Locally convex quasi C*-normed algebras

If A_0 is a C∗-normed algebra and τ a locally convex topology on A_0 making its multiplication separately continuous, then A~_0[τ ] (completion of A_0[τ ]) is a locally convex quasi ∗-algebra over A_0, but it is not necessarily a locally convex quasi ∗-algebra over the C∗-algebra A~0 (completion of A_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 A~0[τ ]; in particular, we study its structure, ∗-representation theory and functional calculus.