CHAPTER 4 - Von Neumann Algebras

Abstract

This chapter provides an overview on von Neumann algebras. The theory of von Neumann algebras is a vast and very well-developed area of the theory of operator algebras. The chapter presents some of the basics and the main results of the von Neumann double commutant theorem and the Kaplansky density theorem. The algebras studied in this chapter, von Neumann algebras, are a class of C*-algebras whose study can be thought of as noncommutative measure theory. With respect to the strong topology, B (H) is a topological vector space, so the operations of addition and scalar multiplication are strongly continuous. If A is a von Neumann algebra on H and p is a projection in A, then pAp is a von Neumann algebra on H. According to the theorem, if A is a nonzero von Neumann algebra, then it is unital.