Maps between Banach algebras preserving the spectrum

A. Bourhim, J. Mashreghi, A. Stepanyan

Abstract: Let $A$ and $B$ be unital semisimple complex Banach algebras, and let $φ_1$ and $φ_2$ be maps from $A$ onto $B$. We show that if the socle of $A$ is an essential ideal of $A$, and $φ_1$ and $φ_2$ satisfy $σ(φ_1(a)φ_2(b))=σ(ab)$, $σ(φ_1(a)φ_2(b))=σ(ab)$ for all $a,b\in A$, then $φ_1φ_2(1)φ_1φ_2(1)$ and $φ_1(1)φ_2φ_1(1)φ_2$ coincide and are Jordan isomorphisms. We also show that a map $φ $ from $A$ onto $B$ satisfies $σ(φ(a)φ(b)φ(a))=σ(aba)$ and $σ(φ(a)φ(b)φ(a))=σ(aba)$ for all $a,b\in A$ if and only if $φ(1)φ(1)$ is a central invertible element of $B$ for which $φ(1)^3=1$ and $φ(1)^2φ $ is a Jordan isomorphism.

Journal: Arch. Math. (2016)

DOI: 10.1007/s00013-016-0960-9