Structure of locally convex quasi C * -algebras

Abstract

The completion of a (normed) $C^{*}$-algebra $A_0[\parallel \parallel ·{\parallel \parallel }_0]$ 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 [10], [15]. In this way, one is led to consideration of locally convex quasi $C^{*}$-algebras, which generalize $C^{*}$-algebras in the context of quasi $*$-algebras. Examples are given and the structure of these relatives of $C^{*}$-algebras is investigated.