Subfactors and Bimodules
David Evans, Yasuyuki Kawahigashi · 1998
Abstract V. F. R. Jones initiated subfactor theory in (Jones 1983) and it has since revolutionized the theory of von Neumann algebras. Furthermore, the discovery of the Jones polynomial, a link invariant, in (Jones 1985) has revealed a truly surprising relation between the operator algebra theory and low-dimensional topology and strengthened links with mathematical physics. We will deal with some of the progress which has been made in this theory in the last decade. In this chapter, we start with the subfactor theory from today’s viewpoint based on bimodules. A bimodule in our setting is a Hilbert space with left and right actions of two von Neumann algebras. The importance of bimodule theory in von Neumann algebras was first emphasized by A. Cannes in his unpublished manuscript on correspondences, and a detailed study of the bimodule theory was given by S. Popa in his unpublished manuscript ‘Correspondences’. It was A. Ocneanu who first realized the importance of bimodules in the Jones theory of subfactors in (Ocneanu 1988), and he obtained several fundamental theorems in the theory, but unfortunately his results have been scattered among his several informal notes and unpublished manuscripts. We will present the basic theory in this chapter as a preliminary to the remainder of this book.