  If $\ca_0[|\cdot|_0]$ is a $\cs$-normed algebra and $\tau$ a locally convex
topology on $\ca_0$ making its multiplication separately continuous, then
$\widetilde{\ca_0}[\tau]$ (completion of $\ca_0[\tau]$) is a locally convex
quasi *-algebra over $\ca_0$, but it is not necessarily a locally convex quasi
*-algebra over the $\cs$-algebra $\widetilde{\ca_0}[|\cdot|_0]$ (completion of
$\ca_0[|\cdot|_0]$). In this article, stimulated by physical examples, we
introduce the notion of a locally convex quasi $\cs$-normed algebra, aiming at
the investigation of $\widetilde{\ca_0}[\tau]$; in particular, we study its
structure, *-representation theory and functional calculus.
