418 15. Group van Neumann AlgebrasAmalgamated free products.
Proposition 15.3.12. Let I' = I'i *A I'2 be an amalgamated free product
and let g = {I'i, I'2}. If both ri are exact and A is amenable, then f'^9 is
amenable and, in particular, r is bi-exact relative to g.Before giving the proof, we point out that the amenability assumption
on A is essential. Indeed, if r i = r~ x A and A is nonamenable, then
r = (I'i * I'~) x A and thus L(I'i * I'i) has a noninjective commutant in
L(I'). More generally, if Si E ri \A normalize A and sias!i = s2as2i for all
a EA, thens= sis2i Er has infinite order and commutes with A.Proof. We first prove that the I'-space f'9 is amenable as a I'i-space. We
prove this for i = 1. Let Ac .€^00 (I'i) be the C-subalgebra of those functions
f such that f = ft for all t E A. Averaging over the right A-action, we
obtain a (left) I'i-equivariant conditional expectation from .€^00 (I'i) onto A.
By Exercise 15.2.2, it suffices to find a I'i-equivariant -homomorphism 1f
from A into C(f'9). Fix a system { e} LJ Sf c ri of representatives of A \ri,
and set
.x = {e} u sg u sgsf u sgsf sg u · · · c r.
Then, every s E I' can be uniquely written in the forms = six, where si E I'i
and x Ex (cf. Appendix E). We define Jr: A-+ .€^00 (I') by 7r(f)(six) = f(si)
for si E I'i and x E .X. Our task is to show 7r(f) -1f(j)t E c 0 (r; g) for every
t E I'i U I'2. Suppose first that t E I'i. Then, for every six E I', one has
either sixri = siri (if x = e) or sixri = siay for some a EA and y Ex
(if x =/= e). Since f is right A-invariant, 7r(f) - 7r(f)t has support in I'i. It
follows that 1r(f) - 1f(j)t E c 0 (r; g). Suppose next that t E r 2. Then, one
has 7r(f) - 1f(j)t = 0 by similar reasoning. Altogether, this implies f'9 is
amenable as a I' i-Space.
Let T =I' /I'i LJ I' /I'2 be the Bass-Serre tree on which I'= I'i A I'2 acts
and let T be its compactification,^4 as defined in Section 5.2. We will find
a I'-equivariant continuous map from f'9 into T, which suffices to show the
amenability of f'9 by Proposition 5.2.1 and Lemma 5.2.6. Choose a base
point o ET and define a I'-equivariant -homomorphism er: C(T)-+ .€^00 (I')
by er(f)(s) = f(so). We will show er(!) - er(f)t E co(I'; {A}) for every
f E C(T) and t E r. Suppose by contradiction that this is not the case.
Then, there exists s > 0 such that the set
n = {s Er: lf(so) - f(srio)I;::::: s} c r
is not small relative to {A}. Hence, there exists a net (sn) inn such that
Sn-+ oo/{A}. We may assume that sno-+ z for some z E T. Since every
edge stabilizer of the r-action on the Bass-Serre tree is an inner conjugate
(^4) Although we use the term "cornpactification", T is not open in T.