Communications in Mathematical Physics

, Volume 335, Issue 2, pp 637–670 | Cite as

Rokhlin Dimension and C*-Dynamics

  • Ilan Hirshberg
  • Wilhelm WinterEmail author
  • Joachim Zacharias
Open Access


We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of \(\mathcal{Z}\)-stable C*-algebras, where \(\mathcal{Z}\) denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve \({\mathcal{Z}}\) -stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic \({\mathbb{Z}^{d}}\)-actions on compact metrizable and finite dimensional spaces.


Hyperbolic Group Nuclear Dimension Irrational Rotation arXiv Preprint Unital Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Archey, D.E.: Crossed product C*-algebras by finite group actions with a generalized tracial Rokhlin property, Ph.D. thesis, University of Oregon (2008)Google Scholar
  2. 2.
    Bartels A., Lück W., Reich H.: Equivariant covers for hyperbolic groups. Geom. Topol. 12(3), 1799–1882 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bartels A., Lück W., Reich H.: The K-theoretic Farrell–Jones conjecture for hyperbolic groups. Invent. Math. 172(1), 29–70 (2008)CrossRefADSzbMATHMathSciNetGoogle Scholar
  4. 4.
    Herman R.H., Jones V.F.R.: Period two automorphisms of UHF C*-algebras. J. Funct. Anal. 45(2), 169–176 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hirshberg I., Kirchberg E., White S.A.: Decomposable approximations of nuclear C*-algebras. Adv. Math. 230, 1029–1039 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hirshberg I., Orovitz J.: Tracially \({\mathcal{Z}}\) -absorbing C*-algebras. J. Funct. Anal. 265(5), 765–785 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hirshberg, I., Phillips, N.C.: Rokhlin dimension: obstructions and permanence properties. (arXiv preprint) arXiv:1410.6581
  8. 8.
    Hirshberg I., Winter W.: Rokhlin actions and self-absorbing C*-algebras. Pac. J. Math. 233(1), 125–143 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Izumi, M.: The Rohlin Property for Automorphisms of C*-Algebras, Mathematical Physics in Mathematics and Physics (Siena, 2000). Fields Institute Communication, vol. 30, American Mathematical Society, Providence, pp. 191–206 (2001)Google Scholar
  10. 10.
    Izumi M.: Finite group actions on C*-algebras with the Rohlin property. I. Duke Math. J. 122(2), 233–280 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Izumi M.: Finite group actions on C*-algebras with the Rohlin property. II. Adv. Math. 184(1), 119–160 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Jiang X., Su H.: On a simple unital projectionless C*-algebra. Am. J. Math. 121(2), 359–413 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kirchberg E.: Central sequences in C*-algebras and strongly purely infinite C*-algebras. Abel Symp. 1, 175–231 (2006)MathSciNetGoogle Scholar
  14. 14.
    Kirchberg E., Rørdam M.: Non-simple purely infinite C*-algebras. Am. J. Math. 122, 637–666 (2000)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kirchberg E., Winter W.: Covering dimension and quasidiagonality. Int. J. Math. 15(1), 63–85 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kishimoto A.: The Rohlin property for automorphisms of UHF algebras. J. Reine Angew. Math. 465, 183–196 (1995)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Lin, H.: AF-embeddings of the crossed products of AH-algebras by finitely generated abelian groups. International Mathematics Research Papers IMRP no. 3 Article ID rpn007 67 (2008)Google Scholar
  18. 18.
    Lindenstrauss E., Weiss B.: Mean topological dimension. Isr. J. Math. 115, 1–24 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Matui H., Sato Y.: \({{\mathcal{Z}}}\) -Stability of crossed products by strongly outer actions II. Am. J. Math. 136(6), 1441–1496 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Matui H., Sato Y.: Strict comparison and \({\mathcal{Z}}\) -absorption of nuclear C*-algebras. Acta Math. 209(1), 179–196 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Matui H., Sato Y.: Decomposition rank of UHF-absorbing C*-algebras. Duke Math. J. 163(14), 2687–2708 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Osaka H., Phillips N.C.: Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property. Ergod. Theory Dyn. Syst. 26(5), 1579–1621 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Osaka H., Phillips N.C.: Crossed products by finite group actions with the Rokhlin property. Math. Z. 270(1–2), 19–42 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Phillips N.C.: The tracial Rokhlin property for actions of finite groups on C*-algebras. Am. J. Math. 133(3), 581–636 (2011)CrossRefzbMATHGoogle Scholar
  25. 25.
    Phillips, N.C.: The tracial Rokhlin property is generic (2012). (arXiv preprint) arXiv:1209.3859
  26. 26.
    Rørdam M., Winter W.: The Jiang–Su algebra revisited. J. Reine Angew. Math. 642, 129–155 (2010)MathSciNetGoogle Scholar
  27. 27.
    Sato Y.: The Rohlin property for automorphisms of the Jiang–Su algebra. J. Funct. Anal. 259(2), 453–476 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Schmidt W.M.: Diophantine Approximation, Lecture Notes in Mathematics, vol. 785. Springer, Berlin (1980)Google Scholar
  29. 29.
    Szabó, G.: The Rokhlin dimension of topological \({\mathbb{Z}^{m}}\) -actions. To appear in Proc. Lond. Math. Soc. (3) (2013). (arXiv preprint) arXiv:math.OA/1308.5418
  30. 30.
    Toms A.S., Winter W.: \({\mathcal{Z}}\) -Stable ASH algebras. Can. J. Math. 60(3), 703–720 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Toms A.S., Winter W.: Minimal dynamics and K-theoretic rigidity: Elliott’s conjecture. Geom. Funct. Anal. 23(1), 467–481 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Winter W.: Decomposition rank and \({\mathcal{Z}}\) -stability. Invent. Math. 179(2), 229–301 (2010)CrossRefADSzbMATHMathSciNetGoogle Scholar
  33. 33.
    Winter W.: Nuclear dimension and \({\mathcal{Z}}\) -stability of pure C*-algebras. Invent. Math. 187(2), 259–342 (2012)CrossRefADSzbMATHMathSciNetGoogle Scholar
  34. 34.
    Winter, W.: Classifying crossed product C*-algebras (2013). (arXiv preprint) arXiv:math.OA/1308.4084
  35. 35.
    Winter W., Zacharias J.: The nuclear dimension of C*-algebras. Adv. Math. 224(2), 461–498 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Ilan Hirshberg
    • 1
  • Wilhelm Winter
    • 2
    Email author
  • Joachim Zacharias
    • 3
  1. 1.Department of MathematicsBen Gurion University of the NegevBeershebaIsrael
  2. 2.Mathematisches InstitutWestfalische Wilhelms-UniversitätMünsterGermany
  3. 3.School of Mathematics and StatisticsUniversity of GlasgowGlasgowScotland

Personalised recommendations