# On the peripheral spectrum of positive elements

## TL;DR

This paper studies the peripheral spectrum of irreducible positive elements in ordered Banach algebras, providing conditions for spectral symmetry and invariance under cyclic group actions.

## Contribution

It offers new conditions determining when the peripheral spectrum forms a cyclic group generated by a root of unity and when the entire spectrum is invariant under this group.

## Key findings

- Peripheral spectrum contains the cyclic group generated by a root of unity under certain conditions.
- The spectrum can be invariant under the action of this cyclic group.
- Conditions are established for the peripheral spectrum to coincide with this cyclic group.

## Abstract

We investigate the peripheral spectrum of irreducible positive elements of an ordered Banach algebra. In particular, we give conditions under which the peripheral spectrum contains (or coincides with) the cyclic group generated by a root of unity, and conditions under which the whole spectrum is invariant under the action of this cyclic group.