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Integral Equations and Operator Theory

, Volume 85, Issue 1, pp 109–126 | Cite as

Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

  • Jonathan H. Brown
  • Gabriel Nagy
  • Sarah Reznikoff
  • Aidan Sims
  • Dana P. Williams
Article

Abstract

The reduced C*-algebra of the interior of the isotropy in any Hausdorff étale groupoid G embeds as a C*-subalgebra M of the reduced C*-algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C*-algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C*-algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, M is a Cartan subalgebra. We prove that for a large class of groupoids G with abelian isotropy—including all Deaconu–Renault groupoids associated to discrete abelian groups—M is a maximal abelian subalgebra. In the specific case of k-graph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.

Keywords

C*-algebra Groupoid Maximal abelian subalgebra Cartan subalgebra 

Mathematics Subject Classification

46L05 (primary) 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Jonathan H. Brown
    • 1
  • Gabriel Nagy
    • 2
  • Sarah Reznikoff
    • 2
  • Aidan Sims
    • 3
  • Dana P. Williams
    • 4
  1. 1.Department of MathematicsUniversity of DaytonDaytonUSA
  2. 2.Department of MathematicsKansas State UniversityManhattanUSA
  3. 3.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  4. 4.Department of MathematicsDartmouth CollegeHanoverUSA

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