A note on the positivity of inverse operators acting on C⋆-algebras

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.