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A Radon-Nikodym theorem for CP-maps on Hilbert pro-C*-modules

Bhumi Amin, Ramesh Golla

TL;DR

This work extends Radon–Nikodým theory to operator-valued CP maps between Hilbert pro-$C^*$-modules by introducing an equivalence relation on CP maps and a corresponding Stinespring-compatibility framework. It defines a preorder on CP maps and proves a Radon–Nikodým-type theorem: whenever $\Psi \preceq \Phi$ there exists a positive contraction $\Delta_\Phi(\Psi) \in \pi_\Phi(E)'$ with $\Psi \sim \Phi_{\sqrt{\Delta_\Phi(\Psi)}}$, yielding a one-to-one correspondence between CP-map equivalence classes and RN derivatives. The results generalize BR’s Stinespring construction for maps between Hilbert pro-$C^*$-modules and connect to prior RN theorems for CP maps on $^*$-algebras, while also providing purity criteria tied to irreducibility of the Stinespring representation. The framework advances structural understanding of CP maps in the pro-$C^*$-algebra setting and offers tools for comparing quantum operations in this generality.

Abstract

We introduce an equivalence relation on the set of all completely positive maps between Hilbert modules over pro-C*-algebras and analyze the Stinespring's construction for equivalent completely positive maps. We then give a preorder relation in the collection of all completely positive maps between Hilbert modules over pro-C*-algebras and obtain a Radon-Nikodym type theorem.

A Radon-Nikodym theorem for CP-maps on Hilbert pro-C*-modules

TL;DR

This work extends Radon–Nikodým theory to operator-valued CP maps between Hilbert pro--modules by introducing an equivalence relation on CP maps and a corresponding Stinespring-compatibility framework. It defines a preorder on CP maps and proves a Radon–Nikodým-type theorem: whenever there exists a positive contraction with , yielding a one-to-one correspondence between CP-map equivalence classes and RN derivatives. The results generalize BR’s Stinespring construction for maps between Hilbert pro--modules and connect to prior RN theorems for CP maps on -algebras, while also providing purity criteria tied to irreducibility of the Stinespring representation. The framework advances structural understanding of CP maps in the pro--algebra setting and offers tools for comparing quantum operations in this generality.

Abstract

We introduce an equivalence relation on the set of all completely positive maps between Hilbert modules over pro-C*-algebras and analyze the Stinespring's construction for equivalent completely positive maps. We then give a preorder relation in the collection of all completely positive maps between Hilbert modules over pro-C*-algebras and obtain a Radon-Nikodym type theorem.
Paper Structure (3 sections, 12 theorems, 76 equations)

This paper contains 3 sections, 12 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results

Key Result

Theorem 2.9

KS Let $\phi:\mathcal{A} \rightarrow \mathcal{B}$ be a continuous completely positive map. Then there exists a Hilbert $\mathcal{B}$-module $X,$ a unital continuous representation $\pi_\phi: \mathcal{A} \rightarrow \mathcal{L}_\mathcal{B}(X),$ and an element $\xi \in X$ such that for all $a \in \mathcal{A}$. Moreover, the set $\chi_\phi = \text{span}\{\pi_\phi(a)(\xi b): a \in \mathcal{A}, b \in

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 32 more