Reconstruction of groupoids and C*-rigidity of dynamical systems

Toke Meier Carlsen, Efren Ruiz, Aidan Sims, Mark Tomforde

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

We show how to construct a graded locally compact Hausdorff étale groupoid from a C⁎-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact étale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded isotropy is abelian and torsion free, then the extended Weyl groupoid of its reduced C⁎-algebra is isomorphic as a graded groupoid to G. In particular, two such groupoids are isomorphic as graded groupoids if and only if there is an equivariant diagonal-preserving isomorphism of their reduced C⁎-algebras. We introduce graded equivalence of groupoids, and establish that two graded groupoids in which the trivially graded isotropy has torsion-free abelian interior are graded equivalent if and only if there is an equivariant diagonal-preserving Morita equivalence between their reduced C⁎-algebras. We use these results to establish rigidity results for a number of classes of dynamical systems, including all actions of the natural numbers by local homeomorphisms of locally compact Hausdorff spaces.
Original languageEnglish
Article number107923
Number of pages55
JournalAdvances in Mathematics
Volume390
DOIs
Publication statusPublished - 29 Oct 2021

Keywords

  • Cartan subalgebra
  • C*-algebra
  • Groupoid
  • Orbit equivalence

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