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

@article{Brown2015CartanSI,
  title={Cartan Subalgebras in C*-Algebras of Haus dorff {\'e}tale Groupoids},
  author={Jonathan H. Brown and Gabriel Nagy and Sarah Reznikoff and Aidan Sims and Dana P. Williams},
  journal={Integral Equations and Operator Theory},
  year={2015},
  volume={85},
  pages={109-126}
}
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… 

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