A bicategorical approach to Morita equivalence for Von Neumann algebras
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.