Isomorphisms of Lattices of Bures-Closed Bimodules over Cartan MASAs

TL;DR

This paper characterizes when the lattices of Bures-closed bimodules over Cartan MASAs in von Neumann algebras are isomorphic, linking it to the cardinality of certain atom pairs, especially in non-atomic cases.

Contribution

It provides a complete characterization of lattice isomorphisms of Bures-closed bimodule lattices over Cartan MASAs based on atom pair cardinalities.

Findings

Lattice isomorphism corresponds to equal cardinalities of atom pair sets.

Non-atomic D_i lead to isomorphic bimodule lattices to projections in L^ ^ ext{infty}([0,1],m).

Characterization holds for von Neumann algebras with separable preduals.

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.