Uniqueness under Spectral Variation in the Socle of a Banach Algebra

TL;DR

This paper investigates spectral conditions in complex semisimple Banach algebras to uniquely characterize prime algebras and minimal socles, extending spectral theory in algebraic structures.

Contribution

It provides new spectral characterizations of prime Banach algebras and minimal socles within Banach algebras with nonzero socles, based on spectral and spectral radius conditions.

Findings

Spectral conditions characterize prime Banach algebras.

Spectral radius inequalities identify minimal socles.

Results extend spectral theory in Banach algebra structures.

Abstract

Let $A$ be a complex semisimple Banach algebra with identity, and denote by $\sigma'(x)$ and $\rho (x)$ the nonzero spectrum and spectral radius of an element $x \in A$, respectively. We explore the relationship between elements $a, b \in A$ that satisfy one of the following conditions: (1) $\sigma' (ax) \subseteq \sigma' (bx)$ for all $x \in A$, (2) $\rho (ax) \leq \rho (bx)$ for all $x \in A$. The latter problem was identified by Bre\v{s}ar and \v{S}penko in [7]. In particular, we use these conditions to spectrally characterize prime Banach algebras amongst the class of Banach algebras with nonzero socles, as well as to obtain spectral characterizations of socles which are minimal two-sided ideals.