Some classes of topological quasi *-algebras

TL;DR

This paper introduces and studies two classes of topological quasi *-algebras, strict CQ*-algebras and HCQ*-algebras, exploring their structure, properties, and relationships, especially their connection to left Hilbert algebras.

Contribution

The paper defines strict CQ*-algebras and HCQ*-algebras, analyzes their properties, and establishes conditions for embedding strict CQ*-algebras into HCQ*-algebras, advancing the theory of topological quasi *-algebras.

Findings

HCQ*-algebras are characterized by containing a left Hilbert algebra with a unit as a dense subspace.

A Hilbert space is a HCQ*-algebra if and only if it contains a left Hilbert algebra with a unit.

A necessary and sufficient condition is provided for embedding a strict CQ*-algebra into a HCQ*-algebra.

Abstract

The completion $\overline{A}[\tau]$ of a locally convex *-algebra $A [ \tau ]$ with not jointly continuous multiplication is a *-vector space with partial multiplication $xy$ defined only for $x$ or $y \in A_{0}$, and it is called a topological quasi *-algebra. In this paper two classes of topological quasi *-algebras called strict CQ$^*$-algebras and HCQ$^*$-algebras are studied. Roughly speaking, a strict CQ$^*$-algebra (resp. HCQ$^*$-algebra) is a Banach (resp. Hilbert) quasi *-algebra containing a C$^*$-algebra endowed with another involution $\sharp$ and C$^*$-norm $\| \|_{\sharp}$. HCQ$^*$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a HCQ$^*$-algebra if and only if it contains a left Hilbert algebra with unit as a dense subspace. Further, we shall give a necessary and sufficient condition under which a strict CQ$^*$-algebra is embedded in a HCQ$^*$-algebra.