On Jordan Ideals and Submodules: Algebraic and Analytic Aspects

Let $\mathcal{A}$ be an algebra, and let $X$ be an arbitrary $\mathcal{A}$-bimodule. A linear space $Y\subset X$ is called a Jordan $\mathcal{A}$-submodule if $Ay+yA\in Y$ for all $A\in\mathcal{A}$ and $y\in Y$. (For $X=\mathcal{A}$, this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over $C^*$-algebras. It turns out that there exist $C^*$-algebras in which not all Jordan ideals are ideals.