Structure of locally convex quasi $C^{\ast }$-algebras

Abstract

The completion of a (normed) $C^{\ast }$-algebra $A_0[\Vert \cdot {\Vert }_0]$ with respect to a locally convex topology $\tau$ on $A_0$ that makes the multiplication of $A_0$ separately continuous is, in general, a quasi *-algebra, and not a locally convex *-algebra. In this way, one is led to consideration of locally convex quasi $C^{\ast }$-algebras, which generalize $C^{\ast }$-algebras in the context of quasi *-algebras. Examples are given and the structure of these relatives of $C^{\ast }$-algebras is investigated.