Locally convex quasi $C^*$-normed algebras

TL;DR

This paper introduces and studies the structure, representation theory, and functional calculus of locally convex quasi C*-normed algebras, motivated by physical examples and extending the theory of C*-algebras.

Contribution

It defines the concept of a locally convex quasi C*-normed algebra and investigates its properties, representation theory, and functional calculus, expanding the framework of C*-algebra theory.

Findings

The structure of locally convex quasi C*-normed algebras is characterized.

Representation theory for these algebras is developed.

Functional calculus applicable to these algebras is established.

Abstract

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.