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

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.

This version of the article has been accepted for publication, after peer review (when applicable) but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s11117-024-01079-8

References in corpus (4)

• Positive maps and entanglement in real Hilbert spaces
• Linear dynamics of cosine operator functions on solid Banach function spaces
• Disjoint linear dynamical properties of elementary operators
• GNS-construction for positive C*-valued sesquilinear maps on a quasi *-algebra

Cited by in corpus (2)

• Banach bimodule-valued positive maps: Inequalities and induced representations
• Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces