Algebraic isomorphisms and J-subspace lattices

The class of J-lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice L on a Banach space X which is also a J-lattice is called a J-subspace lattice, abbreviated JSL. It is demonstrated that every single element of AlgL has rank at most one. It is also shown that AlgL has the strong finite rank decomposability property. Let L-1 and L-2 be subspace lattices that are also JSL's on the Banach spaces X-1 and X-2, respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between AlgL(1) and AlgL(2) preserves rank. Finally we prove that every algebraic isomorphism between AlgL(1) and AlgL(2) is quasi-spatial.