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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Lie $ \sp*$-triple homomorphisms into von Neumann algebras


Author: C. Robert Miers
Journal: Proc. Amer. Math. Soc. 58 (1976), 169-172
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1976-0410406-9
MathSciNet review: 0410406
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Abstract: Let $ M$ and $ N$ be associative $ \ast $-algebras. A Lie $ \ast $-triple homomorphism of $ M$ into $ N$ is a $ \ast $-linear map $ \phi :M \to N$ such that

$\displaystyle \phi [[A,B],C] = [[\phi (A),\phi (B)],\phi (C)].$

(Here $ M$ and $ N$ are considered as Lie $ \ast $-algebras with $ [X,Y] = XY - YX.)$ In this note we prove that if $ N$ is a von Neumann algebra with no central abelian projections and if $ \phi $ is onto, there exists a central projection $ D$ in $ N$ such that $ D\phi $ is a Lie $ \ast $-homomorphism of $ [M,M]$, and $ (I - D)\phi $ is a Lie $ \ast $-antihomomorphism of $ [M,M]$.

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DOI: https://doi.org/10.1090/S0002-9939-1976-0410406-9
Keywords: Lie $ \ast $-triple homomorphism, von Neumann algebra
Article copyright: © Copyright 1976 American Mathematical Society

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