Isomorphisms of some reflexive algebras

Suppose L-1 and L-2 are subspace lattices on complex separable Banach spaces X and Y, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from alg L-1 to alg L-2 is quasi-spatial; in particular, if a subspace lattice L of a complex separable Banach space X contains a sequence E-i such that (E-i)_ not equal X, Ei subset of Ei+1, and V-i=1(infinity) E-i = X then every automorphism of alg L is quasi-spatial.