Journal of Operator Theory

Volume 69, Issue 1, Winter 2013 pp. 117-133.

Centralizers and Jordan derivations for CSL subalgebras of von Neumann algebras

Authors: Pengtong Li (1), Deguang Han (2), and Wai-shing Tang (3)
Author institution: (1) Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China
(2) Department of Mathematics, University of Central Florida, Orlando, 32816, U.S.A.
(3) Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

Summary: We investigate the centralizers and Jordan derivations for commutative subspace lattice algebras in von Neumann algebras. For any CSL subalgebra $\mathcal A$ of a von Neumann algebra, we prove that a (weak) Jordan centralizer $\Phi$ (i.e $\Phi:\mathcal A\rightarrow\mathcal A$ is an additive mapping satisfying $2\Phi(A^2)=\Phi(A)A+A\Phi(A)$ for all $A\in\mathcal A$) is automatically a centralizer. Similarly, we show that every Jordan derivation of $\mathcal A$ is a derivation. Additionally, we obtain concrete characterizations of centralizers for standard subalgebras of CSL algebras, and a stronger result is also obtained for standard subalgebras of nest algebras.

Keywords: centralizers, Jordan derivations, CSL algebras, von Neumann algebras

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