A note on the positivity of inverse operators acting on $C^*$-algebras

TL;DR

This paper establishes criteria for when the inverse of a positive invertible operator on a $C^*$-algebra remains positive, highlighting conditions and counterexamples related to spectral properties.

Contribution

It provides necessary and sufficient conditions for the positivity of inverse operators on $C^*$-algebras, including a counterexample illustrating limitations.

Findings

Criteria for positivity of inverse operators

Counterexample with spectrum on the unit circle

Inverse positivity does not always follow from unitality

Abstract

For a positive and invertible linear operator $T$ acting on a $C^*$-algebra, we give necessary and sufficient criteria for the inverse operator $T^{-1}$ to be positive, too. Moreover, a simple counterexample shows that $T^{-1}$ need not be positive even if $T$ is unital and its spectrum is contained in the unit circle.