Representations and positive functionals

Abstract

We discuss the representation theory of both the locally convex and non-locally convex topological*-algebras. First we discuss the*-representation of topological*-algebras by operators on a Hilbert space. Then we study those topological*-algebras so that every*-representation of which on a Hilbert space is necessarily continuous. It is well-known that each*-representation of aB*-algebra on a Hilbert space is continuous. We show that this is true for a large class of*-algebras more general thanB*-algebras, including certain non-locally convex*-algebras. Finally, we investigate the conditions under which a positive functional on a topological*-algebra is representable.