484 F. Bimodules over von Neumann Algebras
We briefly review Jones's basic construction .. Let A c M be a von
Neumann subalgebra and denote by EA the trace-preserving conditional ex-
pectation from M onto A and by eA E JBS(L^2 (M)) the orthogonal projection
onto L^2 (A). We note that eAXeA = EA(x)eA for every x E M. It follows
that
n
A= {LxkeAYk: n EN, Xk,Yk EM}
k=l
is a *-subalgebra of JBS(L^2 (M)). It is not hard to show that A acts nonde-
generately on L^2 (M) and hence
(M,A) :=A"= (MU {eA})".
The von Neumann algebra (M, A) is called the basic construction of Ac M.
We have
(M,A)' = M' n {eA}' = p(A),
where p(A) is the right action of A on L^2 (M). The von Neumann algebra
(M, A) is semifinite with a canonical faithful normal semifinite trace Tr such
that Tr(xeAy) = T(xy) for x, y E M. (See Exercise F.6.) We note that
if 1t C L^2 (M) is a right A-submodule, then the orthogonal projection PH
onto 1t belongs to (M, A) and dimA 1t = Tr(PH.)· See [158] for more on the
basic construction.
The following very useful theorem is due to Popa.Theorem F.12. Let ACM be finite von Neumann algebras with separable
predual and let p E M be a nonzero projection. Then, for a von Neumann
subalgebra B C pMp, the following are equivalent:.
(1) there is no sequence (wn) of unitary elements^2 in B such that
llEA(bwna)/12 --+.O for every a, b EM;
(2) there exists a positive element d E (M, A) with Tr( d) < oo such
that the ultra weakly closed convex hull of { w dw : w E B unitary}
does not contain O;
(3) there exists a B-A-submodule 1t of pL^2 (M) with dimA 1t < oo;
(4) there exist nonzero projections e E A and f EB, a unital normal
*-homomorphism (): f Bf·__, eAe and a nonzero partial isometry
v E M such that
Vx E f Bf, xv= ve(x)
and such that vv E ()(j Bf)' n eM e and vv E (!Bf)' n f M f.
(^2) A unitary element w in B is a partial isometry in M such that ww = p = ww.