Operator Valued Maps on Hilbert $C^*$-Modules

TL;DR

This paper characterizes operator valued completely bounded maps on Hilbert C*-modules using $$-maps and establishes the uniqueness of non-degenerate $$-maps for completely positive maps on C*-algebras, advancing the understanding of operator module theory.

Contribution

It introduces a new characterization of operator valued completely bounded maps via $$-maps and proves the uniqueness of non-degenerate $$-maps for completely positive maps on C*-algebras.

Findings

Characterization of operator valued completely bounded maps using $$-maps.

Existence and uniqueness (up to unitary) of non-degenerate $$-maps for completely positive maps.

Enhanced understanding of the structure of operator maps on Hilbert C*-modules.

Abstract

We provide a characterization for operator valued completely bounded linear maps on Hilbert $C^*$-modules in terms of $\varphi$-maps. Also, we show that for every operator valued completely positive map $\varphi$ on a $C^*$-algebra $\mathcal{A}$, there is a unique (up to multiplication by a unitary operator) non-degenerate $\varphi$-map on each Hilbert $\mathcal{A}$-module.