Chinese Annals of Mathematics, Series B

, Volume 41, Issue 1, pp 1–26 | Cite as

Persistence Approximation Property for Maximal Roe Algebras

  • Qin Wang
  • Zhen WangEmail author


Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coarse embedding into Hilbert space and X is coarsely uniformly contractible, then C*max(X) has persistence approximation property. The authors also give an application of the quantitative K-theory to the maximal coarse Baum-Connes conjecture.


Quantitative K-theory Persistence approximation property Maximal coarse Baum-Connes conjecture Maximal Roe algebras 

2000 MR Subject Classification

46L80 46L89 51F99 


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The authors would like to thank Hervé Oyono-Oyono for many helpful explanations.


  1. [1]
    Chen, X., Wang, Q. and Wang, X., Characterization of the Haagerup property by fibred coarse embedding into Hilbert space, Bulletin of the London Mathematical Society, 45(5), 2013, 1091–1099.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Chen, X., Wang, Q. and Yu, G., The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space, Adv. Math., 249, 2013, 88–130.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Dell’ Aiera, C., Controlled K-theory for groupoids and applications, J. Funct. Anal., 275, 2018, 1756–1807.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Finn-Sell, M., Fibred coarse embedding, a-T-menability and coarse analogue of the Novikov conjecture, J. Funct. Anal., 267(10), 2014, 3758–3782.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Finn-Sell, M. and Wright, N., The coarse Baum-Connes conjecture, boundary groupoids and expander graphs, Adv. Math. C., 259, 2014, 306–338.CrossRefGoogle Scholar
  6. [6]
    Higson, N. and Kasparov, G., E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math., 144(1), 2001, 23–74.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Higson, N., Lafforgue, V. and Skandalis G., Counterexamples to the Baum-Connes conjecture, Geometric Functional Analysis, 12(2), 2002, 330–354.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Meintrup, D. and Schick, T., A model for the universal space for proper actions of a hyperbolic group, New York J. Math., 8(1–7), 2002.Google Scholar
  9. [9]
    Oyono-Oyono, H. and Yu, G., K-theory for the maximal Roe algebras for certain expanders, J. Funct. Anal., 257(10), 2009, 3239–3292.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Oyono-Oyono, H. and Yu, G., On quantitative operator K-theory, Ann. Inst. Fourier (Grenoble), 65, 2015, 605–674.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Oyono-Oyono, H. and Yu, G., Quantitative K-theory and the kunneth formula for operator algebras, arXiv: e-prints.Google Scholar
  12. [12]
    Oyono-Oyono, H. and Yu, G., Persistence approximation property and controlled K-theory, Munster J. of Math., 10, 2017, 201–268.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Roe, J., Lectures on Coarse Geometry, University Lecture Series, 31, American Mathematical Society, Providence, RI, 2003.zbMATHGoogle Scholar
  14. [14]
    Skandalis, G., Tu, J. L. and Yu, G., The coarse Baum-Connes conjecture and groupoids, Topology, 41(4), 2002, 807–834.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Willett, R. and Yu, G., Higher index theory for certain expanders and Gromov monster groups, I. Adv. Math., 229(3), 2012 1380–1416.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Yu, G., The Novikov conjecture for groups with finite asymtotic dimension, Annals of Mathematics, 147(2), 1998, 325–355.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2020

Authors and Affiliations

  1. 1.Research Center for Operator Algebras, School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghaiChina
  2. 2.Research Center for Operator Algebras, School of Mathematical SciencesEast China Normal UniversityShanghaiChina

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