ABSTRACT
This chapter shows the structure of type one John Von Neumann algebras. Of particular interest are those whose Hilbert spaces are separable. These play a pivotal role in G. Mackey's method of analyzing representations by considering their restrictions to normal subgroups. In a locally convex topology, a convex set's closure is determined by continuous linear functionals. Since von Neumann algebras are convex and strongly closed, a description of strongly continuous linear functionals would be helpful. The chapter shows that each von Neumann algebra is the dual space of the normed closed Banach space of σ-weakly continuous linear functionals defined on it. An important class of Hilbert bundles occurs when one considers disintegrations of measures. The chapter focuses on continuous linear functionals in the topologies just introduced. Though the topologies are different, the continuous linear functionals have fairly explicit and similar descriptions.
