Francois Schulz. Source: Annals of Functional Analysis, Volume 10, Number 2, 218--228.Abstract:
Let $A$ and $B$ be complex Banach algebras, and let $\phi,\phi_{1}$ , and $\phi_{2}$ be surjective maps from $A$ onto $B$ . Denote by $\partial\sigma(x)$ the boundary of the spectrum of $x$ . If $A$ is semisimple, $B$ has an essential socle, and $\partial\sigma(xy)=\partial\sigma(\phi_{1}(x)\phi_{2}(y))$ for each $x,y\in A$ , then we prove that the maps $x\mapsto\phi_{1}(\mathbf{1})\phi_{2}(x)$ and $x\mapsto\phi_{1}(x)\phi_{2}(\mathbf{1})$ coincide and are continuous Jordan isomorphisms. Moreover, if $A$ is prime with nonzero socle and $\phi_{1}$ and $\phi_{2}$ satisfy the aforementioned condition, then we show once again that the maps $x\mapsto\phi_{1}(\mathbf{1})\phi_{2}(x)$ and $x\mapsto\phi_{1}(x)\phi_{2}(\mathbf{1})$ coincide and are continuous. However, in this case we conclude that the maps are either isomorphisms or anti-isomorphisms. Finally, if $A$ is prime with nonzero socle and $\phi$ is a peripherally multiplicative map, then we prove that $\phi$ is continuous and either $\phi$ or $-\phi$ is an isomorphism or an anti-isomorphism.