KSGNS type construction for $\alpha $-completely positive maps on Krein $C^{\ast}$-modules

TL;DR

This paper extends the KSGNS construction to $\

Contribution

It introduces a KSGNS type theorem for $\

Findings

Established a KSGNS theorem for $\

Demonstrated the uniqueness of minimal constructions

Developed a covariant version of the theorem

Abstract

In this paper, we investigate $\Phi$-maps associated to a certain type of $\alpha$-completely positive maps. We then prove a KSGNS (Kasparov--Stinespring--Gel'fand--Naimark--Segal) type theorem for $\alpha $-completely positive maps on Krein $C^*$-modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant $\alpha $-completely positive map and study the structure of minimal covariant KSGNS constructions.