Matrix KSGNS construction and a Radon--Nikodym type theorem

TL;DR

This paper extends the theory of completely positive maps to matrices of linear maps on Hilbert modules over locally C*-algebras, establishing Stinespring and Radon--Nikodym theorems in this context.

Contribution

It introduces a new concept of completely positive matrices of linear maps on Hilbert modules and proves fundamental theorems analogous to classical results.

Findings

Stinespring theorem analogue established

Minimal representations are unitarily equivalent

Radon--Nikodym theorem analogue proved

Abstract

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices.