Abstract
Let L1 and L2 be finite distributive subspace lattices on real or complex Banach spaces. It is shown that every rank-preserving algebraic isomorphism of AlgL1 onto AlgL2 is quasi-spatially induced. If the algebraic isomorphism in question is known only to preserve the rank of rank one operators, then it induces a lattice isomorphism between L1 and L2.
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