Invertibility preserving linear maps on the semi-simple Banach algebras

TL;DR

This paper demonstrates that in semi-simple Banach algebras, invertibility-preserving isomorphisms are necessarily Jordan homomorphisms if the ideal's socle is essential, revealing structural constraints.

Contribution

It establishes a new link between the essentiality of the socle of an ideal and the nature of invertibility-preserving maps in semi-simple Banach algebras.

Findings

Invertibility-preserving isomorphisms are Jordan homomorphisms under certain conditions

Essentiality of the socle influences algebraic structure of maps

Provides conditions for structural characterization of algebra automorphisms

Abstract

In this paper, we show that the essentiality of the scole of an ideal B i a semi-simple Banach algebra A implies that any invertibility preserving isomorphism on A is a Jordan homomorphism. Specially ...