Reconstruction of groupoids and C*-rigidity of dynamical systems
Abstract
We show how to construct a graded locally compact Hausdorff \'etale 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 \'etale 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.
Cite
@article{arxiv.1711.01052,
title = {Reconstruction of groupoids and C*-rigidity of dynamical systems},
author = {Toke Meier Carlsen and Efren Ruiz and Aidan Sims and Mark Tomforde},
journal= {arXiv preprint arXiv:1711.01052},
year = {2022}
}
Comments
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