Fully representable and *-semisimple topological partial *-algebras

TL;DR

This paper investigates the structure of topological *-algebras that are *-semisimple, emphasizing the roles of *-representations and positive forms, and characterizes bounded elements within this framework.

Contribution

It introduces the concept of fully representable partial *-algebras and explores their properties, including the characterization of bounded elements using order relations.

Findings

Invariant positive sesquilinear forms and representable functionals are interchangeable in fully representable algebras.

Characterization of bounded elements via positive cones and order relations.

Extension of previous notions of boundedness to the setting of *-semisimple partial *-algebras.

Abstract

We continue our study of topological partial *-algebras, focusing our attention to *-semisimple partial *-algebras, that is, those that possess a {multiplication core} and sufficiently many *-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous linear functionals and focus on the case where the two notions are completely interchangeable (fully representable partial *-algebras) with the scope of characterizing a *-semisimple partial *-algebra. Finally we describe various notions of bounded elements in such a partial *-algebra, in particular, those defined in terms of a positive cone (order bounded elements). The outcome is that, for an appropriate order relation, one recovers the $\M$-bounded elements introduced in previous works.