Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(2): 277-283

Published online March 31, 2022 https://doi.org/10.4134/BKMS.b200772

Copyright © The Korean Mathematical Society.

Characterizations of Jordan derivable mappings at the unit element

Jiankui Li, Shan Li, Kaijia Luo

East China University of Science and Technology; East China University of Science and Technology; East China University of Science and Technology

Abstract

Let $\mathcal{A}$ be a unital Banach algebra, $\mathcal{M}$ a unital $\mathcal{A}$-bimodule, and $\delta$ a linear mapping from $\mathcal{A}$ into $\mathcal{M}$. We prove that if $\delta$ satisfies $\delta(A)A^{-1}+A^{-1}\delta(A)+A\delta(A^{-1})+\delta(A^{-1})A=0$ for every invertible element $A$ in $\mathcal{A}$, then $\delta$ is a Jordan derivation. Moreover, we show that $\delta$ is a Jordan derivable mapping at the unit element if and only if $\delta$ is a Jordan derivation. As an application, we answer the question posed in \cite[Problem 2.6]{E}.

Keywords: Derivation, Jordan derivation, Triple derivation

MSC numbers: Primary 47B47, 47L35, 47B49

Supported by: This research was partly supported by the National Natural Science Foundation of China (Grant No.11871021).