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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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© Copyright 1982
American Mathematical Society