Locally convex quasi C∗-normed algebras

If A0[‖⋅‖0] is a C∗-normed algebra and τ a locally convex topology on A0 making its multiplication separately continuous, then (completion of A0[τ]) is a locally convex quasi ∗-algebra over A0, but it is not necessarily a locally convex quasi ∗-algebra over the C∗-algebra (completion of A0[‖⋅‖0]). In this article, stimulated by physical examples, we introduce the notion of a locally convex quasi C∗-normed algebra, aiming at the investigation of ; in particular, we study its structure, ∗-representation theory and functional calculus.