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Von Neumann Algebras and Extensions of Inverse Semigroups

Part of: Semigroups Selfadjoint operator algebras

Published online by Cambridge University Press:  19 September 2016

Allan P. Donsig ,
Adam H. Fuller  and
David R. Pitts
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

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.

Keywords

von Neumann algebrabimoduleCartan MASA

MSC classification

Secondary: 46L10: General theory of von Neumann algebras 06E75: Generalizations of Boolean algebras 20M18: Inverse semigroups 20M30: Representation of semigroups; actions of semigroups on sets 46L51: Noncommutative measure and integration
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 57 - 97
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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