Introduction

The theory of algebra of operators on Hilbert space began in the 1930s with a series of papers by von Neumann, and Murray and von Neumann. The principal motivations of these authors were the theory of unitary group representations and certain aspects of the quantum mechanical formalism. They analyzed in great detail the structure of a family of algebras which are referred to nowadays as von Neumann algebras, or W*-algebras. These algebras have the distinctive property of being closed in the weak operator topology and it was not until 1943 that Gelfand and Naimark characterized and partially analyzed uniformly closed operator algebras, the so-called C*-algebras. Despite Murray and von Neumann’s announced motivations the theory of operator algebras had no significant application to group representations for more than fifteen years and its relevance to quantum mechanical theory was not fully appreciated for more than twenty years. Despite this lapse there has been a subsequent fruitful period of interplay between mathematics and physics which has instigated both interesting structural analysis of operator algebras and significant physical applications, notably to quantum statistical mechanics. We intend to describe this theory and these applications. Although these results have also stimulated further important applications of algebraic theory to group representations and relativistic field theory we will only consider these aspects peripherally.