Characterization of derivations on B(X) by local actions

• Published in: Acta mathematica scientia Vol. 37; no. 3; pp. 668 - 678
• Main Authors: XUE, Tianjiao, AN, Runling, HOU, Jinchuan
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2017; Department of Mathematics,Taiyuan University of Technology,Taiyuan 030024,China
• Subjects: 16W25; 47B49; Derivations; Jordan derivable maps; subspace lattice algebras; triangular algebras; 16W25; 47B49; Derivations; Jordan derivable maps; triangular algebras; subspace lattice algebras
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1016/S0252-9602(17)30029-2

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ: A→M is said to be Jordan derivable at a nontrivial idempotent P∈A if δ(A)∘B+A∘δ(B)=δ(A∘B) for any A,B∈A with A ○ B= P, here A ○ B = AB + BA is the usual Jordan product. In this article, we show that if A=AlgN is a Hilbert space nest algebra and M=B(H), or A=M=B(X), then, a linear map δ : A→M is Jordan derivable at a nontrivial projection P∈N or an arbitrary but fixed nontrivial idempotent P∈B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.