A bicategorical approach to Morita equivalence for rings and von Neumann algebras

Abstract

Rings form a bicategory [Rings], with classes of bimodules as horizontal arrows, and bimodule maps as vertical arrows. The notion of Morita equivalence for rings can be translated in terms of bicategories in the following way. Two rings are Morita equivalent if and only if they are isomorphic objects in the bicategory. We repeat this construction for von Neumann algebras. Von Neumann algebras form a bicategory [W*], with classes of correspondences as horizontal arrows, and intertwiners as vertical arrows. Two von Neumann algebras are Morita equivalent if and only if they are isomorphic objects in the bicategory [W*].