# Representable and continuous functionals on Banach quasi *-algebras

## TL;DR

This paper investigates the continuity of representable linear functionals on Banach and Hilbert quasi *-algebras, establishing key properties and correspondences relevant to their structure and representation theory.

## Contribution

It provides new insights into the continuity and full representability of functionals on Banach and Hilbert quasi *-algebras, including a bijective correspondence in the Hilbert case.

## Key findings

- Continuity of representable functionals is characterized in Banach and Hilbert quasi *-algebras.
- Hilbert quasi *-algebras are shown to be fully representable.
- A one-to-one correspondence between positive, bounded elements and continuous representable functionals is established.

## Abstract

In the study of locally convex quasi *-algebras an important role is played by representable linear functionals; i.e., functionals which allow a GNS-construction. This paper is mainly devoted to the study of the continuity of representable functionals in Banach and Hilbert quasi *-algebras. Some other concepts related to representable functionals (full-representability, *-semisimplicity, etc) are revisited in these special cases. In particular, in the case of Hilbert quasi *-algebras, which are shown to be fully representable, the existence of a 1-1 correspondence between positive, bounded elements (defined in an appropriate way) and continuous representable functionals is proved.