Correspondences of von Neumann algebras.

Abstract

In the early 1980's, Alain Connes introduced a morphism between von Neumann algebras in order to define a property formerly defined only for groups and extending Kazhdan's property T. This morphism is called a correspondence. Let ${\cal M}$ and ${\cal N}$ be von Neumann algebras. A correspondence from ${\cal M}$ to ${\cal N}$ is a Hilbert space ${\cal H}$ which is an ${\cal N} - {\cal M}$ bimodule. Equivalently, a correspondence from ${\cal M}$ to ${\cal N}$ is a unital $\sp*$-representation of the algebraic tensor product ${\cal N} \odot\ {\cal M}\sp{o}$ which is normal when restricted to ${\cal N} \odot$ 1 and 1 $\odot\ {\cal M}\sp{o}$. In this thesis, we show that a correspondence is indeed a morphism between von Neumann algebras and we show its link to two other morphisms between von Neumann algebras: normal involutive algebra homomorphisms and completely positive normal linear maps. To achieve this objective, we study some important examples of von Neumann algebras and the Tomita-Takesaki theory.