Skip to main content
SpringerLink
Log in

Atiyah-Jänich Theorem for σ-C*-algebras

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

K-theory for σ-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips (K-Theory 3, 441–478, 1989). We use his representable K-theory to show that the space of Fredholm modular operators with coefficients in an arbitrary unital σ-C*-algebra A, represents the functor X↦RK0(C(X,A)) from the category of countably compactly generated spaces to the category of abelian groups.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M.F.: K-theory. W. A. Benjamin Inc., New York-Amsterdam (1967)

    MATH  Google Scholar 

  2. Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. IHES Publ. Math. 37, 5–26 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arveson, W.: The harmonic analysis of automorphism groups, operator algebras and applications, Part I (Kingston, Ont., 1980). In: Proceedings of the Symposium on Pure Mathematics, vol. 38, pp 199–269. Amer. Math. Soc., Providence, RI (1982)

  4. Bhatt, S.J., Karia, D.J.: Complete positivity, tensor products and C*-nuclearity for inverse limits of C*-algebras. Proc. Indian Acad. Sci. Math. Sci. 101, 149–167 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourbaki, N.: Elements of Mathematics. General Topology. Part 1. Translated from the French Hermann, Publishers in Arts and Science, Publishers in Arts and Science, Paris; Addison-Wesley Publishing Co., Reading, Mass. London-Don Mills, Ont (1966)

  6. Cohen, J.M.: The homotopy groups of inverse limits. Proc. London Math. Soc. 27, 159–177 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  7. El Harti, R., Lukács, G.: Bounded and unitary elements in pro-C*-algebras. Appl. Categ. Structures 14, 151–164 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Emerson, H., Meyer, R.: Dualizing the coarse assembly map. J. Inst. Math. Jussieu 5, 161–186 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Emerson, H., Meyer, R.: Equivariant representable K-theory. J. Topol. 2, 123–156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fragoulopoulou, M.: Topological algebras with involution. North Holland, Amsterdam (2005)

  11. Frank, M., Sharifi, K.: Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C*-modules. J. Operator Theory 64, 377–386 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Franklin, S.P., Smith Thomas, B.V.: A survey of k ω -spaces. Topology Proc. 2, 111–124 (1977)

    MathSciNet  Google Scholar 

  13. Gray, B.I.: Spaces of the same n-type, for all n. Topology 5, 241–243 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guentner, E., Yu, G.: A milnor sequence in operator k-theory, preprints; Department of Mathematical Sciences University of Hawaii at Manoa

  15. Inoue, A.: Locally C*-algebras. Mem. Faculty Sci. Kyushu Univ. Ser. A 25, 197–235 (1971)

    MathSciNet  MATH  Google Scholar 

  16. Jänich, K.: Vektorraumbndel und der Raum der Fredholm-Operatoren. Math. Ann. 161, 129–142 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Joita, M.: Hilbert Modules Over Locally C*-Algebras. University of Bucharest Press (2006)

  18. Kawamura, K.: Some inverse limits of Cuntz algebras, arXiv:1112.2769[math.OA] (2011)

  19. Mahanta, S.: Twisted K-theory, K-homology and bivariant Chern-Connes type character of some infinite dimensional spaces. Kyoto J. Math. 54, 597–640 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Manuilov, V.M., Troitsky, E.V.: Hilbert C ∗-Modules. Translated from the Russian Original by the Authors. Translations of Mathematical Monographs, vol. 226, p 2005. Amer. Math. Soc., Providence, RI (2001)

  21. Milnor, J.: On axiomatic homology theory. Pacific J. Math. 12, 337–341 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mingo, J.A.: K-theory and multipliers of stable C*-algebras. Trans. Amer. Math. Soc. 299, 397–411 (1987)

    MathSciNet  MATH  Google Scholar 

  23. Phillips, N.C.: Inverse limits of C*-algebras. J. Operator Theory 19, 159–195 (1988)

    MathSciNet  MATH  Google Scholar 

  24. Phillips, N.C.: Inverse Limits of C*-Algebras and Applications, Operator Algebras and Applications, vol. 1. LMS Lecture Note Series 135, Cambridge University Press (1988)

  25. Phillips, N.C.: Representable K-theory for σ-C*-algebras. K-Theory 3, 441–478 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Phillips, N.C.: Classifying algebras for the K-theory of C*-algebras. Canad. J. Math. 41(6), 1021–1089 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Phillips, N.C.: K-theory for Fréchet algebras. Internat. J. Math. 2, 77–129 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Phillips, N.C.: The Toeplitz operator proof of noncommutative Bott periodicity. J. Austral. Math. Soc. Ser. A 53, 229–251 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rosenberg, J.: A Minicourse on Applications of Non-Commutative Geometry to Topology. Surveys in Noncommutative Geometry, 1-41, Clay Math Proc., vol. 6. Amer. Math. Soc., Providence, RI (2006)

  30. Schochet, C.L.: Topological methods for C*-algebras. III. Axiomatic homology. Pacific J. Math. 114(2), 399–445 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schochet, C.L.: Equivariant KK-theory for inverse limits of G-C*-algebras. J. Austral. Math. Soc. Ser. A 56(2), 183–211 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schochet, C.L.: The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups. K-Theory 10(1), 49–72 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Schochet, C. L.: A Pext Primer: Pure Extensions and \(\lim ^{1}\) for Infinite Abelian Groups. State University of New York, University at Albany, Albany, NY (2003)

  34. Sharifi, K.: Generic properties of module maps and characterizing inverse limits of C*-algebras of compact operators. Bull. Malays. Math. Sci. Soc. 36, 481–489 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Switzer, R.: Algebraic Topology–Homotopy and Homology. Springer, Berlin (2002)

    MATH  Google Scholar 

  36. Troitsky, E.V.: Classifying spaces for a K-functor connected with a C*-algebra. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 110, 96–98 (1985)

    MathSciNet  Google Scholar 

  37. Voiculescu, D.: Dual algebraic structures on operator algebras related to free products. J. Operator Theory 17(1), 85–98 (1987)

    MathSciNet  MATH  Google Scholar 

  38. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)

  39. Wegge-Olsen, N.E.: K-Theory and C ∗-Algebras: a Friendly Approach. Oxford, Oxford University Press (1993)

    MATH  Google Scholar 

  40. Weidner, J.: KK-Groups for generalized operator algebras. I, II. K-Theory 3(1), 57–77, 79–98 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran

    Kamran Sharifi

Authors
  1. Kamran Sharifi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kamran Sharifi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sharifi, K. Atiyah-Jänich Theorem for σ-C*-algebras. Appl Categor Struct 25, 893–905 (2017). https://doi.org/10.1007/s10485-016-9474-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-016-9474-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation