Spectral isometries onto algebras having a separating family of finite-dimensional irreducible representations

TL;DR

This paper proves that spectral radius-preserving linear maps between certain Banach algebras are Jordan morphisms, extending understanding of algebraic structure preservation under spectral conditions.

Contribution

It establishes that spectral radius-preserving maps onto algebras with finite-dimensional irreducible representations are necessarily Jordan morphisms, generalizing previous spectral isometry results.

Findings

Spectral radius-preserving maps are Jordan morphisms

Results apply to algebras with separating families of finite-dimensional irreducible representations

Extends spectral isometry theory to broader algebra classes

Abstract

We prove that if $\mathcal{A}$ is a complex, unital semisimple Banach algebra and $\mathcal{B}$ is a complex, unital Banach algebra having a separating family of finite-dimensional irreducible representations, then any unital linear operator from $\mathcal{A}$ onto $\mathcal{B}$ which preserves the spectral radius is a Jordan morphism.