Representability of positive functionals on abstract star algebras without identity with applications to locally convex ^*algebras

Abstract

In the context of unbounded representation theory, representable functionals on a star algebra are introduced and are characterized as extendable functionals. This gives faithful representations of certain star algebras as Op^*-algebras. Every representation of a pseudo-complete hermitian locally convex ^*algebra is shown to be essentially selfadjoint. Also proved that a Generalized B^*-algebra (without identity) admits a bounded approximate identity which yields the representability of very positive functional.