Abstract
We show that an isomorphism between two reflexive operator algebras on Hilbert space with commutative subspace lattices (CSL) is automatically continuous and induces an isomorphism between the lattices. For such algebras with completely distributive subspace lattices, CDC algebras, the “quasi-spatially” implemented isomorphisms are shown to be exactly those that preserve the rank of all finite-rank operators. We present some examples of nonspatial isomorphisms and discuss some sufficient conditions on the lattices that ensure that all isomorphisms be spatially implemented. The relationship between isomorphisms and derivations of CSL algebras is also investigated.
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