On representations and topological aspects of positive maps on non-unital quasi *-algebras

TL;DR

This paper develops a representation theory for positive maps on non-unital quasi *-algebras, illustrating with concrete examples and deriving norm inequalities, thus advancing understanding of their structure and topological properties.

Contribution

It introduces a new representation framework for positive maps on non-unital quasi *-algebras and applies it to various concrete non-unital Banach quasi *-algebras.

Findings

Representation of positive sesquilinear maps on non-unital quasi *-algebras

Representation of bounded positive linear maps on non-unital C*-algebras

Norm inequalities for positive maps

Abstract

In this paper, we provide a representation of a certain class of C*-valued positive sesquilinear and linear maps on non-unital quasi *-algebras. Also, we illustrate our results on the concrete examples of non-unital Banach quasi *-algebras, such as the standard Hilbert module over a commutative C*-algebra, Schatten p-ideals, and noncommutative L2 spaces induced by a semifinite, nonfinite trace. As a consequence of our results, we obtain a representation of all bounded positive linear C*-valued maps on non-unital C*-algebras. We also deduce some norm inequalities for these maps. Finally, we consider a noncommutative L2 space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space.