414 15. Group von Neumann AlgebrasExercises
Exercise 15.2.1. Let X be a compact r-space and Prob(X) be the state
space of C(X) equipped with the natural r-action. Prove that Xis amenable
if and only if Prob(X) is amenable.
Exercise 15.2.2. Let X be a compact r-space and assume there is a r-
equivariant u.c.p. map from £^00 (r) into C(X). Prove that X is amenable
provided that r is exact.
15.3. Examples
Hyperbolic groups. We have already seen that f'g and (hence) L~Pr are
amenable for a hyperbolic group r and g = {1 }.
Theorem 15.3.1. Let r be a hyperbolic group and B c L(r) be a diffuse
van Neumann subalgebra. Then B' n L(r) is injective.
Proof. It follows from Theorem 15.1.5 that noninjectivity of B' n L(r) im-
plies that B embeds in Cl inside L(r), which means that B has a nonzero
minimal projection. D
A type Il1-factor N is said to be prime if N ~ N1 ® N2 implies that N1
or N2 is finite-dimensional.
Corollary 15.3.2. Let r be a hyperbolic group and N c L(r) be a subfactor.
Then N is either injective or prime. In particular, the free group factors
L(IF'r) (r;::: 2) are prime..
Proof. Suppose N1 ® N2 ~ N c L(r) is noninjective. Then either N1 or
N2 is noninjective and the other cannot be diffuse by the theorem above.
We note that Ni is a factor since N is a factor and that a finite factor with
a minimal projection is finite-dimensional. D
Direct product of hyperbolic groups.
Lemma 15.3.3. Let ri, ... , r n be groups and r = IJ~=l ri be the direct
product. Let gi be a family of subgroups of ri and define a family g of
subgroups of r by
g = LJ{A x f[rj: A E gi}·
i #iIf each of ri is bi-exact relative to gi, then r is bi-exact relative to g.
We leave the proof as an exercise.