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

Abstract

We introduce S-modules, generalizing the notion of Krein $C^{\ast }$-modules, where a fixed unitary replaces the symmetry of Krein $C^{\ast }$-modules. The representation theory on S-modules is explored and for a given $\ast$-automorphism $\alpha$ on a $C^{\ast }$-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.

17 pages