Invariant ergodic measures and the classification of crossed product \(C^{\ast}\)-algebras. (English) Zbl 1464.37010

Ma, Xin

J. Funct. Anal. 276, No. 4, 1276-1293 (2019).

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).

Reviewer: Hannes Thiel (Münster)

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

Keywords:

dynamical comparison; almost finiteness; small boundary property; classification of \(C^\ast\)-algebras

Citations:

Zbl 1280.46041; Zbl 1077.46054; Zbl 1307.46046; Zbl 1335.46054; Zbl 1453.46055; Zbl 1467.46055; Zbl 1465.37010

References:

[1]	Bosa, J.; Brown, N.; Sato, Y.; Tikuisis, A.; White, S.; Winter, W., Covering dimension of \(C^\ast \)-algebras and 2-coloured classification, Mem. Amer. Math. Soc. (2018), in press, https://arxiv.org/abs/1506.03974
[2]	Buck, J., Smallness and comparison properties for minimal dynamical systems
[3]	Cuntz, J.; Pederson, G. K., Equivalence and traces on \(C^\ast \)-algebras, J. Funct. Anal., 33, 2, 135-164 (1979) · Zbl 0427.46042
[4]	Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra, 38, 29-44 (1976) · Zbl 0323.46063
[5]	Elliott, G. A., On the classification of \(C^\ast \)-algebras of real rank zero, J. Reine Angew. Math., 443, 179-219 (1993) · Zbl 0809.46067
[6]	Elliott, G. A.; Evans, D. E., The structure of the irrational rotation algebra, Ann. Math. (2), 138, 477-501 (1993) · Zbl 0847.46034
[7]	Elliott, G. A.; Niu, Z., The \(C^\ast \)-algebra of a minimal homeomorphism of zero mean dimension, Duke Math. J., 166, 3569-3594 (2017) · Zbl 1410.46046
[8]	Elliott, G. A.; Gong, G.; Lin, H.; Niu, Z., On the classification of simple amenable \(C^\ast \)-algebras with finite decomposition rank, II
[9]	Giol, J.; Kerr, D., Subshifts and perforation, J. Reine Angew. Math., 639, 107-119 (2010) · Zbl 1201.46055
[10]	Gong, G.; Lin, H.; Niu, Z., Classification of finite simple amenable \(Z\)-stable \(C^\ast \)-algebras
[11]	Guentner, E.; Willett, R.; Yu, G., Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and \(C^\ast \)-algebras, Math. Ann., 367, 785-829 (2017) · Zbl 1380.37018
[12]	Gutman, Y.; Lindenstrauss, E.; Tsukamoto, M., Mean dimension of \(Z^k\)-actions, Geom. Funct. Anal., 26, 778-817 (2016) · Zbl 1378.37056
[13]	Hirshberg, I.; Winter, W.; Zacharias, J., Rokhlin dimension and C*-dynamics, Comm. Math. Phys., 335, 637-670 (2015) · Zbl 1333.46055
[14]	Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.04002
[15]	Kerr, D., Dimension, comparison, and almost finiteness · Zbl 1465.37010
[16]	Kerr, D.; Li, H., Ergodic Theory: Independence and Dichotomies (2016), Springer · Zbl 1396.37001
[17]	Kirchberg, E.; Rørdam, M., Central sequence \(C^\ast \)-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math., 695, 175-214 (2014) · Zbl 1307.46046
[18]	Lin, H., Simple nuclear \(C^\ast \)-algebras of tracial topological rank one, J. Funct. Anal., 251, 2, 601-679 (2007) · Zbl 1206.46052
[19]	Lin, H.; Phillips, N. C., Crossed products by minimal homeomorphisms, J. Reine Angew. Math., 641, 95-122 (2010) · Zbl 1196.46047
[20]	Lindenstrauss, E., Mean dimension, small entropy factors, and an embedding theorem, Publ. Math. IHES, 89, 227-262 (1999) · Zbl 0978.54027
[21]	Lindenstrauss, E.; Weiss, B., Mean topological dimension, Israel J. Math., 115, 1-24 (2000) · Zbl 0978.54026
[22]	Matui, H., Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc. (3), 104, 27-56 (2012) · Zbl 1325.19001
[23]	Matui, H.; Sato, Y., Strict comparison and \(Z\)-absorption of nuclear \(C^\ast \)-algebras, Acta Math., 209, 179-196 (2012) · Zbl 1277.46028
[24]	Matui, H.; Sato, Y., Decomposition rank of UHF-absorbing \(C^\ast \)-algebras, Duke Math. J., 163, 14, 2687-2708 (2014) · Zbl 1317.46041
[25]	Phillips, N. C., Crossed products of the Cantor set by free minimal actions of \(Z^d\), Comm. Math. Phys., 256, 1-42 (2005) · Zbl 1084.46056
[26]	Phillips, N. C., An introduction to crossed product \(C^\ast \)-algebras and minimal dynamics (2017)
[28]	Rørdam, M., Structure and classification of \(C^\ast \)-algebras, (Proceedings of the International Congress of Mathematicians. Proceedings of the International Congress of Mathematicians, Madrid 2006, vol. II (2006), EMS Publishing House: EMS Publishing House Zurich), 1581-1598 · Zbl 1104.46033
[29]	Sato, Y., Trace spaces of simple nuclear \(C^\ast \)-algebras with finite-dimensional extreme boundary
[30]	Sato, Y.; White, S.; Winter, W., Nuclear dimension and \(Z\)-stability, Invent. Math., 202, 893-921 (2015) · Zbl 1350.46040
[31]	Shub, M.; Weiss, B., Can one always lower topological entropy?, Ergodic Theory Dynam. Systems, 11, 535-546 (1991) · Zbl 0773.54011
[32]	Suzuki, Y., Almost finiteness for general étale groupoids and its applications to stable rank of crossed products · Zbl 1495.46044
[34]	Szabó, G.; Wu, J.; Zacharias, J., Rokhlin dimension for actions of residually finite groups, Ergodic Theory Dynam. Systems, 1-57 (2017)
[35]	Tikuisis, A.; White, S.; Winter, W., Quasidiagonality of nuclear \(C^\ast \)-algebras, Ann. of Math. (2), 185, 229-284 (2017) · Zbl 1367.46044
[36]	Toms, A. S.; White, S.; Winter, W., \(Z\)-stability and finite-dimensional tracial boundaries, Int. Math. Res. Not., 10, 2702-2727 (2015) · Zbl 1335.46054
[37]	Tu, J.-L., La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory, 17, 215-264 (1999) · Zbl 0939.19001
[38]	Winter, W., Decomposition rank and \(Z\)-stability, Invent. Math., 179, 229-301 (2010) · Zbl 1194.46104
[39]	Winter, W., Nuclear dimension and \(Z\)-stability of pure \(C^\ast \)-algebras, Invent. Math., 187, 259-342 (2012) · Zbl 1280.46041
[40]	Winter, W., Structure of nuclear C*-algebras: from quasidiagonality to classification, and back again · Zbl 1462.46070
[41]	Winter, W.; Zacharias, J., The nuclear dimension of \(C^\ast \)-algebras, Adv. Math., 224, 461-498 (2010) · Zbl 1201.46056