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Classifying crossed product C*-algebras
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 3, June 2016
- pp. 793-820
- 10.1353/ajm.2016.0029
- Article
- Additional Information
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.