Reconstruction of groupoids and C*-rigidity of dynamical systems

TL;DR

This paper constructs a new groupoid from C*-algebras with coactions and uses it to establish rigidity results, showing that certain dynamical systems are uniquely determined by their associated C*-algebras.

Contribution

It introduces the extended Weyl groupoid for C*-algebras with coactions and proves its isomorphism with original groupoids under specific conditions, advancing the understanding of C*-rigidity.

Findings

The extended Weyl groupoid generalizes Renault's Weyl groupoid.

Isomorphism between groupoids corresponds to diagonal-preserving C*-algebra isomorphisms.

Rigidity results apply to various dynamical systems, including actions of natural numbers.

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.