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Rokhlin dimension for actions of residually finite groups

Published online by Cambridge University Press:  28 November 2017

GÁBOR SZABÓ ,
JIANCHAO WU  and
JOACHIM ZACHARIAS
GÁBOR SZABÓ
Affiliation:
Department of Mathematical Sciences, Universitetsparken 5, 2100 København Ø, Denmark email [email protected]
JIANCHAO WU
Affiliation:
Department of Mathematics, Pennsylvania State University, 109 McAllister Building, University Park, PA 16802, USA email [email protected]
JOACHIM ZACHARIAS
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow Q12 8QW, UK email [email protected]
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Abstract

We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 8 , August 2019 , pp. 2248 - 2304
Copyright
© Cambridge University Press, 2017 

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References

Abbaspour, H. and Moskowitz, M.. Basic Lie Theory. World Scientific, Hackensack, NJ, 2007.Google Scholar
Barlak, S., Enders, D., Matui, H., Szabó, G. and Winter, W.. The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras. J. Noncommut. Geom. 9(4) (2015), 1383–1393.Google Scholar
Bartels, A.. Coarse flow spaces for relatively hyperbolic groups. Compositio Math. 153(4) (2017), 745–779.Google Scholar
Bartels, A., Lück, W. and Reich, H.. Equivariant covers for hyperbolic groups. Geom. Topol. 12(3) (2008), 1799–1882.Google Scholar
Bosa, J., Brown, N. P., Sato, Y., Tikuisis, A., White, S. and Winter, W.. Covering dimension of $\text{C}^{\ast }$ -algebras and 2-coloured classification. Preprint, 2016, arXiv:1506.03974. Mem. Amer. Math. Soc., to appear.Google Scholar
Bratteli, O., Kishimoto, A. and Robinson, D. W.. Rohlin flows on the Cuntz algebra 𝓞 ∞ . J. Funct. Anal. 248 (2007), 472–511.Google Scholar
Brown, N., Tikuisis, A. and Zelenberg, A.. Rohklin dimension for $\text{C}^{\ast }$ -correspondences. Preprint, 2016,arXiv:1608.03214, Houston J. Math., to appear.Google Scholar
Busby, R. C. and Smith, H. A.. Representations of twisted group algebras. Trans. Amer. Math. Soc. 149 (1970), 503–537.Google Scholar
Connes, A.. Outer conjugacy classes of automorphisms of factors. Ann. Sci. Éc. Norm. Supér. (4) 4(8) (1975), 383–419.Google Scholar
Connes, A.. Periodic automorphisms of the hyperfinite factors of type II1 . Acta Sci. Math. 39 (1977), 39–66.Google Scholar
Delabie, T. and Tointon, M.. The asymptotic dimension of box spaces of virtually nilpotent groups. Preprint, 2017, arXiv:1706.03730.Google Scholar
Drutu, C. and Kapovich, M.. Lectures on Geometric Group Theory. Colloquium Publications. American Mathematical Society, 2015, https://www.math.ucdavis.edu/∼kapovich/EPR/kapovich_drutu.pdf, to appear.Google Scholar
Elliott, G. A., Gong, G., Lin, H. and Niu, Z.. On the classification of simple $\text{C}^{\ast }$ -algebras with finite decomposition rank, II. Preprint, 2015, arXiv:1507.03437.Google Scholar
Elliott, G. A. and Niu, Z.. On the classification of simple unital C∗ -algebras with finite decomposition rank. Operator Algebras and their Applications: A Tribute to Richard V. Kadison (Contemporary Mathematics, 671) . American Mathematical Society, Providence, RI, 2016.Google Scholar
Finn-Sell, M. and Wu, J.. The asymptotic dimension of box spaces for elementary amenable groups. Preprint, 2015, arXiv:1508.05018v1.Google Scholar
Gardella, E.. Compact group actions with the rokhlin property and their crossed products. Preprint, 2014, arXiv:1408.1946. J. Noncommut. Geom., to appear.Google Scholar
Gardella, E.. Regularity properties and Rokhlin dimension for compact group actions. Houston J. Math. 43(3) (2017), 861–889.Google Scholar
Gardella, E.. Rokhlin dimension for compact group actions. Indiana Univ. Math. J. 66 (2017), 659–703.Google Scholar
Gong, G., Lin, H. and Niu, Z.. Classification of simple amenable ${\mathcal{Z}}$ -stable $\text{C}^{\ast }$ -algebras. Preprint, 2015,arXiv:1501.00135.Google Scholar
Guentner, E., Willett, R. and Yu, G.. Dynamic asymptotic dimension and controlled operator $K$ -theory. Preprint, 2016, arXiv:1609.02093.Google Scholar
Guentner, E., Willett, R. and Yu, G.. Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and C∗ -algebras. Math. Ann. 367 (2017), 785–829.Google Scholar
Gutman, Y.. Mean dimension & Jaworski-type theorems. Proc. Lond. Math. Soc. 111(4) (2015), 831–850.Google Scholar
Herman, R. and Jones, V.. Period two automorphisms of UHF C∗ -algebras. J. Funct. Anal. 45(2) (1982), 169–176.Google Scholar
Herman, R. and Ocneanu, A.. Stability for integer actions on UHF C∗ -algebras. J. Funct. Anal. 59 (1984), 132–144.Google Scholar
Hirshberg, I. and Phillips, N. C.. Rokhlin dimension: obstructions and permanence properties. Doc. Math. 20 (2015), 199–236.Google Scholar
Hirshberg, I., Szabó, G., Winter, W. and Wu, J.. Rokhlin dimension for flows. Comm. Math. Phys. 353(1) (2017), 253–316.Google Scholar
Hirshberg, I. and Winter, W.. Rokhlin actions and self-absorbing C∗ -algebras. Pacific J. Math. 233(1) (2007), 125–143.Google Scholar
Hirshberg, I., Winter, W. and Zacharias, J.. Rokhlin dimension and C∗ -dynamics. Comm. Math. Phys. 335 (2015), 637–670.Google Scholar
Hirshberg, I. and Wu, J.. The nuclear dimension of C∗ -algebras associated to homeomorphisms. Adv. Math. 304 (2017), 56–89. With an appendix by G. Szabó.Google Scholar
Izumi, M.. Finite group actions on C∗ -algebras with the Rohlin property I. Duke Math. J. 122(2) (2004), 233–280.Google Scholar
Izumi, M.. Finite group actions on C∗ -algebras with the Rohlin property II. Adv. Math. 184(1) (2004), 119–160.Google Scholar
Izumi, M. and Matui, H.. ℤ2 -actions on Kirchberg algebras. Adv. Math. 224 (2010), 355–400.Google Scholar
Jennings, S. A.. The group ring of a class of infinite nilpotent groups. Canad. J. Math. 7 (1955), 169–187.Google Scholar
Kawamura, S. and Tomiyama, J.. Properties of topological dynamical systems and corresponding C∗ -algebras. Tokyo J. Math. 13(2) (1990), 251–257.Google Scholar
Khukhro, A.. Box spaces, group extensions and coarse embeddings into Hilbert space. J. Funct. Anal. 263(1) (2012), 115–128.Google Scholar
Kirchberg, E.. Central sequences in C∗ -algebras and strongly purely infinite algebras. Oper. Algebras: The Abel Symposium 1 (2004), 175–231.Google Scholar
Kirchberg, E. and Rørdam, M.. Central sequence C∗ -algebras and tensorial absorption of the Jiang–Su algebra. J. Reine Angew. Math. 695 (2014), 175–214.Google Scholar
Kishimoto, A.. The Rohlin property for automorphisms of UHF algebras. J. Reine Angew. Math. 465 (1995), 183–196.Google Scholar
Kishimoto, A.. A Rohlin property for one-parameter automorphism groups. Comm. Math. Phys. 179(3) (1996), 599–622.Google Scholar
Kishimoto, A.. The Rohlin property for shifts on UHF algebras and automorphisms of Cuntz algebras. J. Funct. Anal. 140 (1996), 100–123.Google Scholar
Kishimoto, A.. Automorphisms of AT algebras with the Rohlin property. J. Operator Theory 40 (1998), 277–294.Google Scholar
Kishimoto, A.. Unbounded derivations in AT algebras. J. Funct. Anal. 160 (1998), 270–311.Google Scholar
Kishimoto, A.. Rohlin flows on the Cuntz algebra 𝓞2 . Internat. J. Math. 13(10) (2002), 1065–1094.Google Scholar
Liao, H.-C.. Rokhlin dimension of Z m -actions on simple C∗ -algebras. Int. J. Math. 28(7) (2017).Google Scholar
Liao, H.-C.. A Rokhlin type theorem for simple C∗ -algebras of finite nuclear dimension. J. Funct. Anal. 270(10) (2016), 3675–3708.Google Scholar
Lindenstrauss, E.. Lowering topological entropy. J. Anal. Math. 67 (1995), 231–267.Google Scholar
Loring, T. A.. Lifting Solutions to Perturbing Problems in C∗ -Algebras (Fields Institute Monographs, 8) . American Mathematical Society, Providence, RI, 1997.Google Scholar
Matui, H. and Sato, Y.. 𝓩-stability of crossed products by strongly outer actions. Comm. Math. Phys. 314(1) (2012), 193–228.Google Scholar
Matui, H. and Sato, Y.. Strict comparison and 𝓩-absorption of nuclear C∗ -algebras. Acta Math. 209(1) (2012), 179–196.Google Scholar
Matui, H. and Sato, Y.. 𝓩-stability of crossed products by strongly outer actions II. Amer. J. Math. 136 (2014), 1441–1497.Google Scholar
Matui, H. and Sato, Y.. Decomposition rank of UHF-absorbing C∗ -algebras. Duke Math. J. 163(14) (2014), 2687–2708.Google Scholar
Nakamura, H.. The Rohlin property for ℤ2 -actions on UHF algebras. J. Math. Soc. Japan 51(3) (1999), 583–612.Google Scholar
Nowak, P. W. and Yu, G.. Large Scale Geometry. EMS, Zurich, 2012.Google Scholar
Orfanos, S.. Generalized Bunce–Deddens algebras. Proc. Amer. Math. Soc. 138 (2010), 299–308.Google Scholar
Osaka, H. and Phillips, N. C.. Crossed products by finite group actions with the Rokhlin property. Math. Z. 270(1–2) (2012), 19–42.Google Scholar
Packer, J. A. and Raeburn, I.. Twisted crossed products of C∗ -algebras. Math. Proc. Cambridge Philos. Soc. 106(2) (1989), 293–311.Google Scholar
Passman, D. S.. The Algebraic Structure of Group Rings (Pure and Applied Mathematics) . Wiley-Interscience (John Wiley & Sons), New York, 1977.Google Scholar
Phillips, N. C.. The tracial Rokhlin property for actions of finite groups on C∗ -algebras. Amer. J. Math. 133(3) (2011), 581–636.Google Scholar
Phillips, N. C.. The tracial Rokhlin property is generic. Preprint, 2012, arXiv:1209.3859.Google Scholar
Robert, L. and Tikuisis, A.. Nuclear dimension and 𝓩-stability for non-simple C∗ -algebras. Trans. Amer. Math. Soc. 369 (2017), 4631–4670.Google Scholar
Roe, J.. Lectures on Coarse Geometry (University Lecture Series, 31) . American Mathematical Society, Providence, RI, 2003.Google Scholar
Rørdam, M. and Stormer, E.. Classification of nuclear C∗ -algebras. Entropy in Operator Algebras (Encyclopaedia of Mathematical Sciences, 126) . Springer, Berlin, 2001.Google Scholar
Rotman, J. J.. An Introduction to the Theory of Groups (Graduate Texts in Mathematics, 148) . 4th edn. Springer, New York, 1995.Google Scholar
Santiago, L.. Crossed products by actions of finite groups with the Rokhlin property. Internat. J. Math. 26 (2015), 31 pages.Google Scholar
Sato, Y.. The Rohlin property for automorphisms of the Jiang–Su algebra. J. Funct. Anal. 259(2) (2010), 453–476.Google Scholar
Sato, Y.. Trace spaces of simple nuclear $\text{C}^{\ast }$ -algebras with finite-dimensional extreme boundary. Preprint, 2013, arXiv:1209.3000.Google Scholar
Sato, Y., White, S. and Winter, W.. Nuclear dimension and 𝓩-stability. Invent. Math. 202 (2015), 893–921.Google Scholar
Sutherland, W. A.. Introduction to Metric and Topological Spaces. Clarendon Press, Oxford, 1975.Google Scholar
Swan, R. G.. Representations of polycyclic groups. Proc. Amer. Math. Soc. 18 (1967), 573–574.Google Scholar
Szabó, G.. Rokhlin dimension and topological dynamics. PhD Thesis, WWU Münster, 2015.Google Scholar
Szabó, G.. The Rokhlin dimension of topological ℤ m -actions. Proc. Lond. Math. Soc. (3) 110(3) (2015), 673–694.Google Scholar
Szabó, G.. Equivariant Kirchberg–Phillips-type absorption for amenable group actions. Preprint, 2016,arXiv:1610.05939.Google Scholar
Szabó, G.. Strongly self-absorbing $\text{C}^{\ast }$ -dynamical systems. Trans. Amer. Math. Soc. Published online 13 July 2017, doi:10.1090/tran/6931.Google Scholar
Tikuisis, A., White, S. and Winter, W.. Quasidiagonality of nuclear C∗ -algebras. Ann. of Math. (2) 185(1) (2017), 229–284.Google Scholar
Tikuisis, A. and Winter, W.. Decomposition rank of 𝓩-stable C∗ -algebras. Anal. PDE 7 (2014), 673–700.Google Scholar
Toms, A. S., White, S. and Winter, W.. 𝓩-stability and finite dimensional tracial boundaries. Int. Math. Res. Not. IMRN 2015(10) (2015), 2702–2727.Google Scholar
Toms, A. S. and Winter, W.. Strongly self-absorbing C∗ -algebras. Trans. Amer. Math. Soc. 359(8) (2007), 3999–4029.Google Scholar
Toms, A. S. and Winter, W.. Minimal dynamics and K-theoretic rigidity: Elliott’s conjecture. Geometric and Functional Analysis 23 (2013), 467–481.Google Scholar
Tu, J.-L.. La conjecture de Baum–Connes pour les feuilletages moyennables. K-theory(3) (1999), 215–264.Google Scholar
Wehrfritz, B. A. F.. Group and Ring Theoretic Properties of Polycyclic Groups (Algebra and Applications, 10) . Springer, London, 2009.Google Scholar
Winter, W.. Covering dimension for nuclear C∗ -algebras II. Trans. Amer. Math. Soc. 361(8) (2009), 4143–4167.Google Scholar
Winter, W.. Decomposition rank and 𝓩-stability. Invent. Math. 179(2) (2010), 229–301.Google Scholar
Winter, W.. Nuclear dimension and 𝓩-stability of pure C∗ -algebras. Invent. Math. 187(2) (2012), 259–342.Google Scholar
Winter, W.. Localizing the Elliott conjecture at strongly self-absorbing C∗ -algebras, with an appendix by H. Lin. J. Reine Angew. Math. 692 (2014), 193–231.Google Scholar
Winter, W.. Classifying crossed product C∗ -algebras. Amer. J. Math. 138 (2016), 793–820.Google Scholar
Winter, W. and Zacharias, J.. Completely positive maps of order zero. Münster J. Math. 2 (2009), 311–324.Google Scholar