1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

15.2. On bi-exactness 411

15.2. On bi-exactness

Definition 15.2.1. Let r be a group and g be a family of subgroups of

I'. For f E .e^00 (r) and t E r, we define the right translation ft E .e^00 (r) by

ft(s) = f(st-^1 ). Note that (jl)t' = ftt'. Now define a compact space f'g by

C(f'g) = {f E .e^00 (r) : f - ft E co(r; Q) for every t Er}

and view it as a r-space, where r acts by left translation. We define another

compact r-space ~gr c f'g by

C(~gr) = C(f'g)/co(r; Q)

and we call it the Q-boundary of r.

Remark 15.2.2. It is not hard to see that x E f'g belongs to ~g if and
only if there is a net (sn) in r such that Sn---+ x and Sn---+ oo/Q.

It is possible that Q = 0 and co(r; Q) = {O}, but otherwise we have
co(r) c co(r; Q) c C(f'g) and f'g is an equivariant compactification of
r.^2 By Gelfand duality, there is a one-to-one correspondence between equi-
variant compactifications f' of r and intermediate C*-subalgebras c 0 (r) c
C(f') c .e^00 (r) which are left translation invariant. It is possible that r E g
and c 0 (r; Q) = .e^00 (r) and ~gr= 0.

Note that f E C(f'g) if f - ft E co(r; Q) for all t in some generating
subset of r, since f - ftt' = (f - ft') + (f - p)t'.

Proposition 15.2.3. Let r be a countable group and g be a nonempty
family of subgroups of r. Then the following are equivalent:

( 1) r is bi-exact relative to g;

(2) the Q-boundary ~gr is amenable;^3

(3) the Gelfand spectrum of .e^00 (r) I co (r; Q) is amenable as a r x r-

space (with the left-times-right translation action).

Proof. Assume condition (1) and let μ: r ---+ Prob(r) be a map as in
Definition 15.1.2. Then, the u.c.p. map μ: .e^00 (r) ---+ .e^00 (r) defined by
μ(f)(x) = (!, μ(x)) has the property that μ(s.f) - s.μ(f)t E c 0 (r; Q). In
particular, μ(!) E C(f'g). Let Q: C(f'g) ---+ C(~gr) be the quotient map.
Then, Q o μ is a r-equivariant u.c.p. map from .e^00 (r) into C(~gr). One
can now deduce the amenability of ~gr from Exercise 15.2.2.

Next, we assume condition (2) and let X denote the Gelfand spectrum
of .e^00 (r)/c 0 (r; Q). The inclusion C(f'g) c .e^00 (r) induces a continuous map

2 A compactification is a compact space f containing r as an open dense subset; it is equi-
variant if the left translation action of r on r extends continuously to f. (This is the same as
Definition 5.3.16, where equivariance was assumed.)

(^3) By convention, we say that the empty r-space 0 is amenable if r is exact.