Bimodules over von
Neumann Algebras
Appendix F
In this appendix we study bimodules over von Neumann algebras, restrict-
ing our attention to the finite case. Hence, every von Neumann algebra
M appearing in this appendix comes together with a distinguished faithful
normal tracial state T (or TM, if the context isn't clear). For an inclusion of
von Neumann algebras N C M, we assume TN = TM IN.
Examples.
Definition F.1. Let M and N be von Neumann algebras. A (left) M-
module is a Hilbert space 1t together with a normal unital -homomorphism
1l': M ---+ $(1t). (NB: Injectivity of 1l' is not assumed.) A right M -module is a
Hilbert space 1t together with a normal unital -homomorphism p: M^0 P ---+
$(1t). To reduce clutter, we rarely write 1l' and p (but keep them in mind,
of course).
An M-N -bimodule^1 is a Hilbert space 1t together with normal unital *-
homomorphisms 1l': M ---+ $(1t) and p: N°P ---+ $(1t) whose ranges commute.
We refer to 1l' as the left action and to p as the right action. We use the
intuitive notation
aex := 1f(a)p(x^0 P)e
for a E M, x E N and e E H.
By the representation theory of von Neumann algebras, every M-module
has a rather simple structure - it's isomorphic to a submodule of E9 L^2 (M)
(with the diagonal M-action).
(^1) Also called a correspondence from N to M.
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