1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

15.1. Noninjective relative commutants 409

(.\ ® .\)(s)(l ® p)(t) (cf. Fell's absorption principle). Letμ: r --t Prob(I') be

a map as in the definition of bi-exactness and define an isometry V: £^2 (r) --t

£^2 (r) ® £^2 (r) by V b"x = U (μ( x)^112 ® b"x). Then, it is routine to check that

V*(.\(s) ® p(t))Vb"x = (.\(s)(μ(x)^112 ), μ(sxr^1 )^112 ).\(s)p(t)6x.

Since

lim llA.(s)(μ(x)^1 l^2 ) - μ(sxC^1 )^112 ll~.::::; lim lls.μ(x) - μ(sxC^1 )111 = 0,

x-->oo/Q x-->oo/Q

we have V*(.\(s) ® p(t))V - .\(s)p(t) E IK(r; 9) for every s, t Er. D

Here is the main theorem of this section. (See Definition F.13 for the

terminology "N embeds in L(A) inside L(I')".)

Theorem 15.1.5. Let r be a countable group and g be a countable family

of subgroups of r. Assume that the group r is bi-exact relative to g. Let

p E L(r) be a projection and N c pL(I')p be a von Neumann subalgebra. If

the relative commutant N' n pL(I')p is noninjective, then there exists A E g

such that N embeds in L(A) inside L(I').

The proof of this result requires some preparation. Let MC IBl(H) be a

von Neumann algebra and consider the *-homomorphism

<l>p: M 8 M' 3 L ak ®bk rt L Ep(pakp)bkp E IBl(pH).

k k

Suppose that there are weakly dense C* -subalgebras Cz C M and Cr C M'

such that Cz is exact and <l>p is min-continuous on Cz 8 Cr. Then P is

injective.

Proof. By Lemma 9.2.9, our assumptions imply that <l>p is min-continuous

on M 8 M'. By The Trick, <I>PIM extends to a u.c.p. map 'ljJ from IBl(H)

into (pM')' = pMp. (Note that the argument for The Trick only requires

<I> p lic1@M' to be *-homomorphic.) It follows that Epo'lj; IB(y?-l) is a conditional

expectation from IBl(pH) onto P. D