KSGNS construction for $\tau$-maps on S-modules and $\mathfrak{K}$-families

TL;DR

This paper generalizes Krein $C^*$-modules to S-modules with a fixed unitary, develops a KSGNS construction for $ au$-maps, and provides a decomposition theorem for $rak K$-families, advancing the representation theory of these structures.

Contribution

It introduces S-modules as a generalization of Krein $C^*$-modules and extends the KSGNS construction to $ au$-maps and $rak K$-families, broadening the theoretical framework.

Findings

Established the representation theory for S-modules.

Proved the KSGNS construction for $ au$-maps.

Decomposed $rak K$-families within this framework.

Abstract

We introduce S-modules, generalizing the notion of Krein $C^*$-modules, where a fixed unitary replaces the symmetry of Krein $C^*$-modules. The representation theory on S-modules is explored and for a given $*$-automorphism $\alpha$ on a $C^*$-algebra the KSGNS construction for $\alpha$-completely positive maps is proved. An extention of this theorem for $\tau$-maps is also achieved, when $\tau$ is an $\alpha$-completely positive map, along with a decomposition theorem for $\mathfrak K$-families.