Lie and Jordan Ideals in Reflexive Algebras

Abstract

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace $${\mathfrak{I}}$$ of a CDCSL algebra $${\mathcal{A}}$$ , that $${\mathfrak{I}}$$ is a Lie ideal if and only if $${{A\mathfrak{I}A}}^{-1} \subseteq \mathfrak{I}$$ for all invertibles A in $${\mathcal{A}}$$ , and that $${\mathfrak{I}}$$ is a Jordan ideal if and only if it is an associative ideal.