Characterization theorems in von Neumann algebras

Abstract

The aim of this thesis is to study the characterization theorems in von Neumann algebras. This class of operator algebras was defined for the first time in 1930 by J von Neumann in terms of a representation on a Hilbert space. After the studies of Gelfand, Naimark and Segal, von Neumann algebras were defined as *-subalgebras of bounded operators on a Hilbert space which are weak operator closed. Von Neumann himself was intrigued by the question how to characterize van Neumann algebras in a more abstract, hence representation- independent way. By studying the features of von Neumann algebras, Kadison and Sakai almost simultaneously solved this problem in the mid-fifties. Chapter one contains important results on projections and operators that are needed to prove the characterization theorems later. The well-know spectral theory and a few important facts on Borel calculus are also stated here. By using a theorem of Baire we extend the Gelfand-Naimark *- isomorphism to a *- homomorphism between all the bounded complex Borel functions on the spectrum of an operator T and the von Neumann algebra generated by T and I.