Hilbert Modules - Square Roots of Positive Maps

TL;DR

This paper explores the concept that square roots of positive maps in *-algebras can be understood as Hilbert modules, providing insights into positivity notions and their implications in algebraic structures.

Contribution

It offers a comprehensive review and analysis showing that square roots of positive maps are naturally modeled as Hilbert modules, advancing understanding in algebraic positivity.

Findings

Square roots of positive maps are Hilbert modules.

Discussion of positivity requirements in *-algebras.

Derivation of basic consequences of these notions.

Abstract

We reflect on the notions of positivity and square roots. We review many examples which underline our thesis that square roots of positive maps related to *-algebras are Hilbert modules. As a result of our considerations we discuss requirements a notion of positivity on a *-algebra should fulfill and derive some basic consequences.