Abstract
We study isomorphisms mapping one CSL algebra to another. An isomorphism is spatial if it is implemented by a similarity transformation given by a bounded invertible operator. An isomorphism is quasi-spatial if the implementing operator is densely defined and one-to-one, but not necessarily bounded. In an earlier paper, Frank Gilfeather and the current author showed that if £ 1 and £ 2 are commutative and £ 1 is completely distributive, and if there is a sequence E n of projections in Alg£ 1 such that E n → I and (E n ) - ¬= I, then every isomorphism from Alg £ 1 onto Alg £ 2 is quasi-spatial. In this paper, we prove a theorem with the same conclusion, but without the hypothesis of complete distributivity. Along the way, we introduce the notion of a quilting. An isomorphism which is not implemented spatially or quasi-spatially may still have a quilting, which is a way of locally implementing the isomorphism on subspaces in the lattice.
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