If $\ca_0[|\cdot|_0]$ is a $\cs$-normed algebra and $\tau$ a locally convextopology on $\ca_0$ making its multiplication separately continuous, then$\widetilde{\ca_0}[\tau]$ (completion of $\ca_0[\tau]$) is a locally convexquasi *-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, weintroduce the notion of a locally convex quasi $\cs$-normed algebra, aiming atthe investigation of $\widetilde{\ca_0}[\tau]$; in particular, we study itsstructure, *-representation theory and functional calculus.
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