A note on peripherally multiplicative maps on Banach algebras

Abstract

Let $A$ and $B$ be complex Banach algebras, and let $\varphi ,{\varphi }_1$, and ${\varphi }_2$ 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 $\partial \sigma (xy)=\partial \sigma ({\varphi }_1(x){\varphi }_2(y))$ for each $x,y\in A$, then we prove that the maps $x\mapsto {\varphi }_1(\mathbf{1}){\varphi }_2(x)$ and $x\mapsto {\varphi }_1(x){\varphi }_2(\mathbf{1})$ coincide and are continuous Jordan isomorphisms. Moreover, if $A$ is prime with nonzero socle and ${\varphi }_1$ and ${\varphi }_2$ satisfy the aforementioned condition, then we show once again that the maps $x\mapsto {\varphi }_1(\mathbf{1}){\varphi }_2(x)$ and $x\mapsto {\varphi }_1(x){\varphi }_2(\mathbf{1})$ 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 $\varphi$ is a peripherally multiplicative map, then we prove that $\varphi$ is continuous and either $\varphi$ or $-\varphi$ is an isomorphism or an anti-isomorphism.