Operator Algebras

Abstract

The theory of operator algebras was initiated by J. von Neumann in the early 1930s, in large part out of a desire to find a rigorous mathematical foundation for the newly emerging field of quantum physics. In very simple terms, the work of von Neumann was concerned with linear transformations, which we shall here call operators, on complex inner-product spaces (usually infinite-dimensional, and usually complete with respect to a natural metric). A related concept is that of a C*-algebra, introduced in 1943 by I. Gelfand and M.A. Neumark, which refers to algebras that are defined abstractly rather than as algebras of linear transformations. The term operator algebra generally applies to any algebra of operators that acts on an inner-product space and is closed under the canonical involution, or to any (possibly abstract) C*-algebra.