Banach Journal of Mathematical Analysis

Characterizations of Jordan left derivations on some algebras

Guangyu An, Yana Ding, and Jiankui Li

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Abstract

A linear mapping δ from an algebra A into a left A-module M is called a Jordan left derivation if δ(A2)=2Aδ(A) for every AA. We prove that if an algebra A and a left A-module M satisfy one of the following conditions—(1) A is a C-algebra and M is a Banach left A-module; (2) A=AlgL with {L:LJL}=(0) and M=B(X); and (3) A is a commutative subspace lattice algebra of a von Neumann algebra B and M=B(H)—then every Jordan left derivation from A into M is zero. δ is called left derivable at GA if δ(AB)=Aδ(B)+Bδ(A) for each A,BA with AB=G. We show that if A is a factor von Neumann algebra, G is a left separating point of A or a nonzero self-adjoint element in A, and δ is left derivable at G, then δ0.

Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 466-481.

Dates
Received: 1 April 2015
Accepted: 18 August 2015
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1463153911

Digital Object Identifier
doi:10.1215/17358787-3599675

Mathematical Reviews number (MathSciNet)
MR3504180

Zentralblatt MATH identifier
1351.47027

Subjects
Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 47L35: Nest algebras, CSL algebras 47C15: Operators in $C^*$- or von Neumann algebras

Keywords
$C^{*}$-algebra Jordan left derivation left derivable point left separating point

Citation

An, Guangyu; Ding, Yana; Li, Jiankui. Characterizations of Jordan left derivations on some algebras. Banach J. Math. Anal. 10 (2016), no. 3, 466--481. doi:10.1215/17358787-3599675. https://projecteuclid.org/euclid.bjma/1463153911


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