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Isomorphisms Of Lattices Of Bures-Closed Bimodules Over Cartan Masas, Adam H. Fuller, David R. Pitts

Department of Mathematics: Faculty Publications

For i = 1; 2, let (Mi;Di) be pairs consisting of a Cartan MASA Di in a von Neumann algebra Mi, let atom(Di) be the set of atoms of Di, and let Si be the lattice of Bures-closed Di bimodules in Mi. We show that when Mi have separable preduals, there is a lattice isomorphism between S1 and S2 if and only if the sets

f(Q1;Q2) 2 atom(Di) atom(Di) : Q1MiQ2 6= (0)g

have the same cardinality. In particular, when Di is nonatomic, Si is isomorphic to the lattice of projections in L1([0; 1];m) where m is Lebesgue measure, regardless …