Introduction to von Neumann algebras, Lecture 7 (von Neumann algebras as dual spaces, various topologies)
Until this point in the course, we concentrated on constructions of von Neumann algebras, examples, and properties of von Neumann algebras as algebras. In this lecture we turn to study subtler topological and Banach-space theoretic aspects of von Neumann algebras. We begin by showing that every von Neumann algebra is the Banach-space dual of a Banach space. For this to have any hope of being true, it must be true for the von Neumann algebra ; we therefore look there first.
(The reference for this lecture is mostly Takesaki, Vol. I, Chapters 2 and 3).