von Neumann algebras

We begin by gathering some preliminary facts about II ₁ factors, which are special kinds of von Neumann algebras. Recall that a von Neumann algebra is a C*-algebra M which admits a pre-dual. i.e., there exists some Banach space, say X, such that M is, as a Banach space, isometrically isomorphic to the Banach dual space X* It is a fact that such a pre-dual is unique up to isometric isomorphism, and any choice of a pre-dual is usually denoted by M _*. It follows that M has a canonical topology, usually called the σ-weak topology, which is the smallest vector space topology with respect to which evaluations at members of M _* are continuous. It is customary to call a linear map between von Neumann algebras normal if it is continuous with respect to the σ-weak topologies on domain and range. The natural morphisms in the category of von Neumann algebras are the normal *-homomorphisms.