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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Lie triple derivations of von Neumann algebras


Author: C. Robert Miers
Journal: Proc. Amer. Math. Soc. 71 (1978), 57-61
MSC: Primary 46L10; Secondary 17B65
MathSciNet review: 0487480
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Abstract: A Lie triple derivation of an associative algebra M is a linear map $ L:M \to M$ such that

$\displaystyle L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ]$

for all $ X,Y,Z \in M$. (Here $ [X,Y] = XY - YX$ and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator $ A \in M$ such that $ L(X) = [A,X] + \lambda (X)$ where $ \lambda :M \to {Z_M}$ is a linear map which annihilates brackets of operators in M.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0487480-9
PII: S 0002-9939(1978)0487480-9
Keywords: Lie triple derivation, von Neumann algebra
Article copyright: © Copyright 1978 American Mathematical Society