Representations and positive functionals

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.The research of the first-named author was partially supported by an NSERC grant. This work was done by the second-named author when he was a post-doctoral fellow at McMaster University.