Completely Positive Maps: Pro-C*-algebras and Hilbert modules over Pro-C*-algebras

Abstract

In this paper, we begin by presenting a construction for induced representations of Hilbert modules over pro-$C^{\ast }$-algebras for a given continuous ${}^{\ast }$-morphism between pro-$C^{\ast }$-algebras. Subsequently, we describe the structure of completely positive maps between two pro-$C^{\ast }$-algebras using Paschke's GNS construction for CP-maps on pro-$C^{\ast }$-algebras. Furthermore, through our construction, we establish a structure theorem for a $\ddot{I}•-$map between two Hilbert modules over pro-$C^{\ast }$-algebras, where $\ddot{I}•$ is a continuous CP-map between pro-$C^{\ast }$-algebras. We also discuss the minimality of these representations.

A computation gap in line 13 of page 13 of the earlier version is corrected. For this we corrected the definition of v in Lemma 3.8