Generalized derivations preserving quasinilpotent elements in Banach algebras

Abstract

Let A be a complex Banach algebra, let be the set of all quasinilpotent elements in A and let be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that provided that A has the property . The class of Banach algebras with the property is quite large: it includes -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].

Keywords:

• Banach algebra
• radical
• spectrum
• quasinilpotent
• generalized derivation

AMS Subject Classifications:

• 47B47
• 47B48
• 46H15

Acknowledgements

The authors are thankful to the referee for the very thorough reading of the paper and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the MOST of Taiwan, R.O.C. [MOST 104-2115-M-018-001].