Bilattices and Morita equivalence of masa bimodules

2014/10/01 by Eleftherakis, G. K.
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1410.0194

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 S¹, S² are bilattices which correspond to reflexive masa bimodules U₁, U₂ and f: S¹→ S² is an onto bilattice homomorphism, then: (i) If U₁ is synthetic, then U₂ is synthetic. (ii) If U₂ contains a nonzero compact (or a finite or a rank 1) operator, then U₁ also contains a nonzero compact (or a finite or a rank 1) operator.

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