Algebraic isomorphisms and $\mathcal{J}$-subspace lattices

The class of $\mathcal{J}$-lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice $\mathcal{L}$ on a Banach space $X$ which is also a $\mathcal{J}$-lattice is called a $\mathcal{J}$-subspace lattice, abbreviated JSL. It is demonstrated that every single element of $Alg\mathcal{L}$ has rank at most one. It is also shown that $Alg\mathcal{L}$has the strong finite rank decomposability property. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ 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 $Alg\mathcal{L}_1$ and $Alg\mathcal{L}_2$ preserves rank. Finally we prove that every algebraic isomorphism between $Alg\mathcal{L}_1$ and $Alg\mathcal{L}_2$ is quasi-spatial.