REPRESENTABILITY OF POSITIVE FUNCTIONALS ON ABSTRACT STAR ALGEBRAS WITHOUT IDENTITY WITH APPLICATIONS TO LOCALLY CONVEX *ALGEBRAS

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