Isomorphisms of lattices of Bures-closed bimodules over Cartan MASAs

Authors

Adam H. Fuller, University of Nebraska - LincolnFollow
David R. Pitts, University of Nebraska - LincolnFollow

Date of this Version

3-1-2013

Citation

2013 Authors

Comments

New York J. Math. 19 (2013) 657-668.

Abstract

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 of the isomorphism classes of M1 and M2.