Classifying crossed product C*-algebras

TL;DR

This paper advances the classification of certain crossed product C*-algebras by their invariants, integrating recent structural and dynamical results, and extends classification results to new classes of actions and spaces.

Contribution

It provides new classification results for crossed product C*-algebras associated with free minimal Z^d-actions and certain minimal homeomorphisms, broadening the scope of Elliott invariant classification.

Findings

Crossed products of free minimal Z^d-actions on the Cantor set are classified by their K-theory.

Classification extends to finite-dimensional compact metrizable spaces under certain conditions.

Crossed products from specific minimal homeomorphisms of odd-dimensional spheres are determined by invariant measures.

Abstract

I combine recent results in the structure theory of nuclear C*-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In particular, transformation group C*-algebras associated to free minimal Z^d-actions on the Cantor set with compact space of ergodic measures are classified by their ordered K-theory. In fact, the respective statement holds for finite dimensional compact metrizable spaces, provided that projections of the crossed products separate tracial states. Moreover, C*-algebras associated to certain minimal homeomorphisms of odd dimensional spheres are only determined by their spaces of invariant Borel probability measures (without a condition on the space of ergodic measures). Finally, I show that for a large collection of classifiable C*-algebras, crossed products by Z^d-actions are generically again classifiable.