Classifiable C∗-algebras from minimal Z-actions and their orbit-breaking subalgebras.

In this paper we consider the question of what abelian groups can arise as the K-theory of C ∗ -algebras arising from minimal dynamical systems. We completely characterize the K-theory of the crossed product of a space X with finitely generated K-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible K-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their C ∗ -algebras are classified by their Elliott invariants. We also investigate the K-theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups G 0 and G 1 and any Choquet simplex Δ with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated C ∗ -algebra has K-theory given by this pair of groups and tracial state space affinely homeomorphic to Δ . We also improve on the second author's previous results by using our orbit-breaking construction to C ∗ -algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both K 0 and K 1 . These results have important applications to the Elliott classification program for C ∗ -algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C ∗ -algebras associated to étale equivalence relations. [ABSTRACT FROM AUTHOR]