# Operator system structures and extensions of Schur multipliers

## TL;DR

This paper explores the structures of operator systems related to C*-algebras, introduces operator-valued Schur multipliers, and characterizes positive extension problems within this framework.

## Contribution

It establishes maximal and minimal operator $ ext{A}$-system structures, characterizes dual operator systems, and analyzes positive extensions of operator-valued Schur multipliers.

## Key findings

- Existence of maximal and minimal operator $ ext{A}$-system structures.
- Characterization of dual operator $ ext{A}$-systems.
- Conditions for positive extension of operator-valued Schur multipliers.

## Abstract

For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal and minimal operator $\mathcal{A}$-system structures on an AOU $\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an abstract characterisation of dual operator $\mathcal{A}$-systems, and study the maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU $\mathcal{A}$-space. We introduce operator-valued Schur multipliers, and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $\varphi$ and, under some richness conditions, characterise its affirmative solution in terms of the equality between the canonical and the maximal dual operator $\mathcal{A}$-system structures on an operator system naturally associated with the domain of $\varphi$.