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Banach bimodule-valued positive maps: Inequalities and induced representations

Giorgia Bellomonte, Stefan Ivkovic, Camillo Trapani

TL;DR

This work develops a generalized representation theory for Banach bimodule-valued positive and completely positive sesquilinear maps on normed quasi *-algebras, extending classical results from $C^*$-algebras to a wider Banach-analytic setting. It establishes a GNS-like framework via Cauchy–Schwarz inequalities and Kojima–Stinespring-type dilations for $\mathfrak Y$-valued maps, including a Radon–Nikodym-type theorem and unitary-equivalence results for the resulting representations. The authors provide both abstract structural results and concrete operator-valued examples (e.g., trace-class and dual von Neumann algebra contexts) to illustrate the applicability of the dilations and inequalities. The work broadens the toolkit for operator-algebraic representations in non-C*-algebra environments and has potential implications for noncommutative analysis and quantum information in generalized Banach-module settings.

Abstract

In this paper, we consider representations induced by general positive and completely positive sesquilinear maps with values in ordered Banach bimodules, such as the space of trace-class operators and the spaces of bounded linear operators from a von Neumann algebra into the dual of another von Neumann algebra. Also, we deduce some new inequalities for these maps.

Banach bimodule-valued positive maps: Inequalities and induced representations

TL;DR

This work develops a generalized representation theory for Banach bimodule-valued positive and completely positive sesquilinear maps on normed quasi *-algebras, extending classical results from -algebras to a wider Banach-analytic setting. It establishes a GNS-like framework via Cauchy–Schwarz inequalities and Kojima–Stinespring-type dilations for -valued maps, including a Radon–Nikodym-type theorem and unitary-equivalence results for the resulting representations. The authors provide both abstract structural results and concrete operator-valued examples (e.g., trace-class and dual von Neumann algebra contexts) to illustrate the applicability of the dilations and inequalities. The work broadens the toolkit for operator-algebraic representations in non-C*-algebra environments and has potential implications for noncommutative analysis and quantum information in generalized Banach-module settings.

Abstract

In this paper, we consider representations induced by general positive and completely positive sesquilinear maps with values in ordered Banach bimodules, such as the space of trace-class operators and the spaces of bounded linear operators from a von Neumann algebra into the dual of another von Neumann algebra. Also, we deduce some new inequalities for these maps.

Paper Structure

This paper contains 5 sections, 12 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representations induced by positive maps
  4. Representations induced by completely positive sesquilinear maps
  5. Applications to operator-valued maps

Key Result

Proposition 3.1

Let ${\mathfrak X}$ be a complex vector space. Assume that $\mathfrak Y$, $\mathfrak{Y}_0$, $\mathfrak K$ and $\mathfrak{C}$ are as above. Let $\Phi : {\mathfrak X} \times {\mathfrak X} \to \mathfrak Y$ be a positive sesquilinear map. Let $\Omega, \Upsilon$ be locally compact Hausdorff spaces, ${\ma

Theorems & Definitions (41)

  • Definition 2.1
  • Example 2.2
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • ...and 31 more