412 15. Group von Neumann Algebras
'Pl: X ---+ b.gr which is r x r-equivariant, where the right action of r on
b.gr is trivial. By symmetry, there exists a continuous r x r-equivariant
map 'Pr : X ---+ b.?r, where b.?r is amenable as a 1 x r-space and is trivial as
a r x 1-space. Thus, the r x r-space b.gr x b.?r is amenable and 'Pl x 'Pr is a
r x r-equivariant continuous map from X into it. Therefore, Xis amenable.
Finally, assume condition (3) and define a C* -algebra D by
D = C*(,\(r), p(r), £^00 (r)) + IK(r; Q) c IIB(£^2 (r)).It is not hard to see that JK(r; Q) is an ideal in D and D /IK(r; Q) is a quotient
of the crossed product of £^00 (r)/co(r; Q) by r x r (actually, it's isomorphic
to this crossed product). By assumption, the canonical -homomorphism
C~ (r) 0 c; (r) ---+ D /IK(r; Q) is min-continuous and D /IK(r; Q) is nuclear.
Hence, the quotient map from D to D /IK(r; Q) has a u.c.p. splitting on any
separable C -subalgebra, by the Choi-Effros Lifting Theorem. Thanks to
Lemma 15.1.4, we are done. D
It will be more convenient to work with b._9r than the original defini-
tion of bi-exactness. This allows us to exploit the technology developed in
previous chapters.
Definition 15.2.4. Let fi be an equivariant compactification of r. We say
fi is small at infinity relative tog if the following holds: If (sn) is a net in r
such that Sn ---+ x E fi and Sn ---+ oo / g, then snt ---+ x for evei;y t E r.
One should check that an equivariant compactification fi ·of r is small
at infinity relative to g if and only if the identity map on r extends to a
continuous map from fi9 onto fi. The image of b.gr under this map is the
set of x E fi such that there is a net ( sn) in r with the property that Sn ---+ x
and Sn---+ oo/Q.
Example 15.2.5. In the examples below, amenability of b.gr follows from
that of fi9..
(1) Let Q be the empty family. Then, c 0 (r; Q) = {O} and fi9 is a
one-point set. Hence fi9 is amenable if and only if r is amenable.
(2) Let Q = {1 }, where 1 is the trivial subgroup consisting of the
neutral element. Then, c 0 (r; Q) = c 0 (r) and fi9 is the universal
compactification which is small at infinity. Rec.all from Section 5.3
that if r is a hyperbolic group, then there is a r-equivariant con-
tinuous map from fi9 onto the Gromov compactification - hence
fi9 is amenable for a hyperbolic group (Corollary 5.3.19).
(3) Suppose r E Q. Then co(r; Q) = C(fi9) = £^00 (r) and fi9 = ,Br.
Hence fi9 is amenable if and only if r is exact.It is often useful to ignore amenable subgroups.