## Abstract

Let A and B be complex unital Banach algebras, and let σ(x) and σ^ (x) denote the spectrum of x and its polynomially convex hull, respectively. A spectrally additive map ϕ: A→ B is a surjective function which satisfies σ(x+ y) = σ(ϕ(x) + ϕ(y)) for each x, y∈ A . By establishing a new additive characterization of rank one elements in a Banach algebra, we prove that any spectrally additive map acting on a semisimple domain preserves rank one elements in both directions. This settles an open question raised in [1], which ultimately then classifies spectrally additive maps on a large class of Banach algebras as Jordan-isomorphisms. By refining the techniques in [1] even further, we are able to prove the following more general result: If A is semisimple and either A or B has an essential socle, then any surjective map ϕ: A→ B with the property that σ^ (x+ y) = σ^ (ϕ(x) + ϕ(y)) for each x, y∈ A is a continuous Jordan-isomorphism.

Original language | English |
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Pages (from-to) | 194-208 |

Number of pages | 15 |

Journal | Acta Mathematica Hungarica |

Volume | 170 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 2023 |

## Keywords

- Jordan-isomorphism
- rank
- semisimple Banach algebra
- socle
- spectrum
- spectrum-preserving map

## ASJC Scopus subject areas

- General Mathematics