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 $\hat{I}¦:A\to B$ satisfies $[\hat{I}¦(x^2),\hat{I}¦(x)]=0$ for all $x\in A$, then there exist a surjective direct sum of an additive homomorphism and an additive anti-homomorphism $\hat{I}¨:A\to B$, an invertible element $\hat{I}»\in \mathcal{Z}(B)$, and an additive mapping $\hat{I}\P :A\to \mathcal{Z}(B)$ such that $\hat{I}¦(x)=\hat{I}»\hat{I}¨(x)+\hat{I}\P (x)$ for all $x\in A$.