Lie $^*$-triple homomorphisms into von Neumann algebras
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- by C. Robert Miers
- Proc. Amer. Math. Soc. 58 (1976), 169-172
- DOI: https://doi.org/10.1090/S0002-9939-1976-0410406-9
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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 \[ \phi [[A,B],C] = [[\phi (A),\phi (B)],\phi (C)].\] (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 $[M,M]$.References
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Bibliographic Information
- © Copyright 1976 American Mathematical Society
- 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