Continuity of Positive Functionals and Representations

This dissertation is mainly concerned with the study of the continuity of positive functionals on topological *-algebras and of the representations of topological *-algebras by operators on a Hilbert space. In each case, both the locally convex and the non-locally convex topological *-algebras are considered. First we examine the general situation; then we discuss the results for the various known classes of topological *-algebras as special cases of our general considerations. These algebras include bounded algebras, MQ*-algebras, uniformly A-convex algebras and Banach algebras for the locally convex case, and F-algebras and p-normed algebras for the non-locally convex case. Meanwhile we relax the condition for the requirement of an identity element and the condition on the continuity of the involution map in the algebra. In this way we partially generalize some previous results, and the known results in some cases follow from ours. Those that are not particular cases of our results will also be discussed. Doctor of Philosophy (PhD)