 4.3.8 and the
universal property of the Stone-Cech compactification. Now, it is easy to
see that condition (2) is equivalent to condition (4) in Theorem 5.1.6 via the
correspondenceμ<---+ m given by mt(x) = μt(x-^1 t). Hence, we get (1) =? (2).
(This also follows from the identification C (,Br) = £^00 (r) of r-C* -algebras,
Proposition 5.1.3 and Theorem 4.4.3.) The implication (2) =? (3) is immedi-
ate; (3) =? (1) follows from the inclusion C~(r) c C(X) ><Irr and Theorem
4.4.3.
The countability /metrizability assertion is standard fare once we recall
that C (Y) is separable if and only if Y is metrizable. Indeed, assume r
acts on C(X) amenably and let Tn: r-+ C(X) be a sequence of functions
satisfying the definition of amenable action. Since r is countable, the C* -
algebra A generated by the functions {Tn(g) : n E N,g Er} is separable.
The algebra generated by A and all its translates by r is separable, abelian
and r-invariant. Finally, the Tn's take values in this algebra, so r acts
amenably on it. 0
There are several different ways of proving exactness for free groups. For
example, it follows from work of Choi that C~(JF2) embeds into a Cuntz al-
gebra ([36]). Since subalgebras of nuclear C*-algebras are exact, this shows
exactness of all free groups.^4 Here is a straightforward proof based on mea-
sures.
Proposition 5.1.8. Free groups are exact.
Proof. It suffices to show lF2 is exact. Fort E lF2 of length l with reduced
form t =ti·· ·tz, we denote t(k) =ti·· ·tk fork::::; land t(k) = t fork> l.
4Perhaps you haven't seen the fact that all free groups contain one another as subgroups?
It suffices to show lF2 contains a copy of JF 00 • Let a, b E lF2 be free generators and check that the
group elements bnab-n are all free.