\(C^*\)-algebraic characterization of bounded orbit injection equivalence for minimal free Cantor systems (Q422013)

A remarkable result by \textit{T. Giordano, I. Putnam} and \textit{C. F. Skau} [J. Reine Angew. Math. 469, 51--111 (1995; Zbl 0834.46053)] asserts that one can classify free minimal Cantor dynamical systems (that is, free minimal actions of \(\mathbb{Z}\) on a Cantor space) in terms of isomorphism classes of their \(C^*\)-crossed products via the concept of ``strong orbit equivalence'', meaning the existence of a homeomorphism of the underlying Cantor sets for the actions that maps orbits to orbits in an almost continuous way (there is at most one point of discontinuity). Moreover, as shown by \textit{G. A. Elliott} [J. Reine Angew. Math. 443, 179--219 (1993; Zbl 0809.46067)], this class of crossed-product \(C^*\)-algebras can also be classified by \(K\)-theory invariants (including the canonical order and distinguished elements).NEWLINENEWLINEOne is tempted to extend these results to free minimal Cantor actions of \(\mathbb{Z}^d\), but the situation is much more complex if \(d>1\). The present article studies some aspects of this problem. More precisely, the authors study bounded injection equivalence, a relation introduced by \textit{S. J. Lightwood} and \textit{N. S. Ormes} in [Ergodic Theory Dyn. Syst. 27, No. 1, 153--182 (2007; Zbl 1152.37007)], and show that this is closely related to a strengthened form of Morita-Rieffel equivalence between the associated crossed-product \(C^*\)-algebras. In this direction, the main result of the paper gives equivalent conditions for the existence of a bounded orbit injection between two free minimal Cantor systems in terms of their crossed-product \(C^*\)-algebras. Roughly speaking, this result asserts that there exists such a bounded orbit injection if and only if one of the crossed-product \(C^*\)-algebras can be embedded into the other via a monomorphism that ``remembers the underlying Cantor spaces''. The image of such a monomorphism is a corner by some projection (the image of the unit by the monomorphism). In particular this implies that the crossed products are Morita-Rieffel equivalent. As a consequence of their results, they prove that bounded orbit equivalence is, indeed, an equivalence relation among free minimal Cantor systems.NEWLINENEWLINEIn a second part of the article, the authors attach to each free minimal Cantor system an ordered group and prove that this group is an invariant for bounded orbit equivalence, that is, if two such systems are bounded orbit equivalent, then their associated ordered groups are isomorphic. The authors use Voronoi tilings to study this group. For \(\mathbb{Z}\)-actions, the Pimsner-Voiculescu sequence implies that this is the ordered \(K_0\)-group, but they are different for general \(\mathbb{Z}^d\)-actions even if \(d=2\).