1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

408 15. Group von Neumann Algebras

Definition 15.1.2. We say the group r is bi-exact relative to g if it is exact

and there exists a map

μ: r _____, Prob(r)

such that for every s, t E r, one has

lim JJμ(sxt) - s.μ(x)JI = 0.

x-+oo/Q

Remark 15.1.3. A countable exact group r is bi-exact relative to g if for

every finite subset E c r and s > 0, there exists μ: r _____, Prob(r) such that

for every s, t EE the subset {x : JJμ(sxt) - s.μ(x)JI ;:::: s} is small relative

to Q. Since we won't need this fact, we won't prove it. However, the main

points are laid out in Exercise 15.1.1.

Let JK(r; 0) be the hereditary C* -subalgebra of IIB( .e^2 (r)) generated by

co(r; Q):

JK(r; Q) =the norm closure of c 0 (r; Q)IIB(R^2 (r))c 0 (r; Q).

Since the left and right regular representations A and, respectively, p nor-

malize co(r; Q), the reduced group C*-algebras C1(r) and c;(r) are in the

multipliers of JK(r; 9).

Lemma 15.l.4. Let r be an exact group and g be a nonempty family of

subgroups of r. Then r is bi-exact relative to Q if and only if there exists a

u.c.p. map

e: c1(r) @c;(r) _____, IIB(.e^2 (r))

such that B(a@ b) - ab E JK(r; Q) for every a E C1(r) and b E c;(r).

Proof. We first prove the "if'' direction. Let e be a u.c.p. map such that

B(a@b) - ab E JK(r; Q). By Voiculescu's Theorem (Theorem 1.7.8), there is

an isometry V: R^2 (r) _____, R^2 (r x r) such that B(a@b)-V*(a@b)V E JK(R^2 (r))

for every a and b. It follows that

V*(.:X.(s) Q9 p(t))V - .:X.(s)p(t) E JK(r; Q)

for every s, t E r. Define a map μ: r -----t Prob(r) by

μ(x)(y) = L J(V5x)(y, z)J^2 •

It follows that

JJμ(sxt) - s.μ(x)JJ1 ~ JJ IV5sxtl^2 - J(.:X.(s) @p(t))-^1 V5xJ^2 JJ1

~ 2JIV5sxt - (.:X.(s)@ p(t))-^1 V5xJJ2 -----t 0

as x _____, oo/Q.

Now we prove the "only if'' direction. Define a unitary operator U on
R^2 (r) @ R^2 (r) by U(5x @ 5y) = Ox @ Ox-iy, so that U*(.:X.(s) @ p(t))U =