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

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.