Modules, completely positive maps, and a generalized KSGNS construction

TL;DR

This paper presents a broad generalization of the KSGNS dilation theorem applicable to modules over C*-algebras, utilizing Kolmogorov decompositions, with specific applications under additional assumptions.

Contribution

It introduces a generalized KSGNS construction for right modules over C*-algebras, expanding the theoretical framework beyond Hilbert modules.

Findings

Established a general KSGNS dilation theorem for right modules

Utilized Kolmogorov decompositions for positive-definite kernels

Derived functional analytic applications with added assumptions

Abstract

A very general KSGNS type dilation theorem in the context of right (not necessarily Hilbert) modules over $C^*$-algebras is presented. The proof uses Kolmogorov type decompositions for positive-definite kernels with values in spaces of sesquilinear maps. More specific functional analytic applications are obtained by adding assumptions.