Jordan Modules and Jordan Ideals of Reflexive Algebras

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Let \({\mathcal{L}}\) be a completely distributive subspace lattice on a Banach space and alg \({\mathcal{L}}\) the associated reflexive algebra. Suppose that the following $$\mbox{Condition A:}\dim(F/F\wedge F_-)\ne1\;\; \mbox{for all}\;\;F\in\mathcal{L}$$ holds; note that if \({\mathcal{L}}\) is an atomic Boolean subspace lattice, this condition means that every atom of \({\mathcal{L}}\) has dimension at least two. It is shown that every reflexive Jordan Alg \({\mathcal{L}}\) -module is an associative Alg \({\mathcal{L}}\) -module. We give an example which shows that if the Condition A is removed, then the conclusion is not necessarily true. Moreover, we prove that all reflexive Jordan ideals of Alg \({\mathcal{L}}\) are associative ideals in the case that no the Condition A is assumed. The same conclusions hold for weakly closed Jordan modules and weakly closed Jordan ideals if the rank one subalgebra of Alg \({\mathcal{L}}\) is weakly dense in Alg \({\mathcal{L}}\) .