The Spectral extension property in the unitization of Banach Algebras

TL;DR

This paper investigates the spectral extension property in the unitization of Banach algebras, establishing conditions under which the property in the unitized algebra implies or is implied by the original algebra.

Contribution

It provides necessary and equivalent conditions relating the spectral extension property of a Banach algebra and its unitization.

Findings

Identifies conditions for the spectral extension property to transfer between a Banach algebra and its unitization.

Shows that if the unitization has SEP, then the original algebra also has SEP.

Explores the converse implications and necessary conditions for SEP in the unitization.

Abstract

Let $A$ be a non-unital Banach algebra and let $A_e = A \oplus {\mathbb C}1$ be the unitization of $A$. It is true that if $A_e$ has the spectral extension property (SEP), then $A$ has the same. Does the converse hold? In this paper, we give some necessary as well as some equivalent conditions.