## Abstract

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a C^{⋆}-algebra onto a Banach algebra. We then use this result to show that a C^{⋆}-algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function ϕ: A→ B satisfying (i) σ(ϕ(x) ϕ(y) ϕ(z)) = σ(xyz) for all x, y, z∈ A (where σ denotes the spectrum), and (ii) ϕ is continuous at 1. In particular, if (in addition to (i) and (ii)) ϕ(1) = 1, then ϕ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Brešar and Špenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

Original language | English |
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Pages (from-to) | 391-398 |

Number of pages | 8 |

Journal | Archiv der Mathematik |

Volume | 113 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Oct 2019 |

## Keywords

- Banach algebra
- C-algebra
- Isomorphism
- Spectrum

## ASJC Scopus subject areas

- General Mathematics

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