482 F. Bimodules over van Neumann Algebrasthe right M--action, we have PE Iffi(R^2 ) ® M, where M acts on L^2 (M) from
the left. The isometry V determines a *-isomorphism N ~ P(Iffi(R^2 ) ® M)P.
The von Neumann algebra Iffi(R^2 ) ® M is of type II 00 and has a canonical
faithful normal semifinite trace 7, defined by
+([xi,j]i,j) = L T(xi,i) E [O, oo],
i
where positive elements x E JIB(.€^2 ) ®Mare viewed as infinite matrices [xi,j]i,j
with entries in M. We define dimM 1i = i(P) E [O, oo] and call it the
dimension of the right M-module 1i. It depends on the choice of tracial
state T but not on the choice of V. Indeed, if W: 1i ~ £^2 Q9 L^2 (M) is
another right M-module isometry, then we have WV* E JIB(R^2 ) ®Mand
i(VV*) = i(VW*WV*) = i(WV*VW*) = i(WW*).
If dimM 1i < oo, then there exists an increasing sequence of projections
Zn E Z(M) converging strongly to 1 such that P(l Q9 Zn);:) En@ lM, where
En is a rank-n projection in Iffi(R^2 ). To see this, take Zn E Z(M) to be the
maximal projection such that I:i ctr(~,iZn) :::; n. (It might be easier to
consider Z(M) as a function space on the spectrum of Z(M).) We note
that 1izn is isomorphic to a right M-submodule off~ Q9 L^2 (M).
Proposition F .10. Let 1i be an N-M-bimodule with dimM 1i < oo. Then,
there exist a nonzero projection f E N and a nonzero f N f-M -sub-bimodule
K of f1i such that K is isomorphic, as a right M -module, to a right M -
submodule of L^2 (M).Proof. Let 1i be an N-M-bimodule with dimM 1i < oo. Truncating by a
central projection if necessary, we may identify 1i with P(R~ Q9 L^2 (M)), as
a right M-module, for some n and some projection PE Mn(M). It follows
that N C PMn(M)P. Let z E Z(N) be the projection such that N z is of
type II and N(l - z) is of type I.
Suppose first that z i= 0 and let qi, ... , qn E N be mutually equivalent
projections with sum z. Then, f = qi E N and K = f1i satisfies the
requirement. Suppose next that z = 0 and N is of type I. Then, there exists
a nonzero projection f E N such that f N f is abelian. Choose j such that
(ei,j@ lM)f i= 0 and let p be the right support of (ei,j@ lM)f. It follows
that p E fMn(M)f and p ;:) ei,i Q9 lM in Mn(M). By Corollary F.8, the
projection pis equivalent to a projection q E (f NJ)' nfMn(M)f. (Consider
a maximal abelian subalgebra of fMn(M)f which contains f NJ.) Now,
f EN and K = q1i satisfies the requirement. In any case, we are done. D
L^2 (M) as unbounded operators and Popa's Theorem. Let M be a von
Neumann algebra with a distinguished faithful normal tracial state T. An
important fact about finite von Neumann algebra is that one can interpret