Von Neumann Algebra - Bimodules and Subfactors

Bimodules and Subfactors

A bimodule (or correspondence) is a Hilbert space H with module actions of two commuting von Neumann algebras. Bimodules have a much richer structure than that of modules. Any bimodule over two factors always gives a subfactor since one of the factors is always contained in the commutant of the other. There is also a subtle relative tensor product operation due to Connes on bimodules. The theory of subfactors, initiated by Vaughan Jones, reconciles these two seemingly different points of view.

Bimodules are also important for the von Neumann group algebra M of a discrete group . Indeed if V is any unitary representation of, then, regarding as the diagonal subgroup of x, the corresponding induced representation on l2 (,V) is naturally a bimodule for two commuting copies of M. Important representation theoretic properties of can be formulated entirely in terms of bimodules and therefore make sense for the von Neumann algebra itself. For example Connes and Jones gave a definition of an analogue of Kazhdan's Property T for von Neumann algebras in this way.