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Invariant ergodic measures and the classification of crossed product $$C^{\ast}$$-algebras. (English) Zbl 1464.37010
The Toms-Winter conjecture predicts that three regularity conditions of very different nature coincide for the class of unital, separable, simple, nuclear, non-elementary $$C^\ast$$-algebras:
(1) Finite nuclear dimension, namely a noncommutative analog of covering dimension;
(2) Z-stability, namely the tensorial absorption of the Jiang-Su algebra Z;
(3) Strict comparison, namely a comparison property of the Cuntz semigroup of the $$C^\ast$$-algebra.
By now, it is known that (1) and (2) are equivalent [W. Winter, Invent. Math. 187, No. 2, 259–342 (2012; Zbl 1280.46041); J. Castillejos et al., Invent. Math. 224, No. 1, 245–290 (2021; Zbl 1467.46055)], that (2) implies (3), see [M. Rørdam, Int. J. Math. 15, No. 10, 1065–1084 (2004; Zbl 1077.46054)], and that (3) implies (2) in many particular cases, see for example [E. Kirchberg and M. Rørdam, J. Reine Angew. Math. 695, 175–214 (2014; Zbl 1307.46046); A. S. Toms et al., Int. Math. Res. Not. 2015, No. 10, 2702–2727 (2015; Zbl 1335.46054); H. Thiel, Commun. Math. Phys. 377, No. 1, 37–76 (2020; Zbl 1453.46055)].
In the context of dynamical systems, analogues of the three above conditions have been introduced and studied in [D. Kerr, J. Eur. Math. Soc. (JEMS) 22, No. 11, 3697–3745 (2020; Zbl 1465.37010)]. They are:
(1) Finite tower dimension;
(2) Almost finiteness;
(3) Dynamical comparison.
The main result of the paper is that almost finiteness and dynamical comparison are equivalent if the space of invariant ergodic probability measures of the dynamical system is compact and zero-dimensional (see Theorem 1.2 and Corollary 3.6 in the paper).

MSC:
 37A55 Dynamical systems and the theory of $$C^*$$-algebras 46L35 Classifications of $$C^*$$-algebras 46L45 Decomposition theory for $$C^*$$-algebras 46L87 Noncommutative differential geometry 46L55 Noncommutative dynamical systems
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