\(C^*\)-algebraic characterization of bounded orbit injection equivalence for minimal free Cantor systems
type: Item
instance of: scholarly article
title: \(C^*\)-algebraic characterization of bounded orbit injection equivalence for minimal free Cantor systems
zbMATH Open document ID: 1284.37017
DOI: 10.1216/RMJ-2012-42-1-157
author: Frédéric Latrémolière
author: Nicholas Ormes
published in: Rocky Mountain Journal of Mathematics
publication date: 2012-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0903.1881
full work available at URL: https://projecteuclid.org/euclid.rmjm/1327329317
reviewed by: Alcides Buss
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37A55
Mathematics Subject Classification ID: 37B50
Mathematics Subject Classification ID: 37D05
zbMATH DE Number: 6035431
zbMATH Keywords: minimal Cantor systems
zbMATH Keywords: orbit injection
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: Morita-Rieffel equivalence
MaRDI profile type: Publication
arXiv ID: 0903.1881
OpenAlex ID: W2962755655
review text: 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 particula…
description: scientific article; zbMATH DE number 6035431