We relate Morita equivalence for von Neumann algebras to the ``Connes fusion'' tensor product between correspondences. In the purely algebraic setting, it is well known that rings are Morita equivalent if and only if they are equivalent objects in a bicategory whose 1-cells are bimodules. We present a similar result for von Neumann algebras. We show that von Neumann alg ebras form a bicategory, having Connes's correspondences as $1$-morphisms, and (bounded) intertwiners as $2$-morphisms. Further, we prove that two von Neumann algebras are Morita equivalent if and only if they are equivalent objects in the bicategory. The proofs make extensive use of the Tomita-Takesaki modular theory.

None of the above, but in MSC2010 section 46Lxx (msc 46L99), categories, bicategories and generalizations (msc 18D05)
CWI
CWI. Probability, Networks and Algorithms [PNA]
Stochastic Dynamics and Discrete Probability

Brouwer, R.M. (2003). A bicategorical approach to Morita equivalence for Von Neumann algebras. CWI. Probability, Networks and Algorithms [PNA]. CWI.