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Lie and Jordan ideals of operators on Hilbert space


Authors: C. K. Fong, C. R. Miers and A. R. Sourour
Journal: Proc. Amer. Math. Soc. 84 (1982), 516-520
MSC: Primary 47D25
DOI: https://doi.org/10.1090/S0002-9939-1982-0643740-0
MathSciNet review: 643740
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Abstract: A linear manifold $ \mathfrak{L}$ in $ \mathfrak{B}(\mathfrak{H})$ is a Lie ideal in $ \mathfrak{B}(\mathfrak{H})$ if and only if there is an associative ideal $ \mathfrak{J}$ such that $ [\mathfrak{J},\mathfrak{B}(\mathfrak{H})] \subseteq \mathfrak{L} \subseteq \mathfrak{J} + {\mathbf{C}}I$. Also $ \mathfrak{L}$ is a Lie ideal if and only if it contains the unitary orbit of every operator in it. On the other hand, a subset of $ \mathfrak{B}(\mathfrak{H})$ is a Jordan ideal if and only if it is an associative ideal.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0643740-0
Article copyright: © Copyright 1982 American Mathematical Society

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