A note on Huijsmans-de Pagter problem in ordered Banach algebras

TL;DR

This paper provides a specific example in ordered Banach algebras demonstrating that a positive element can have a spectrum of {1} without being greater than or equal to the algebra's unit element.

Contribution

It presents a counterexample in ordered Banach algebras addressing the Huijsmans-de Pagter problem, highlighting a nuanced spectral property.

Findings

Positive element with spectrum {1} not ≥ unit element

Counterexample to a spectral ordering conjecture

Clarifies limitations of spectral orderings in Banach algebras

Abstract

We give an example of a positive element $a$ in some ordered Banach algebra $A$ such that its spectrum is equal to $\{1\}$ and it is not greater than or equal to the unit element of $A$.