### Abstract

Let A be a complex Banach algebra, let Q_{A} be the set of all quasinilpotent elements in A and let Q_{A}^{r} be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that g(Q_{A}) ⊆ Q_{A}^{r} provided that A has the property β. The class of Banach algebras with the property β is quite large: it includes C*-algebras, group algebras of locally compact groups, commutative Banach algebras, Banach algebras of all bounded linear operators on Banach spaces and so on. Our theorems are natural generalizations of the recent results for derivations obtained by Alaminos et al. [Bull. London Math. Soc. 46:379–384, 2014].

Original language | English |
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Pages (from-to) | 1888-1908 |

Number of pages | 21 |

Journal | Linear and Multilinear Algebra |

Volume | 66 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2018 Sep 2 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

Liu, C. K., & Liau, P. K. (2018). Generalized derivations preserving quasinilpotent elements in Banach algebras.

*Linear and Multilinear Algebra*,*66*(9), 1888-1908. https://doi.org/10.1080/03081087.2017.1376613