Lie and Jordan ideals of operators on Hilbert space

Abstract

A linear manifold L \mathfrak {L} in B ( H ) \mathfrak {B}(\mathfrak {H}) is a Lie ideal in B ( H ) \mathfrak {B}(\mathfrak {H}) if and only if there is an associative ideal J \mathfrak {J} such that [ J , B ( H ) ] ⊆ L ⊆ J + C I [\mathfrak {J},\mathfrak {B}(\mathfrak {H})] \subseteq \mathfrak {L} \subseteq \mathfrak {J} + {\mathbf {C}}I . Also L \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 B ( H ) \mathfrak {B}(\mathfrak {H}) is a Jordan ideal if and only if it is an associative ideal.