Quotient-bounded elements in locally convex algebras. II

Abstract

Consideration of quotient-bounded elements in a locally convex GB^*-algebra leads to the study of proper GB ^*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a proper GB ^*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem for C^*-algebra and two other representation theorems for b^*-algebras (also called LMC^*-algebras), one representinga b ^*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutative L _p-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras and L^w-integral of a measurable field of C^*-algebras are discussed briefly.