## The nuclear dimension of $$C^\ast$$-algebras associated to homeomorphisms.(English)Zbl 1376.46042

The paper under review studies the nuclear dimension of $$C^\ast$$-algebras associated to groups and dynamical systems. The main result is Theorem 5.1: Given a locally compact Hausdorff space $$X$$ with finite covering dimension and an arbitrary homeomorphism $$\alpha$$ of $$X$$, the crossed product of $$C_0(X)$$ by the automorphism induced by $$\alpha$$ has finite nuclear dimension.
The literature contains many results estimating the nuclear dimension of a crossed product in terms of the Rokhlin dimension of the action (see [I. Hirshberg et al., Comm. Math. Phys. 335, No. 2, 637–670 (2015; Zbl 1333.46055)], [G. Szabó, Proc. Lond. Math. Soc. (3) 110, No. 3, 673–694 (2015; Zbl 1330.54047)]). However, such results are limited to free homeomorphisms. By removing this assumption, the paper provides a substantial generalization.
The authors deduce estimates for the nuclear dimension of certain group $$C^\ast$$-algebras. It follows in particular that the $$C^\ast$$-algebra of the lamplighter group has finite nuclear dimension, although it has infinite decomposition rank.

### MSC:

 46L05 General theory of $$C^*$$-algebras 46L55 Noncommutative dynamical systems

### Keywords:

$$C^\ast$$-algebras; nuclear dimension; $$C^\ast$$-dynamics

### Citations:

Zbl 1333.46055; Zbl 1330.54047
Full Text:

### References:

 [1] Arveson, W., Notes on extensions of $$C^\ast$$-algebras, Duke Math. J., 44, 2, 329-355, (1977) · Zbl 0368.46052 [2] Blackadar, B., Operator algebras and non-commutative geometry, III, (Operator Algebras: Theory of $$C^\ast$$-Algebras and von Neumann Algebras, Encyclopaedia Math. Sci., vol. 122, (2006), Springer-Verlag Berlin) [3] Carrión, J. R., Classification of a class of crossed product $$C^\ast$$-algebras associated with residually finite groups, J. Funct. Anal., 260, 9, 2815-2825, (2011) · Zbl 1220.46042 [4] Carrión, J. R.; Dadarlat, M.; Eckhardt, C., On groups with quasidiagonal $$C^\ast$$-algebras, J. Funct. Anal., 265, 1, 135-152, (2013) · Zbl 1287.46043 [5] Eckhardt, C.; McKenney, P., Finitely generated nilpotent group $$C^\ast$$-algebras have finite nuclear dimension, J. Reine Angew. Math., (2014), in press; preprint [6] Engelking, R., Dimension theory, North-Holland Math. Libr., vol. 19, (1978), North-Holland Publishing Co./PWN—Polish Scientific Publishers Amsterdam-Oxford-New York/Warsaw, translated from the Polish and revised by the author [7] Gutman, Y., Mean dimension and jaworski-type theorems, Proc. Lond. Math. Soc. (3), 111, 4, 831-850, (2015) · Zbl 1352.37017 [8] Gutman, Y., Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory Dynam. Systems, (2015), in press; preprint [9] Hirshberg, I.; Phillips, N. C., Rokhlin dimension: obstructions and permanence properties, Doc. Math., 20, 199-236, (2015) · Zbl 1350.46043 [10] Hirshberg, I.; Winter, W.; Zacharias, J., Rokhlin dimension and $$C^\ast$$-dynamics, Comm. Math. Phys., 335, 2, 637-670, (2015) · Zbl 1333.46055 [11] Kirchberg, E.; Winter, W., Covering dimension and quasidiagonality, Internat. J. Math., 15, 1, 63-85, (2004) · Zbl 1065.46053 [12] Kulesza, J., Zero-dimensional covers of finite-dimensional dynamical systems, Ergodic Theory Dynam. Systems, 15, 5, 939-950, (1995) · Zbl 0882.54034 [13] Lindenstrauss, E., Lowering topological entropy, J. Anal. Math., 67, 231-267, (1995) · Zbl 0849.54031 [14] Pears, A. R., Dimension theory of general spaces, (1975), Cambridge University Press Cambridge, England-New York-Melbourne · Zbl 0312.54001 [15] Szabó, G.; Wu, J.; Zacharias, J., Rokhlin dimension for actions of residually finite groups, (2014), preprint [16] Szabó, G., Rokhlin dimension and topological dynamics, (2015), Westfälische Wilhelms-Universität, Ph.D. dissertation · Zbl 1342.46003 [17] Szabó, G., The rokhlin dimension of topological $$\mathbb{Z}^m$$-actions, Proc. Lond. Math. Soc. (3), 110, 3, 673-694, (2015) · Zbl 1330.54047 [18] Toms, A. S.; Winter, W., Minimal dynamics and K-theoretic rigidity: Elliott’s conjecture, Geom. Funct. Anal., 23, 1, 467-481, (2013) · Zbl 1280.46046 [19] Winter, W., Decomposition rank of subhomogeneous $$C^\ast$$-algebras, Proc. Lond. Math. Soc. (3), 89, 2, 427-456, (2004) · Zbl 1081.46049 [20] Winter, W., Covering dimension for nuclear $$C^\ast$$-algebras. II, Trans. Amer. Math. Soc., 361, 8, 4143-4167, (2009) · Zbl 1178.46070 [21] Winter, W.; Zacharias, J., The nuclear dimension of $$C^\ast$$-algebras, Adv. Math., 224, 2, 461-498, (2010) · Zbl 1201.46056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.