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, thus extending the results from Bellomonte (GNS-construction for positive \(C^*-\)valued sesquilinear maps on a quasi \(*-\)algebra, Mediterr. J. Math., 21 166 (22 pp) (2024)) to the case of 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 \(L^2\)-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 \(L^2\)-space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space.