Von Neumann algebras

Abstract Continuing on from Chapter 11, Chapter 12 furthers the discussion of C*-algebras. This chapter is devoted to a particular class of C*-algebras called von Neumann algebras. The chapter presents the two great foundations of von Neumann algebra theory, which are the double commutant theorem of von Neumann and the density theorem of Kaplansky with respect to the strong operator topology. In addition, it discusses the polar decomposition of operators, W*-algebras, Hilbert–Schmidt operators, and trace-class operators. It then explores the topics of commutative von Neumann algebras, as well as the enveloping von Neumann algebra of a C*-algebra. Finally, the chapter offer exercises to challenge the reader.