Spectral decomposition of doubly power-bounded elements in Banach algebras

TL;DR

This paper characterizes doubly power-bounded elements with finite spectrum in Banach algebras, providing a spectral decomposition and extending classical theorems to this setting.

Contribution

It introduces a spectral decomposition for such elements and generalizes existing theorems, advancing understanding of their structure in Banach algebras.

Findings

Spectral decomposition for doubly power-bounded elements with finite spectrum

Extension of Gelfand's theorem to these elements

Generalization of Koehler and Rosenthal's theorem in Banach algebras

Abstract

We establish a characterization of doubly power-bounded elements with finite spectrum in Banach algebras. In particular, we present a spectral decomposition for such elements, extending a classical theorem of Gelfand concerning doubly power-bounded elements with singleton spectrum. Furthermore, we generalize a theorem of Koehler and Rosenthal for doubly power-bounded elements to the setting of Banach algebras. In the final section, we are initiating a study to investigate whether the properties of doubly power-bounded elements can offer insight into the commutativity of Banach algebras.