Completely positive maps on Hilbert modules over pro-C*-algebras

TL;DR

This paper extends foundational theorems like GNS, Stinespring, and Radon-Nikodym to the setting of Hilbert modules over pro-C*-algebras, broadening the theoretical framework for operator algebras.

Contribution

It develops new versions of GNS, Stinespring, and Radon-Nikodym theorems specifically for Hilbert modules over pro-C*-algebras, which were not previously established.

Findings

Derived Paschke's GNS construction for pro-C*-algebras

Established an analogue of Stinespring theorem in this setting

Obtained a Radon-Nikodym type theorem for operator-valued maps

Abstract

We derive Paschke's GNS construction for completely positive maps on unital pro-C*-algebras from the KSGNS construction, presented by M. Joita [J. London Math. Soc. {\bf 66} (2002), 421--432], and then we deduce an analogue of Stinespring theorem for Hilbert modules over pro-C*-algebras. Also, we obtain a Radon-Nikodym type theorem for operator valued completely positive maps on Hilbert modules over pro-C*-algebras.