### 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 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Linear and Multilinear Algebra*,

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

}

*Linear and Multilinear Algebra*, vol. 66, no. 9, pp. 1888-1908. https://doi.org/10.1080/03081087.2017.1376613

**Generalized derivations preserving quasinilpotent elements in Banach algebras.** / Liu, Cheng-Kai; Liau, Pao Kuei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized derivations preserving quasinilpotent elements in Banach algebras

AU - Liu, Cheng-Kai

AU - Liau, Pao Kuei

PY - 2018/9/2

Y1 - 2018/9/2

N2 - Let A be a complex Banach algebra, let QA be the set of all quasinilpotent elements in A and let QA r be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that g(QA) ⊆ QA 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].

AB - Let A be a complex Banach algebra, let QA be the set of all quasinilpotent elements in A and let QA r be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that g(QA) ⊆ QA 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].

UR - http://www.scopus.com/inward/record.url?scp=85029719617&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029719617&partnerID=8YFLogxK

U2 - 10.1080/03081087.2017.1376613

DO - 10.1080/03081087.2017.1376613

M3 - Article

VL - 66

SP - 1888

EP - 1908

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 9

ER -