Banach algebra mappings preserving the invertibility of linear pencils

Abstract

Let $A$ and $B$ be complex unital Banach algebras, and let $\varphi, ψ: A \to B$ be surjective mappings. If $A$ is semisimple with an essential socle and $\varphi$ and $ψ$ preserves the invertibility of linear pencils in both directions, that is, for any $x, y \in A$ and $λ\in \mathbb{C}$, $λx+y$ is invertible in $A$ if and only if $λ\varphi(x) + ψ(y)$ is invertible in $B$, then we show that there exists an invertible element $u$ in $B$ and a Jordan isomorphism $J: A \to B$ such that $\varphi(x) = ψ(x) = uJ(x)$ for all $x \in A$.

Theorems & Definitions (22)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4: aupetitmoutontrace
• Proposition 2.5
• proof
• Theorem 3.1