Von Neumann Algebras and Extensions of Inverse Semigroups

Allan P. Donsig; Adam H. Fuller; David R. Pitts

doi:10.1017/S0013091516000183

Von Neumann Algebras and Extensions of Inverse Semigroups

Part of: Selfadjoint operator algebras Semigroups

Published online by Cambridge University Press: 19 September 2016

Allan P. Donsig ,

Adam H. Fuller and

David R. Pitts

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Allan P. Donsig: Affiliation:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA (adonsig1@math.unl.edu; afuller7@math.unl.edu; dpitts2@math.unl.edu)
Adam H. Fuller: Affiliation:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA (adonsig1@math.unl.edu; afuller7@math.unl.edu; dpitts2@math.unl.edu)
David R. Pitts: Affiliation:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA (adonsig1@math.unl.edu; afuller7@math.unl.edu; dpitts2@math.unl.edu)

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Abstract

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Abstract

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In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.