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

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.