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arXiv · 1303.1062

Isomorphisms of Lattices of Bures-Closed Bimodules over Cartan MASAs

Adam H. Fuller·David R. Pitts
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Abstract

For i=1,2, let (M_i,D_i) be pairs consisting of a Cartan MASA D_i in a von Neumann algebra M_i, let atom(D_i) be the set of atoms of D_i, and let S_i be the lattice of Bures-closed D_i bimodules in M_i. We show that when M_i have separable preduals, there is a lattice isomorphism between S_1 and S_2 if and only if the sets {(Q_1, Q_2) \in atom(D_i) x atom(D_i): Q_1 M_i Q_2 \neq (0)} have the same cardinality. In particular, when D_i is non-atomic, S_i is isomorphic to the lattice of projections in L^\infty([0,1],m) where m is Lebesgue measure, regardless of the isomorphism classes of M_1 and M_2.

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Adam H. Fuller, David R. Pitts. 2013-03-05. Isomorphisms of Lattices of Bures-Closed Bimodules over Cartan MASAs. https://arxiv.org/abs/1303.1062

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