Reconstruction of groupoids and C*-rigidity of dynamical systems

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.

v2: 45 pages; removed notion of "weakly Cartan subalgebra" as it is equivalent to a sigma-unital abelian subalgebra (see Lemma 4.1) and updated numerous statements accordingly. Thanks to R. Meyer and B. Kwasniewski for pointing this out. Numerous other minor corrections and improvements. This version to appear in Advances in Mathematics

References in corpus (3)

• Étale groupoids and their $C^{\ast }$-algebras
• Renault's Equivalence Theorem for Reduced Groupoid C*-algebras
• C*-algebras, groupoids and covers of shift spaces

Cited by in corpus (12)

• Graded C*-algebras and twisted groupoid C*-algebras
• An Algebraic Approach to the Weyl Groupoid
• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• C*-algebras, groupoids and covers of shift spaces
• Refined moves for structure-preserving isomorphism of graph C*-algebras
• Topological full groups of ultragraph groupoids as an isomorphism invariant
• Groupoid algebras as covariance algebras
• Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras
• A note on the core of Steinberg algebras
• Orbit equivalence of higher-rank graphs
• Continuous orbit equivalence for automorphismhi systems of equivalence relations
• $C^{\ast }$-rigidity of dynamical systems and étale groupoids