1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

F. Bimodules over von Neumann Algebras 489

j,k,k'

for every x EB (and, in particular, uu* = L,k,k' Wk,k'eowk,k' = 1). D

Lemma F.19. Let M be a finite van Neumann algebra. For every x EM

and projections ek E M with L, ek = 1, one has

IT(x)l^2 :::; L T(ekx*ekx).

k

Proof. We may assume that T(y) =(ye, e) for y EM. Then, we have

IT(x)l^2 =IL T(ekxek)l^2 = i L(ekxeke, eke)l^2

k k

:::; L liekxekell^2 L liekell^2 = L T(ekx*ekx)

k k k

by the Cauchy-Schwarz inequality. D

A subgroup A c r is called malnormal if for every s E r \ A one has

sAs-^1 n A= {e}.

Theorem F.20. Let Ac r be a malnormal subgroup and Ao c L(A) be a

diffuse van Neumann subalgebra. Then, Aci n L(r) c L(A). More generally,

if u E L(r) is a unitary element such that uAou* c L(A), then u E L(A).

Proof. Let u E L(r) be a unitary element such that uAou c L(A). It
suffices to show T(A.(s)u) = 0 for every s E, r \A. Let s Er\ A be given
and take £ > 0 arbitrary. Let A' = sAs-^1. We observe that An A' = { e}
implies that T(ax) = T(a)T(x) for every a E L(A) and x E L(A'). Let's apply
this observation to uAou c L(A) and >..(s)Ao>..(s) c L(A'). Since Ao is
diffuse, we may find projections ek E Ao such that L, ek = 1 and T( ek) < £
for every k. Then, we have

T(eku*>..(s)ek>..(s)*u) = T(ueku*>..(s)ek>..(s)*)

= T(ueku*)T(>..(s)ek>..(s)*) = T(ek)^2.

Therefore, by Lemma F.19, we have

IT(.X.(s)u)l^2 :::; L T(eku >..(s)ek>..(s)u) = L T(ek)^2 < £.
k k
Since£> 0 was arbitrary, we conclude that T(A.(s)u) = 0. D
Remark F .21. It can be shown that the subgroup A c r is malnormal in
the following cases: A = ri c r 1 * r2 = r, and A c Y Z A = r (wreath
product).