Skip to main content
placeholder image

Cartan subalgebras in C*-algebras of Hausdorff étale groupoids

Journal Article


Download full-text (Open Access)

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.

UOW Authors


  •   Brown, Jonathan H. (external author)
  •   Nagy, Gabriel (external author)
  •   Reznikoff, Sarah (external author)
  •   Sims, Aidan
  •   Williams, Dana P. (external author)

Publication Date


  • 2016

Citation


  • Brown, J. H., Nagy, G., Reznikoff, S., Sims, A. & Williams, D. P. (2016). Cartan subalgebras in C*-algebras of Hausdorff étale groupoids. Integral Equations and Operator Theory, 85 (1), 109-126.

Scopus Eid


  • 2-s2.0-84961820363

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6477&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5449

Has Global Citation Frequency


Number Of Pages


  • 17

Start Page


  • 109

End Page


  • 126

Volume


  • 85

Issue


  • 1

Place Of Publication


  • Switzerland

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.

UOW Authors


  •   Brown, Jonathan H. (external author)
  •   Nagy, Gabriel (external author)
  •   Reznikoff, Sarah (external author)
  •   Sims, Aidan
  •   Williams, Dana P. (external author)

Publication Date


  • 2016

Citation


  • Brown, J. H., Nagy, G., Reznikoff, S., Sims, A. & Williams, D. P. (2016). Cartan subalgebras in C*-algebras of Hausdorff étale groupoids. Integral Equations and Operator Theory, 85 (1), 109-126.

Scopus Eid


  • 2-s2.0-84961820363

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6477&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5449

Has Global Citation Frequency


Number Of Pages


  • 17

Start Page


  • 109

End Page


  • 126

Volume


  • 85

Issue


  • 1

Place Of Publication


  • Switzerland