1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

486 F. Bimodules over van Neumann Algebras

may assume that uo = f o = f. Then, for every unitary element w E B, we

have

L llEA(viwvo)ll~ ~ L llv*vEe(viwvo)ll~

i i

= L llv*vEe(v*v)e(uiwuo)ll~

i

= L llEe(v*v)v*uiwuovll~

i

= L T(v*u 0 w*uivEe(v*v)2v*uiwuov)

i

= L T(u 0 w*uiuiwuovE 0 (v*v)^2 v*)

i

= L T(vEe(v*v)^2 v*)

i

= T(Ee(v*v)^3 ).

Since I:i llvi'll~ :::; 1 and llEA(v;'wvo)ll2 :::; llvi'lb we can choose a finite

subset i of {Vi} such that

L llEA(v;'wvo)ll~ ~ T(Ee(v*v)^3 )/2 > 0

ViEff

for all unitary elements w in B. This completes the proof. D

Definition F.13. Let A c M and B c pMp be finite van Neumann al-

gebras. We say B embeds in A inside M if one of (and hence all of) the

conditions in Theorem F.12 holds.

Nate that if there is a nonzero projection Po E B such that poBPo embeds

in A inside JYI, then B embeds in A inside M (as condition (4) in Theorem

F.12 evidently implies). Recall that a (nonzero) projection f EB is minimal

if and only if f Bf= <Cf, and a van Neumann algebra Bis diffuse if it has

no minimal projections.

Corollary F.14. Let M be a finite van Neumann algebra with separable

predual and (An) be a sequence of van Neumann subalgebras. Let NC pMp

be a van Neumann subalgebra such that N does not embed in An inside M

for any n. Then, there exists a diffuse abelian van Neumann subalgebra

BC N such that B does not embed in An inside M for any n.

Proof. We may assume that each Ak appears infinitely often in the sequence

(An). Let { Xn} be a 11 II 2-norm dense sequence in the closed unit ball of

M. We will construct an increasing sequence B 1 c B 2 c · · · of finite-

dimensional abelian van Neumann subalgebras of N with unitary elements