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On the Range of Certain ASH Algebras of Real Rank Zero

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Abstract

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, mod-p K-theory, Math. Ann., 305, 1996, 601–616].

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References

  1. An, Q. and Elliott, G. A., On the KK-theory of Elliott-Thomsen algebras, J. Operator Theory, 78(2), 2017, 435–472.

    MathSciNet  MATH  Google Scholar 

  2. An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202.

    Article  MathSciNet  MATH  Google Scholar 

  3. Blackadar, B., The homotopy lifting theorem for semiprojective C*-algebras, Math. Scand., 118(2), 2016, 291–302.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bratteli, O., Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc., 171, 1972, 195–234.

    MathSciNet  MATH  Google Scholar 

  5. Dadarlat, M. and Gong, G., A classification result for approximately homogeneous C*-algebras of real rank zero, Geom. Funct. Anal., 7(4), 1997, 646–711.

    Article  MathSciNet  MATH  Google Scholar 

  6. Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, mod-p K-theory, Math. Ann., 305, 1996, 601–616.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dixmier, J., On some C*-algebras considered by Glimm, J. Funct. Anal., 1, 1967, 182–203.

    Article  MathSciNet  MATH  Google Scholar 

  8. Effros, E., Handelman, D. and Shen, C. L., Dimension groups and their affine representations, Amer. J. Math., 102, 1980, 385–407.

    Article  MathSciNet  MATH  Google Scholar 

  9. Eilers, S., A complete invariant for AD algebras with bounded torsion in K1, J. Funct. Anal., 139(2), 1996, 325–348.

    Article  MathSciNet  MATH  Google Scholar 

  10. Eilers, S., Loring, T. A. and Pedersen, G. K., Fragility of subhomogeneous C*-algebras with one-dimensional spectrum, Bull. London Math. Soc., 31(3), 1999, 337–344.

    Article  MathSciNet  MATH  Google Scholar 

  11. Elliott, G. A., On the classification of inductive limits of sequence of semisimple finite-dimensional algebras, J. Algebra, 38, 1976, 29–44.

    Article  MathSciNet  MATH  Google Scholar 

  12. Elliott, G. A., On the classification of C*-algebras of real rank zero, J. Reine Angew. Math., 443, 1993, 179–219.

    MathSciNet  MATH  Google Scholar 

  13. Elliott, G. A., Gong, G., Lin, H. and Niu, Z., On the classification of simple amenable C*-algebras with finite decomposition rank II, 2015, arXiv: 1507.03437v2.

  14. Elliott, G. A., Gong, G. and Su, H., On the Classification of C*-algebras of Real Rank Zero, IV., Reduction to Local Spectrum of Dimension Two, 20, Amer. Math. Soc., Providence, RI, 1998.

    MATH  Google Scholar 

  15. Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc., 95, 1960, 318–340.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gong, G., On the classification of simple inductive limit C*-algebras. I. The reduction theorem. Doc. Math., 7, 2002, 255–461.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gong, G., Lin, H. and Niu, Z., A classification of finite simple amenable \(\cal{Z}\)-stable C*-algebras, I: C*-algebras with generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can., 42(3), 2020, 63–450.

    MathSciNet  Google Scholar 

  18. Gong, G., Lin, H. and Niu, Z., A classification of finite simple amenable \(\cal{Z}\)-stable C*-algebras, II: C*-algebras with rational generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can., 42(4), 2020, 451–539.

    MathSciNet  Google Scholar 

  19. Jiang, X. and Su, H., On a simple unital projectionless C*-algebra, Amer. J. Math., 121(2), 1999, 359–413.

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, Z., On the simple inductive limits of splitting interval algebras with dimension drops, Canad. J. Math., 64(3), 2012, 544–572.

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, Z., A decomposition theorem for real rank zero inductive limits of 1-dimensional non-commutative CW complexes, J. Topol. Anal., 11(1), 2019, 181–204.

    Article  MathSciNet  MATH  Google Scholar 

  22. Loring, T. A., C*-algebras generated by stable relations, J. Funct. Anal., 112, 1993, 159–201.

    Article  MathSciNet  MATH  Google Scholar 

  23. Mygind, J., Classification of certain simple C*-algebras with torsion in K1, Canad. J. Math., 53(6), 2001, 1223–1308.

    Article  MathSciNet  MATH  Google Scholar 

  24. Thomsen, K., Limits of certain subhomogeneous C*-algebras, Mem. Soc. Math. Fr. (N.S.), 71, 1997, pp. 125.

    MathSciNet  MATH  Google Scholar 

  25. Tikuisis, A., White, S. and Winter, W., Quasidiagonality of nuclear C*-algebras, Ann. Math., 185(1), 2017, 229–284.

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, S., A Riesz decomposition property and ideal structure of multiplier algebras, J. Operator Theory, 24(2), 1990, 209–225.

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Professor Guihua Gong for helpful discussions. The authors would like to thank the referee for his/her valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Qingnan An

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116024, China

    Zhichao Liu

Authors
  1. Qingnan An
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  2. Zhichao Liu
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Corresponding author

Correspondence to Zhichao Liu.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 12101113, 12101102, 11920101001) and the Fundamental Research Funds for the Central Universities (Nos. 2412021QD001, DUT20RC(3)064).

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An, Q., Liu, Z. On the Range of Certain ASH Algebras of Real Rank Zero. Chin. Ann. Math. Ser. B 44, 279–288 (2023). https://doi.org/10.1007/s11401-023-0014-0

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  • DOI: https://doi.org/10.1007/s11401-023-0014-0

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