Classifiable $\mathrm{C}^*$-algebras from minimal $\mathbb{Z}$-actions and their orbit-breaking subalgebras

Abstract

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 G0 and G1 and any Choquet simplex ∆ 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 ∆. 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 K0 and K1. 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.