F. Bimodules over von Neumann Algebras 485Proof. (1) =?- (2): By condition (1), there exist a finite 'subset ~ C M
and E > 0 such that maxa,bE~ llEA(bwa)ll2 ;:: E for all unitary elements
w E B. Let eA E IIB(L^2 (M)) be the orthogonal projection onto L^2 (A) and
let d = ~bE~beAb E (M, A). One checks that Tr(d) = ~bE~T(bb) < oo
and
'I)wdwa,a) = L (eA~,~) = L llEA(bwa)ll~;:: c^2
aE~ a,bE~ a,b
for all unitary elements w EB. This yields condition (2).
(2) =?- (3): Let d be as in condition (2) and let C be the ultraweakly
closed convex hull of {wdw: w EB unitary}. Since C can be regarded as a
closed convex subset in L^2 ( (M, A), Tr), we can consider the circumcenter do
of C (cf. Exercises D .1 and F. 3). By uniqueness of the circumcenter, we have
do E p(M, A)p n B'. It can be shown that Tr(do) :::; Tr(d) < oo. (NB: This
fact is not entirely trivial since Tr is not bounded.) Therefore, there exists
a nonzero spectral projection q of do such that Tr(q) < oo. It follows that
1-{ = qL^2 (M) is a nonzero B-A-submodule such that dimA 1-{ = Tr(q) < oo.
(3) =?- (4): By Proposition F.10, there exists a nonzero projection f E
B, a nonzero f Bf-A-sub-bimodule /(,of 1-{ and a right A-module isometry
V: K -----t L^2 (A). Let x E f Bf be given. Since VxV commutes with
the right A-action, we have V x V E eAe, where e = VV E A. Hence
(} ( x) = V x V defines a unit al normal -homomorphism from f Bf into eAe.
Let e = VlA EK and observe that e-=/= 0, since ve = e. Then we have
xe = VB(x)lA = Vi(;) = V*lAB(x) = ee(x)
for every x E f Bf. Since J(, c L^2 ( M), we m~y view e as a square .integrable
operator Le affiliated with Mand we have xLe = LeB(x) for every unitary
element x E f Bf (see Exercise F .4). It follows that
1Lel^2 = (xLe)*(xLe) = (LeB(x))*(LeB(x)) = B(x)*ILel^2 B(x)
f~r every unitary element x E fBJ; hence ILel com:mutes with B(f Bf). Let
Le= vlLel be the polar decomposition of Le. Then we have ·
xvlLel = xLe = LeB(x) = vlLelB(x) = vB(x)ILel
and hence xv = vB(x) for every (unitary) element x E f Bf. The claims
that v*v E B(f Bf)' n eM e and vv* E (f BJ)' n f M f are automatic.
(4) =?-(1): Let e, f and v be as in condition (4). Let Ee be the trace-
preserving conditional expectation from eMe onto B(f Bf). We note that
Ee ( v*v) is a nonzero positive element in the center of (} (f Bf) and that
vEe(v*v)^2 v* E (f Bf)' n f Mf. Let {fi} be a maximal family of mutually
orthogonal projections in B such that fi ;::$ f in B. Then, ~ fi coincides
with the central support off in B (cf. [183, Lemma V.1.7]). Let Ui EB be
a partial isometry such that uiui = fi and uiui :::; f, and set Vi= uiv. We