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.