A bicategorical approach to Morita equivalence for von Neumann algebras

Abstract

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 iff 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 algebras 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 iff they are equivalent objects in the bicategory. The proofs make extensive use of the Tomita–Takesaki modular theory.