Skip to main content
SpringerLink
Log in

The Classification Problem for Amenable C*-Algebras

  • Conference paper
Proceedings of the International Congress of Mathematicians

Abstract

For thirty-five years, one of the most interesting and rewarding classes of operator algebras to study has been the approximately finite-dimensional C*-algebras of Glimm and Bratteli ([39], [7]).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. B. Blackadar, A simple C-algebra with no nontrivial projections, Proc. Amer. Math. Soc. 78 (1980), 504–508.

    MathSciNet  MATH  Google Scholar 

  2. B. Blackadar, A simple unital projectionless C-algebra, J. Operator Theory 5 (1981), 63–71.

    MathSciNet  MATH  Google Scholar 

  3. B. Blackadar, Symmetries of the CAR algebra, Ann. of Math. (2) 131 (1990), 589–623.

    Article  MathSciNet  Google Scholar 

  4. B. Blackadar, O. Bratteli, G. A. Elliott, and A. Kumjian, Reduction of real rank in inductive limits of C-algebras, Math. Ann. 292 (1992), 111–126.

    Article  MathSciNet  Google Scholar 

  5. B. Blackadar and J. Cuntz, The structure of stable algebraically simple C-algebras, Amer. J. Math. 104 (1982), 813–822.

    Article  MathSciNet  Google Scholar 

  6. B. Blackadar and M. Rørdam, Extending states on preordered semigroups and the existence of quasitraces on C-algebras, J. Algebra 152 (1992), 240–247.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto, Actions of finite groups on AF algebras obtained by folding the interval, K-Theory, to appear.

    Google Scholar 

  9. O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto, On the classification of C-algebras of real rank zero, III: The infinite case, preprint.

    Google Scholar 

  10. O. Bratteli, G. A. Elliott, and A. Kishimoto, The temperature state space of a C-dynamical system, II, Ann. of Math. (2) 123 (1986), 205–263.

    Article  MathSciNet  Google Scholar 

  11. O. Bratteli, D. E. Evans, and A. Kishimoto, Crossed products of totally disconnected spaces by ℤ2ℤ2, Ergodic Theory Dynamical Systems 13 (1994), 445–484.

    MathSciNet  MATH  Google Scholar 

  12. O. Bratteli, D. E. Evans, and A. Kishimoto, The Rokhlin property for quasi-free automorphisms of the Fermion algebra, Proc. London Math. Soc., to appear.

    Google Scholar 

  13. O. Bratteli and A. Kishimoto, Non-commutative spheres, III: Irrational rotations, Comm. Math. Phys. 147 (1992), 605–624.

    Article  MathSciNet  Google Scholar 

  14. O. Bratteli, A. Kishimoto, M. Rørdam, and E. Størmer, The crossed product of a UHF-algebra by a shift, Ergodic Theory Dynamical Systems 13 (1993), 615–626.

    MathSciNet  MATH  Google Scholar 

  15. M.-D. Choi and E. G. Effros, Separable nuclear C-algebras and injectivity, Duke Math. J. 43 (1976), 309–322.

    Article  MathSciNet  Google Scholar 

  16. A. Connes, Classification of injective factors, Ann. of Math. (2) 104 (1976), 73–115.

    Article  MathSciNet  Google Scholar 

  17. A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248–253.

    Article  MathSciNet  Google Scholar 

  18. J. Cuntz, K-theoryfor certain C-algebras, Ann. of Math. (2) 113 (1981), 181–197.

    Article  MathSciNet  Google Scholar 

  19. M. Dadarlat, Approximately unitarily equivalent morphisms and inductive limit C- algebras, preprint.

    Google Scholar 

  20. M. Dadarlat, Reduction to dimension three of local spectra of real rank zero C- algebras, preprint.

    Google Scholar 

  21. J. Dixmier, On some C-algebras considered by Glimm, J. Funct. Anal. 1 (1967), 182–203.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. G. A. Elliott, Dimension groups with torsion, Internat. J. Math. 1 (1990), 361–380.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. G. A. Elliott, A classification of certain simple C-algebras, Quantum and Non- Commutative Analysis (editors, H. Araki et al.), Kluwer, Dordrecht, 1993, pages 373–385.

    Chapter  Google Scholar 

  26. G. A. Elliott, A classification of certain simple C-algebras, II, Ann. of Math. (2), to appear.

    Google Scholar 

  27. G. A. Elliott, An invariant for simple C-algebras, in preparation.

    Google Scholar 

  28. G. A. Elliott and D. E. Evans, The structure of the irrational rotation C-algebra, Ann. of Math. (2) 138 (1993), 477–501.

    Article  MathSciNet  Google Scholar 

  29. G. A. Elliott and G. Gong, On inductive limits of matrix algebras over the two-torus, preprint.

    Google Scholar 

  30. G. A. Elliott and G. Gong, On the classification of C-algebras of real rank zero, II, preprint.

    Google Scholar 

  31. G. A. Elliott, G. Gong, and L. Li, On simple inductive limits of matrix algebras over higher dimensional spaces, II, in preparation.

    Google Scholar 

  32. G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, Homomorphisms, homotopies, and approximations by circle algebras, C. R. Math. Rep. Acad. Sci. Canada 16 (1994), 45–50.

    MathSciNet  MATH  Google Scholar 

  33. G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, Abelian C-subalgebras of C- algebras of real rank zero and inductive limit C-algebras, preprint.

    Google Scholar 

  34. G. A. Elliott and M. Rørdamrdam, The automorphism group of the irrational rotation C-algebra, Comm. Math. Phys. 155 (1993), 3–26.

    Article  MathSciNet  Google Scholar 

  35. G. A. Elliott and M. Rørdamrdam, Classification of certain infinite simple C-algebras, II, preprint.

    Google Scholar 

  36. G. A. Elliott and K. Thomsen, The state space of the K0-group of a simple separable C-algebra, Geom. and Functional Anal. 4 (1994), 522–538.

    Article  MathSciNet  Google Scholar 

  37. D. E. Evans and A. Kishimoto, Compact group actions on UHF algebras obtained by folding the interval, J. Funct. Anal. 18 (1991), 346–360.

    Article  MathSciNet  Google Scholar 

  38. T. Giordano, I. F. Putnam, and C. Skau, Topological orbit equivalence and C- crossed products, preprint.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. G. Gong, On inductive limits of matrix algebras over higher dimensional spaces, Part II, preprint.

    Google Scholar 

  41. G. Gong, On simple inductive limits of matrix algebras over higher dimensional spaces, preprint.

    Google Scholar 

  42. U. Haagerup, All nuclear C-algebras are amenable, Invent. Math. 74 (1983), 305–319.

    Article  MathSciNet  Google Scholar 

  43. U. Haagerup, Every quasi-trace on an exact C-algebra is a trace, preprint.

    Google Scholar 

  44. P. de la Harpe, Moyennabilité du groupe unitaire et propriété P de Schwartz des algèbres de von Neumann, Algèbres d’opérateurs (Séminaire, Les Plans-sur-Bex, Suisse 1978) (editor, P. de la Harpe), Lecture Notes in Math. 725, Springer-Verlag, Berlin and New York, 1979, pages 220 –227.

    Google Scholar 

  45. R. H. Herman and A. Ocneanu, Stability for integer actions on UHF C-algebras, J. Funct. Anal. 59 (1984), 132–144.

    Article  MathSciNet  Google Scholar 

  46. R. H. Herman, I. F. Putnam, and C. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamical systems, Internat. J. Math. 3 (1992), 827–864.

    Article  MathSciNet  Google Scholar 

  47. E. Kirchberg, Lecture at ICM Satellite Meeting, Geneva, 1994.

    Google Scholar 

  48. A. Kumjian, An involutive automorphism of the Bunce-Deddens algebra, C. R. Math. Rep. Acad. Sci. Canada 10 (1988), 217–218.

    MathSciNet  MATH  Google Scholar 

  49. L. Li, Ph.D. thesis, University of Toronto, 1994.

    Google Scholar 

  50. H. Lin, Approximation by normal elements with finite spectra in C-algebras of real rank zero, preprint.

    Google Scholar 

  51. H. Lin, Homomorphisms from C(X) into C-algebras, preprint.

    Google Scholar 

  52. H. Lin, Almost commuting self-adjoint matrices and applications, preprint.

    Google Scholar 

  53. H. Lin and N. C. Phillips, Classification of direct limits of even Cuntz-circle algebras, preprint.

    Google Scholar 

  54. H. Lin and N. C. Phillips, Approximate unitary equivalence of homomorphisms from O∞, preprint.

    Google Scholar 

  55. A. L. T. Paterson, Nuclear C-algebras have amenable unitary groups, Proc. Amer. Math. Soc. 124 (1992), 719–721.

    MathSciNet  MATH  Google Scholar 

  56. I. F. Putnam, On the topological stable rank of certain transformation group C- algebras, Ergodic Theory Dynamical Systems 10 (1990), 187–207.

    Article  MathSciNet  Google Scholar 

  57. M. Rørdam, Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math. 440 (1993), 175–200.

    MathSciNet  MATH  Google Scholar 

  58. M. Rørdam, Classification of Cuntz-Krieger algebras, K-Theory, to appear.

    Google Scholar 

  59. M. Rørdam, Classification of certain infinite simple C-algebras, J. Funct. Anal., to appear.

    Google Scholar 

  60. H. Su, Ph.D. thesis, University of Toronto, 1992.

    Google Scholar 

  61. K. Thomsen, Inductive limits of interval algebras: The tracial state space, Amer. J. Math. 116 (1994), 605–620.

    Article  MathSciNet  Google Scholar 

  62. K. Thomsen, On the range of the Elliott invariant, J. Funct. Anal., to appear.

    Google Scholar 

  63. K. Thomsen, On the ordered K0-group of a simple C-algebra, preprint.

    Google Scholar 

  64. J. Villadsen, The range of the Elliott invariant, J. Reine Angew. Math., to appear.

    Google Scholar 

  65. S. G. Walters, Inductive limit automorphisms of the irrational rotation C-algebra, Comm. Math. Phys., to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Matematisk Institut, Københavns Universitet, DK-2100, København Ø, Denmark

    George A. Elliott

  2. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 1A1

    George A. Elliott

Authors
  1. George A. Elliott
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäuser Verlag, Basel, Switzerland

About this paper

Cite this paper

Elliott, G.A. (1995). The Classification Problem for Amenable C*-Algebras. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_85

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_85

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation