The subject of this article lies at the intersection between topological dynamics and the Elliott classification program of simple \(C^\ast\)-algebras. Along with the emergence of what one would consider the modern fine structure theory of \(C^\ast\)-algebras today, the transformation group \(C^\ast\)-algebras \(\mathcal C(X)\rtimes G\) associated to topological dynamical systems \(G\curvearrowright X\) have always served as a rich and motivating source of examples. The price to pay for a rich class of examples coming from the crossed product construction, however, is that it tends to be notoriously hard to pin down specific structural properties of the \(C^\ast\)-algebra at the level of the original dynamical system.
Ever since historically fundamental examples of crossed products have been shown to fall into the realm of \(C^\ast\)-algebra classification (see for example [G. A. Elliott and D. E. Evans, Ann. Math. (2) 138, No. 3, 477–501 (1993; Zbl 0847.46034)]), a lot of interest has been going out to investigate under what conditions a dynamical system \(G\curvearrowright X\) gives rise to a classifiable \(C^\ast\)-algebra \(\mathcal C(X)\rtimes G\). This means that it belongs to a natural class of \(C^\ast\)-algebras among which its K-theory and traces capture its isomorphism class. In this context one typically assumes \(X\) to be a compact metrizable space and \(G\) to be a countable discrete amenable group. It is well known that the simplicity of the crossed product is ensured by assuming, for instance, that the action \(G\curvearrowright X\) is free and minimal. A great deal of theory has been built to combat the question what properties of the action ensures that the crossed product is in addition classifiable. Due to the many recent advances in the abstract structure and classification theory of \(C^\ast\)-algebras (see [W. Winter, in: Proceedings of the international congress of mathematicians 2018, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Volume III. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 1801–1823 (2018; Zbl 1462.46070); J. Castillejos et al., Invent. Math. 224, No. 1, 245–290 (2021; Zbl 1467.46055)]), we now know that it is enough to investigate the question when the crossed product \(\mathcal C(X)\rtimes G\) is Jiang-Su stable, i.e., it tensorially absorbs the so-called Jiang-Su algebra \(\mathcal Z\) [X. Jiang and H. Su, Am. J. Math. 121, No. 2, 359–413 (1999; Zbl 0923.46069)]. A sufficient dynamical condition, called almost finiteness, was introduced by D. Kerr for this purpose to great success in [J. Eur. Math. Soc. (JEMS) 22, No. 11, 3697–3745 (2020; Zbl 1465.37010)], which was modeled after an earlier concept by H. Matui for ample groupoids [Proc. Lond. Math. Soc. (3) 104, No. 1, 27–56 (2012; Zbl 1325.19001)]. Its power and versatility in comparison to earlier approaches has since been demonstrated in [D. Kerr and G. Szabó, Commun. Math. Phys. 374, No. 1, 1–31 (2020; Zbl 1446.46040); D. Kerr and P. Naryshkin, “Elementary amenability and almost finiteness”, Preprint, arXiv:2107.05273; P. Naryshkin, “Polynomial growth, comparison, and the small boundary property”, Preprint, arXiv:2108.04670].
In this paper the authors suggest the first systematic approach to study the converse direction of this problem, namely to link \(C^\ast\)-algebraic properties back to the structure of the original dynamical system. Instead of starting with the structure of just the abstract \(C^\ast\)-algebra \(\mathcal C(X)\rtimes G\), it is natural in this context to first try to deduce something interesting from the behavior of the Cartan pair \(\mathcal C(X) \subset \mathcal C(X)\rtimes G\). The authors introduce and study a kind of Cartan version of Cuntz subequivalence between positive functions in \(\mathcal C(X)\), through which it is possible for them to link the resulting comparison property to the earlier known concept of dynamical comparison for the action \(G\curvearrowright X\), which is in turn known to be closely tied to almost finiteness. Furthermore, the authors introduce tracial \(\mathcal Z\)-stability for Cartan pairs, which is a natural strengthening of tracial \(\mathcal Z\)-stability for \(C^\ast\)-algebras and is the actual condition that appears as a consequence of almost finiteness in Kerr’s theorem from [D. Kerr, J. Eur. Math. Soc. (JEMS) 22, No. 11, 3697–3745 (2020; Zbl 1465.37010)]. The main result in the article asserts that if one assumes the small boundary property for \(G\curvearrowright X\), then dynamical comparison and/or almost finiteness is equivalent to tracial \(\mathcal Z\)-stability of the Cartan pair \(\mathcal C(X)\subset\mathcal C(X)\rtimes G\). Since the publication of the article, some convincing preliminary evidence has appeared supporting the suspicion that, in great contrast to the small boundary property, the dynamical comparison property may be a redundant assumption. A natural and interesting open problem arising from this article is therefore whether tracial \(\mathcal Z\)-stability of the Cartan pair \(\mathcal C(X)\subset\mathcal C(X)\rtimes G\) implies the small boundary property for the action \(G\curvearrowright X\).