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

TL;DR

This paper develops a framework for understanding completely positive maps and representations of Hilbert modules over pro-C*-algebras, extending classical concepts to this broader algebraic setting.

Contribution

It introduces a construction for induced representations and a structure theorem for $ ext{phi}$-maps over pro-C*-algebras, generalizing Paschke's GNS construction.

Findings

Established a structure theorem for $ ext{phi}$-maps between Hilbert modules over pro-C*-algebras.

Described the structure of CP-maps between pro-C*-algebras using GNS construction.

Presented a minimality discussion for the constructed representations.

Abstract

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