## Abstract

Let A and B be complex Banach algebras, and let ϕ ϕ_{1}, and ϕ_{2} be surjective maps from A onto B. Denote by ∂σ(x) the boundary of the spectrum of x. If A is semisimple, B has an essential socle, and ∂σ(xy) = ∂σ(ϕ_{1} (x)ϕ_{2}(y)) for each x, y ∈ A, then we prove that the maps x & map; ϕ_{1}(1)ϕ_{2}(x) and x ↦ ϕ_{1}(x)ϕ_{2}(1) coincide and are continuous Jordan isomorphisms. More- over, if A is prime with nonzero socle and ϕ_{1} and ϕ_{2} satisfy the aforementioned condition, then we show once again that the maps x ↦ϕ_{1}(1)ϕ_{2}(x) and x ↦ ϕ_{1}(x)ϕ_{2}(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 ϕ is a peripherally multiplicative map, then we prove that ϕ is continuous and either ϕ or -ϕ is an isomorphism or an anti-isomorphism.

Original language | English |
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Pages (from-to) | 218-228 |

Number of pages | 11 |

Journal | Annals of Functional Analysis |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 May 2019 |

## Keywords

- Banach algebras
- Nonlinear preservers
- Peripheral spectrum
- Peripherally multiplicative maps
- Spectrum

## ASJC Scopus subject areas

- Analysis
- Anatomy
- Algebra and Number Theory