Dualizability and index of subfactors

In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with finite dimensional centers, the Haagerup L^2 -space and Connes fusion are functorial with respect to homorphisms of finite index. Along the way, we describe a string diagram notation for maps between bimodules that are not necessarily bilinear.