Citation: Yamagami, Shigeru. "Vector Bundles and Bimodules." Quantum and Non-Commutative Analysis: Past, Present and Future Perspectives (1993): 321-329.
Web: https://link.springer.com/content/pdf/10.1007/978-94-017-2823-2_25.pdf
Tags: Mathematical, Subfactors
In this lovely little paper, Shigeru Yamagami gives a construction of subfactors based on the action of a group on a factor, and highlights some categorical aspects of this construction. This present paper is a shortened of the longer article
> Kosaki, Hideki, and Shigeru Yamagami. "Irreducible bimodules associated with crossed product algebras." International Journal of Mathematics 3.05 (1992): 661-676.
In the first part of this paper, Yamagami remarks on a simple point which I had never considered before. Suppose that H is a Hilbert space with the structure of an A-B bimodule given two factors A,B. That is, an action of A and B^* on H such that the actions of A and B commute. The action of B induces a subspace B' of H, given by the commutant of H. Thus, the bimodule induces an inclusion of factors. Conversely, given a subfactor A -> B, taking any representation of B will induce an A-B' bimodule. This allows one to go back and forth between subfactors and bimodules. In the present situation, it is nicer to work with bimodules because their categorical structure is plainer to see.
The authors then introduce the following notation. They fix a big group Gamma, and consider finite subgroups of Gamma. They define a H-K bundle to be a vector bundle over Gamma with a left and right action of H and K and such that the actions are compatible with the regular left/right actions of H and K on Gamma. Taking the fiber-wise relative tensor product, the space of H-K bundles with H,K varying naturally forms a 2-category.
The main construction is now stated as follows. Let N be a factor, and let Gamma -> Aut(N) be an action of Gamma on N. The authors construct a functorial fully-faithful function from the 2-category of bimodules over finite subgroups of Gamma, and the 2-category of bimodules over N. This construction sends an H-K bundle to an H'-K' bundle where H' is a von Neumann algebra obtained by taking a twisted product of H with N and K' is obtained by taking a twisted product of K with N. This twisting is given by the action of Gamma on N.