Skip to main content
Log in

The classification problem for simple unital finite rank dimension groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The Borel complexity of the isomorphism problem for finite-rank unital simple dimension groups increases with rank. This implies that the isomorphism problems for the corresponding classes of Bratteli diagrams and LDA-groups also increase with rank.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. K. R. Davidson, C*-Algebras by Example, Fields Institute Monographs, Vol. 6, American Mathematical Society, Providence, RI, 1996.

    Google Scholar 

  2. E. G. Effros, Dimensions and C*-Algebras, CBMS Regional Conference Series in Mathematics, Vol. 46, Conference Board of the Mathematical Sciences, Washington, DC, 1981.

    Google Scholar 

  3. E. Effros, D. Handelman and C.-L. Shen, Dimension groups and their affine representations, American Journal of Mathematics 102 (1980), 191–204.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras, Journal of Algebra 38 (1976), 29–44.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. A. Elliott, On totally ordered groups and K0, in Ring Theory, (Proc. Conf. Univ. Waterloo, Waterloo, 1978), Lecture Notes in Mathematics, Vol. 734, Springer, Berlin, 1979, pp. 1–49.

    Google Scholar 

  6. K. Goodearl, Partially Ordered Abelian Groups With Interpolation, Mathematical Surveys and Monographs, Vol. 20, American Mathematical Society, Providence, RI, 1986.

    Google Scholar 

  7. D. Handelman, Ultrasimplicial dimension groups, Archic der Mathematik 40 (1983), 109–115.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Hartley and A. E. Zalesskii, Confined subgroups of simple locally finite roups and ideals of group rings, Journal of the London Mathematical Society 55 (1997), 210–230

    Article  Google Scholar 

  9. G. Hjorth, Around nonclassifiability for countable torsion-free abelian groups, in Abelian Groups and Modules (Dublin, 1998), Trends in Mathematics, Birkhäuser, Basel, 1999, pp. 269–292.

    Chapter  Google Scholar 

  10. S. Jackson, A. S Kechris and A. Louveau, Countable Borel equivalence relations, Annals of Pure and Applied Logic 82 (1996), 221–272.

    Article  MathSciNet  MATH  Google Scholar 

  11. V. Kanovei, Borel Equivalence Relations, University Lecture Series, Vol. 44, American Mathematical Society, Providence, RI, 2008.

    Google Scholar 

  12. Y. Lavrenyuk and V. Nekrashevych, On classification of inductive limits of direct products of alternating groups, Journal of the London Mathematical Society 75 (2007), 146–162.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. A. Springer, Linear Algebraic Groups, Progress in Mathematics, Vol. 9, Birkhäuser, Boston, MA, 1998.

    Google Scholar 

  14. S. Thomas, On the complexity of the classification problem for torsion-free abelian groups of finite rank, Bulletin of Symbolic Logic 7 (2001), 329–344.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Thomas, The classification problem for torsion-free abelian groups of finite rank, Journal of the American Mathematical Society 16 (2003), 233–258.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Thomas, Popa superrigidity and countable Borel equivalence relations, Annals of Pure and Applied Logic 158 (2009), 175–189.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Thomas, The classification problem for p-local torsion-free abelian groups of finite rank, preprint, available at http://sites.math.rutgers.edu/~sthomas/local7.pdf.

  18. A. E. Zalesskii, Group rings of inductive limits of alternating groups, Algebra i Analiz 2 (1990), 132–149; English translation: Leningrad Mathematical Journal 2 (1991), 1287–1303.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Paul Ellis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ellis, P. The classification problem for simple unital finite rank dimension groups. Isr. J. Math. 226, 419–445 (2018). https://doi.org/10.1007/s11856-018-1700-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1700-6

Navigation