Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids
- 127 Downloads
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.
KeywordsC*-algebra Groupoid Maximal abelian subalgebra Cartan subalgebra
Mathematics Subject Classification46L05 (primary)
Unable to display preview. Download preview PDF.
- 3.Blackadar, B.: Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122. Theory of C*-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin (2006)Google Scholar
- 8.Dixmier, J: C*-algebras, North-Holland Publishing Co., Amsterdam-New York- Oxford (1977) (Translated from the French by Francis Jellett; North-Holland Mathe- matical Library, Vol. 15.)Google Scholar
- 19.Renault J.: A groupoid approach to C*-algebras, Lecture Notes in Mathematics. vol. 793. Springer-Verlag, New York (1980)Google Scholar
- 21.Renault, J: Topological amenability is a Borel property (2013). (arXiv:1302.0636 [math.OA]).
- 23.Sims, A., Williams, D.P.: The primitive ideals of some étale groupoid C*- algebras. Algebr. Represent. Theory 18, 1–20 (2015) (arXiv:1501.02302 [math.OA])
- 25.Williams D.P.: Crossed products of C*-algebras, Mathematical Surveys and Mono- graphs. vol. 134. American Mathematical Society, Providence (2007)Google Scholar
- 26.Yang, D.: Periodic higher rank graphs revisited (2014) (arXiv:1403.6848 [math.OA])
- 27.Yang, D.: Cycline subalgebras are Cartan (2014)Google Scholar