Bilattices and Morita equivalence of masa bimodules

TL;DR

This paper introduces spatial Morita equivalence for masa bimodules, showing it corresponds to isomorphic bilattices, and explores how properties like synthesis and compact operators are preserved under bilattice homomorphisms.

Contribution

It establishes a new equivalence relation for masa bimodules and links it to bilattice isomorphisms, providing insights into their structural properties.

Findings

Spatial Morita equivalence iff bilattices are isomorphic

Synthesis property is preserved under bilattice homomorphisms

Presence of compact or finite rank operators is preserved

Abstract

We define an equivalence relation between bimodules over maximal abelian selfadjoint algebras (masa bimodules) which we call spatial Morita equivalence. We prove that two reflexive masa bimodules are spatially Morita equivalent iff their (essential) bilattices are isomorphic. We also prove that if S^1, S^2 are bilattices which correspond to reflexive masa bimodules U_1, U_2 and f: S^1\rightarrow S^2 is an onto bilattice homomorphism, then: (i) If U_1 is synthetic, then U_2 is synthetic. (ii) If U_2 contains a nonzero compact (or a finite or a rank 1) operator, then U_1 also contains a nonzero compact (or a finite or a rank 1) operator.