Jordan Derivations of Reflexive Algebras

Let ${\mathcal L}Let L{\mathcal L} be a subspace lattice on a Banach space X and suppose that ú{L ? L: L_- < X}=X{\vee\{L\in\mathcal L: L_- < X\}=X} or ${\land\{L_- : L \in \mathcal L, L>(0)\}=(0)}${\land\{L_- : L \in \mathcal L, L>(0)\}=(0)} . Then each Jordan derivation from AlgL{\mathcal L} into B(X) is a derivation. This result can apply to completely distributive subspace lattice algebras, J{\mathcal J} -subspace lattice algebras and reflexive algebras with the non-trivial largest or smallest invariant subspace.