Bounded elements in certain topological partial *-algebras

TL;DR

This paper investigates the structure of topological partial *-algebras, emphasizing the role of bounded elements and their impact on operator representations and the interplay of different partial multiplications.

Contribution

It extends the analysis of bounded elements in topological partial *-algebras, linking operator multiplications with invariant positive sesquilinear forms and representation theory.

Findings

Bounded elements called $ ext{M}$-bounded are crucial in the structure.

The link between strong and weak multiplications is clarified.

Representation in terms of partial GC*-algebras is discussed.

Abstract

We continue our study of topological partial *algebras, focusing our attention to the interplay between the various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link between the strong and the weak multiplications, on one hand, and invariant positive sesquilinear (ips) forms, on the other. Then the analysis is extended to abstract topological partial *algebras, emphasizing the crucial role played by appropriate bounded elements, called $\M$-bounded. Finally, some remarks are made concerning representations in terms of the so-called partial GC*-algebras of operators.