Hermitian spectral theory, automatic continuity and locally convex ^*-algebras with a C^*-enveloping algebra

Abstract

A considerable number of non-normed topological ^*-algebras admit a C^*-enveloping algebra. Various characterizations of such algebras are given in terms of diverse properties of decisive features of a given topological ^*-algebra, like the sets of its continuous ^*-representations, its continuous positive functionals, the hermitian spectrum, etc. Interconnections of these sorts of algebras with hermiticity and C^*-spectrality are also discussed. An analogy in structure, between (not necessarily involutive) commutative Q-algebras and topological ^*-algebras having a C^*-enveloping algebra, leads to the study of some automatic continuity results. Furthermore, different examples are presented concerning the question: When does a dense Fréchet ^*-subalgebra of a C^*-algebra have a C^*-enveloping algebra? Some questions and conjectures are stated that arise from the whole study.