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.