Editing Von Neumann algebra (section)

==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 [[Alain Connes|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 Γ. Indeed, if ''V'' is any [[unitary representation]] of Γ, then, regarding Γ as the diagonal subgroup of Γ × Γ, the corresponding [[induced representation]] on ''l''<sup>2 </sup>(Γ, ''V'') is naturally a bimodule for two commuting copies of ''M''. Important [[representation theory|representation theoretic]] properties of Γ 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.