## Annals of Functional Analysis

### A note on peripherally multiplicative maps on Banach algebras

Francois Schulz

#### 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.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 218-228.

Dates
Received: 22 April 2018
Accepted: 10 September 2018
First available in Project Euclid: 19 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1552960866

Digital Object Identifier
doi:10.1215/20088752-2018-0025

Mathematical Reviews number (MathSciNet)
MR3941383

Zentralblatt MATH identifier
07083890

#### Citation

Schulz, Francois. A note on peripherally multiplicative maps on Banach algebras. Ann. Funct. Anal. 10 (2019), no. 2, 218--228. doi:10.1215/20088752-2018-0025. https://projecteuclid.org/euclid.afa/1552960866

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