Commutativity preserving mappings in Banach algebras

Abstract

Let $A$ and $B$ be unital complex Banach algebras having no quotients isomorphic to $\mathbb{C}$ or $M_2(\mathbb{C})$. Assume additionally that $B$ is semisimple. If a surjective additive mapping $\Phi\colon A\to B$ satisfies $[\Phi(x^2),\Phi(x)] = 0$ for all $x\in A$, then there exist a surjective direct sum of an additive homomorphism and an additive anti-homomorphism $\Psi\colon A\to B$, an invertible element $\lambda\in\mathcal{Z}(B)$, and an additive mapping $\zeta\colon A\to\mathcal{Z}(B)$ such that $\Phi(x)=\lambda\Psi(x)+\zeta(x)$ for all $x\in A$.