Additive derivations of certain reflexive algebras

Let L be a T-subspace lattice on a Banach space X, AlgL be the associated reflexive algebra and A be a subalgebra of AlgL containing all finite rank operators in AlgL. If either dimK = infinity or dimK(-)(perpendicular to) = infinity for every K is an element of L with K not equal (0) and K- not equal X, then every additive derivation D from A into AlgL is linear and quasi-spatial, that is, there exists a densely defined, closed linear operator T : Dom(T) subset of X -- \u3e X with its domain Dom(T) invariant under every element of A, such that D(A)x = (TA - AT)x for all A is an element of A and X is an element of Dom(T). This result can apply to those reflexive algebras with atomic Boolean subspace lattices and pentagon subspace lattices, respectively