1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

192 5. Exact Groups and Related Topics

We've seen that the left-translation action of a hyperbolic group r on

.e= (r) is amenable, but much more is true: the action of r x r on .€^00 (r)

(given by the left and right translations) is amenable mod co(r).

Corollary 5,3.19. If r is hyperbolic, then r x r acts amenably on the

quotient algebra .e= (I')/ co (r).

Proof. The previous proposition ensures that we can find a (rxr)-invariant

subalgebra A c .€^00 (I')/c 0 (r) such that the restriction of the r x r action

to A is amenable on r x { e} and trivial on { e} x r (just let A be the

image of C(f') under the quotient map). By symmetry, we can also find

B c .e= (r) / c 0 (r) such that the restriction of the r x r action to B is trivial

on r x { e} and amenable on { e} x r. The result now follows from Exercise

4.3.1. D

For free groups, the following fact was discovered by Akemann and Os-

trand in [2].

Corollary 5.3.20. Let r be hyperbolic, .\ and p be the left and, respectively,

right regular representations and 7r: IIB(.€^2 (r)) ----+ JIB( .€^2 (r)) /JK(.€^2 (r)) be the

quotient map. Then, the *-homomorphism

C~(r) 0 c;(r) 3 L ak ® Xk f-+ 7r(L akxk) E IIB(.€^2 (r))/JK(.€^2 (r))

k k

is continuous with respect to the minimal tensor norm.

Proof. This is in fact an immediate corollary of Corollary 5.3.19 and The-
orem 4.3.4, but we give a different proof here. It suffices to show that there
exists a nuclear C-algebra A c IIB(.€^2 (r)) such that C~(r) c A and 7r(A)
commutes with 7r( c; (r)). Indeed, if such A exists, then we have an inclusion
C~(r)@C;(r) c A@c;(r) = A®maxc;(r) and a natural -homomorphism
A ®max c;(r)----+ IIB(.€^2 (r))/JK(.€^2 (r)).

So, let f' = r U or be the Gromov compactification and embed C(f') c
.e^00 (r) as above. By Theorems 5.3.15 and 4.4.3, the C-subalgebra A of
the uniform Roe algebra generated by C(f') and C~(r) is nuclear. (It is
-isomorphic to C(f') >4r r by Proposition 5.1.3.) By Proposition 5.3.18, we
have

p;f pt - f = l - f E co(I') C lK(.€^2 (r))

for any f E C(f') and any t E r, which implies that 7r(A) commutes with
7r(c;(r)), as desired. D