Abstract

I combine recent results in the structure theory of nuclear ${\rm C}^*$-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In particular, transformation group ${\rm C}^*$-algebras associated to free minimal ${\Bbb Z}^{d}$-actions on the Cantor set with compact space of ergodic measures are classified by their ordered ${\rm K}$-theory. In fact, the respective statement holds for finite dimensional compact metrizable spaces, provided that projections of the crossed products separate tracial states. Moreover, ${\rm C}^*$-algebras associated to certain minimal homeomorphisms of spheres $S^{2n+1}$ 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 ${\rm C}^*$-algebras, crossed products by ${\Bbb Z}^{d}$-actions are generically again classifiable.

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