Local Operator Multipliers and Positivity

TL;DR

This paper extends Stinespring's Theorem to unbounded cases, characterizes positive local Schur and operator multipliers, and explores their structure and positivity properties in non-commutative settings.

Contribution

It introduces local operator multipliers as a non-commutative analogue of local Schur multipliers and provides their characterization and positivity criteria.

Findings

Extended Stinespring's Theorem to unbounded maps

Characterized positive local Schur multipliers

Described positive local operator multipliers via approximation

Abstract

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and obtain a characterisation that extends earlier results concerning operator multipliers and local Schur multipliers. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones.