1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

488 F. Bimodules over von Neumann Algebras

Since Bis maximal abelian, we have Ji := vv* E B. Let N = B(f Bf)' neM e.

Since Ae C N is maximal abelian, Corollary F.8 implies that there exists a

partial isometry w1 EN such that w1wi = v*v and el := wiw1 E Ae. Let

v1 = vw1. For every x E B and a E A, we have w1 awi E N and

xv1avi = vB(fxf)w1awiv* = vw1awiB(fxj)v* = v1avix.

It follows that v1Avi c B' n f1M fI = B f1. Hence,

Bfi = Bvv* = vB(fBJ)v* = v1B(f Bf)vi C v1Avi C Bfi.

Consequently el = viv1 E A, Ji = v1vi E B and v1Avi = Bfi. Trim-

ming el if necessary, we may assume that r(e1) = 1/k for some k E N.

Take equivalent projections e2, ... , ek E A (resp. h, ... , fk E B) such that

L:7=l ej = 1 (resp. L:7=l fj = 1). By Lemma F.16, we can find partial

isometries v2, ... , Vk EM such that ej = vjvj, fj = Vjvj and VjAvj = Bfj·

Setting u = 2=7=l Vj, we are done. D

Another case where local embedding implies global embedding is the
following.

Lemma F.18. Let A, B c M be diffuse finite van Neumann algebras such

that A and B' nM are factors. (This implies that M and B are also factors.)

Assume that A~ n MC A for any diffuse van Neumann subalgebra Ao CA.

If B embeds in A inside M, then there exists a unitary element u E M such

that uBu* c A.

Proof. Let e E A, f E B, (}, v E M be given as in condition ( 4) of The-
orem F.12. Since vv E (!Bf)' n f Mf = f(B' n M)f (cf. Exercise F.2),
vv =ff' for some projection f' EB' n M. We take projections Jo E fBf
and fo E f'(B' n M)f' such that r(fo) = 1/n and r(f 0 ) = 1/n' for some
n, n' EN. Let vo = f 0 fov and eo = B(fo) EA. Then, for any x E foBfo, we
have

xvo = fofoxv = fofovB(x) = voB(x).

This implies that v 0 vo E A~ n M c A, where Ao = B(foB fo) EB e6-Ae6-is a
diffuse subalgebra of A. Let u1, ... , Un E B be partial isometries such that
ukuk = fo and 2:: u'kuk = 1, likewise for fo E B' n M and u~, ... , u~, E
B' n M. We note that r(vov 0 ) = r(fof 0 ) = (nn')-^1. Let wk,k' E A be
partial isometries such that wk,k'wk,k' = v 0 vo and I: wk,k'wk,k' = 1. Then,
u = 2:: wk,k'vouku~, is the desired unitary. Indeed, one has

UXU * = '""""' ~ Wj,j'VoUjUj'X * I ( Uk' I )* UkVOWk,k' * *

j,j',k,k'

= '""""' L_; Wj,k'VoUjXUkVOWk,k' * * *

j,k,k'