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Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups

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Abstract

In this paper we study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions on certain growth rates for discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm. We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups. For a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra. For a group with subexponential growth, we show that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra.

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References

  1. Bédos, E., Conti, R.: On twisted Fourier analysis and convergence of Fourier series on discrete groups. J. Fourier Anal. Appl. 15(3), 336–365 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bickel, P., Lindner, M.: Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics. Theory Probab. Appl. 56(1), 1–20 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brown, N.P., Ozawa, N.: \({\rm C}^*\)-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics. American Mathematical Society, Providence (2008)

    Google Scholar 

  4. Chen, X., Fu, B.: Non-commutative Fejér theorems. Integr. Equ. Oper. Theory 74(3), 301–312 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, X., Wang, Q.: Ideal structure of uniform Roe algebra of coarse spaces. J. Funct. Anal. 216(1), 191–211 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, X., Wei, S.: Spectral invariant subalgebras of reduced crossed product C\(^*\)-algebras. J. Funct. Anal. 197(1), 228–246 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Connes, A., Moscovici, H.: Cyclic cohomology and Novikov conjecture and hyperbolic groups. Topology 29(3), 345–388 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. de la Harpe, P.: Groupes hyperboliques, algèbres d’opérateures et un théorèmes de Jolissaint. C. R. Acad. Sci. Paris, Sér. I 307, 771–774 (1998)

    MathSciNet  Google Scholar 

  9. Fendler, G., Gröchenig, K., Leinert, M.: Convolution-dominated operators on discrete groups. Integr. Equ. Oper. Theory 61(4), 493–509 (2008)

    Article  MATH  Google Scholar 

  10. Grigorchuk, R.I.: Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48(5), 939–985 (1984)

    MathSciNet  Google Scholar 

  11. Haagerup, U.: An example of a non nuclear C\(^*\)-algebras, which has the metric approximation property. Invent. Math. 50, 279–293 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jolissaint, P.: Rapidly decreasing functions in reduced C\(^*\)-algebras of groups. Trans. Am. Math. Soc. 317(1), 167–196 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Lafforgue, V.: A proof of property (RD) for cocompact lattices of \(SL(3,\mathbb{R})\) and \(SL(3,\mathbb{C})\). J. Lie Theory 10(2), 255–267 (2000)

    MATH  MathSciNet  Google Scholar 

  14. Lafforgue, V.: K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math 149(1), 1–95 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ozawa, N.: Amenable actions and exactness for discrete groups. C. R. Acad. Sci. Paris Sér. I Math. 330(8), 691–695 (2000)

    Article  MATH  Google Scholar 

  16. Yu, G.: The coarse Baum-Connes conjecture of spaces which admit a uniformly embedding into Hilbert spaces. Invent. Math. 139(1), 201–240 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors wish to thank the referees for their huge amount of corrections, explanations, and valuable comments on the original version of this paper. The authors are also very grateful to Taxas A&M University for its support and hospitality during their visits. The second author wishes to thank the Erwin Schrödinger International Institute for Mathematical Physics, Universität Wien, for its support and hospitality during his visit in April, 2014, in the Workshop on “Geometry of Computation in Groups”. The third author is also indebted to the China Scholarship Council for its support. The authors are supported in part by NSFC (No. 11231002, 11420101001, 10901033, 10971023).

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Xiaoman Chen

  2. Research Center for Operator Algebras, Department of Mathematics, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, People’s Republic of China

    Qin Wang

  3. College of Mathematics and Statistics, Chongqing University (at Huxi Campus), Chongqing, 401331, People’s Republic of China

    Xianjin Wang

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  1. Xiaoman Chen
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  2. Qin Wang
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  3. Xianjin Wang
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Correspondence to Xianjin Wang.

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Communicated by Hans G. Feichtinger.

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Chen, X., Wang, Q. & Wang, X. Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups. J Fourier Anal Appl 21, 555–574 (2015). https://doi.org/10.1007/s00041-014-9380-z

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  • DOI: https://doi.org/10.1007/s00041-014-9380-z

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