Skip to main content
SpringerLink
Log in

Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L 2-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra (usually the full group C*-algebra C*(π) of the fundamental group π=π1(M) of a manifold M).

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. Alberti, P. M. and Matthes, R.: Connes trace formula and Dirac realizations of Maxwell and Yang-Mills action, Preprint math-ph/9910011 at xxx.lanl.gov, 1999.

  2. Atiyah, M. F.: Elliptic operators, discrete groups and von Neumann algebras, Colloque 'Analyse et Topologie' en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque 32-33, Soc. Math. France, Paris, 1976, 43-72.

    Google Scholar 

  3. Baillet, M., Denizeau, Y. and Havet, J.-F.: Indice d'une esperance conditionelle, Compositio Math. 66 (1988), 199-236.

    Google Scholar 

  4. Burghelea, D., Friedlander, L., Kappeler, T. and McDonald, P.: Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Anal. Funct. Anal. 6 (1996), 751-859.

    Google Scholar 

  5. Burghelea, D., Friedlander, L. and Kappeler, T.: Torsion for manifolds with boundary and glueing formulas, J. Funct. Anal. 137 (1996), 320-363.

    Google Scholar 

  6. Burghelea, D., Friedlander, L. and Kappeler, T.: Relative torsion, Comm. Contemp. Math. 3 (2001), 15-85.

    Google Scholar 

  7. Carey, A. L. and Mathai, V.: L 2-torsion invariants, J. Funct. Anal. 110 (1992), 377-409.

    Google Scholar 

  8. Carey, A. L., Farber, M. and Mathai, V.: Determinant lines, von Neumann algebras and L 2-torsion, J. Reine Angew. Math. 484 (1997), 153-181.

    Google Scholar 

  9. Carey, A. L., Mathai, V. and Mishchenko, A. S.: On analytic torsion over C*-algebras, In: Warsaw Conference on 'Dynamical Zeta Functions, Nielson Theory and Reidemeister Torsion', Warsaw, Poland, 1996, Banach Center Publications 49 (1999), pp. 43-67.

  10. Cheeger, J. and Gromov, M.: Bounds on the von Neumann dimension of L 2-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), 1-34.

    Google Scholar 

  11. Cheeger, J. and Gromov, M.: L 2-cohomology and group cohomology, Topology 25 (1986), 189-215.

    Google Scholar 

  12. Dixmier, J.: Les C*-Algèbres et Leurs Représentations, Gauthier-Villars, Paris, 1964.

    Google Scholar 

  13. Farber, M.: Abelian categories, Novikov-Shubin invariants, and Morse inequalities, C. R. Acad. Sci. Paris Sér. I, Math. 321 (1995), 1593-1598.

    Google Scholar 

  14. Farber, M.: Von Neumann categories and extended L 2-homology, Theory 15 (1998), 347-405.

    Google Scholar 

  15. Frank, M.: Self-duality and C*-reflexivity of Hilbert C*-modules, Z. Anal. Anwendungen 9 (1990), 165-176.

    Google Scholar 

  16. Frank, M.: Beiträge zur Theorie der Hilbert-C*-Modulen (Habilitation Thesis, Universität Leipzig, Leipzig, F.R.G., October 1997, 190 pp.), Shaker Verlag, Aachen-Maastrich, 1997.

    Google Scholar 

  17. Frank, M., Manuilov, V. M. and Troitsky, E. V.: On conditional expectations arising from group actions, Z. Anal. Anwendungen 16 (1997), 831-850.

    Google Scholar 

  18. Frank, M. and Larson, D. R.: Frames in Hilbert C*-modules and C*-algebras, Preprint, 1998.

  19. Manuilov, V. M., Troitsky, E. V. and Frank, M.: Conditional expectations connected with group actions (in Russian), Vestn. Mosk. Univ. Ser. Math. Mech. 3 (1998), 30-34.

    Google Scholar 

  20. Frank, M.: Geometrical aspects of Hilbert C*-modules, Positivity 3 (1999), 215-243.

    Google Scholar 

  21. Frank, M.: The commutative case: spinors, Dirac operator and de Rham algebra, In: Proceedings of the Workshop 'The Standard Model of Elementary Particle Physics from a Mathematical-Geometrical Viewpoint', Ev.-Luth. Volkshochschule Hesselberg, March 14-19, 1999 / Preprint math-ph/0002045 at xxx.arxiv.org.

  22. Frank, M. and Larson, D. R.: A module frame concept for Hilbert C*-modules, In: D. R. Larson and L. W. Baggett (eds), Functional and Harmonic Analysis of Wavelets (San Antonio, TX, Jan. 1999), Amer. Math. Soc., Providence, RI, Contemp. Math. 247 (2000), 207-233.

    Google Scholar 

  23. Kaplansky, I.: Modules over operator algebras, Amer. J. Math. 75 (1953), 839-858.

    Google Scholar 

  24. Kasparov, G. G.: Hilbert C*-modules: The theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), 133-150.

    Google Scholar 

  25. Lance, E. C.: Hilbert C*-Modules-a Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, Cambridge, England, 1995.

    Google Scholar 

  26. Lott, J. and Lück, W.: L 2-topological invariants of 3-manifolds, Invent. Math. 120 (1995), 15-60.

    Google Scholar 

  27. Lück, W. and Rothenberg, M.: Reidemeister torsion and the K-theory of von Neumann algebras, K-Theory 5 (1991), 213-264.

    Google Scholar 

  28. Lück, W.: Analytic and topological torsion for manifolds with boundary and symmetries, J. Differential Geom. 37 (1993), 263-322.

    Google Scholar 

  29. Lück, W.: L 2-Betti numbers of mapping tori and groups, Topology 33 (1994), 203-214.

    Google Scholar 

  30. Lück, W.: L 2-torsion and 3-manifolds, In: K. Johannson (ed.), Low-Dimensional Topology, Knoxville 1992, International Press, Cambridge, MA, 1994, pp. 75-107.

    Google Scholar 

  31. Lück, W.: Approximating L 2-invariants by their finite-dimensional analogues, Geom. Anal. Funct. Anal. 4 (1994), 455-481.

    Google Scholar 

  32. Lück, W.: Hilbert modules and modules over finite von Neumann algebras, and applications to L 2-invariants, Math. Ann. 309 (1997), 247-285.

    Google Scholar 

  33. Lück, W.: Dimension theory of arbitrary modules over finite von Neumann algebras and L 2-Betti numbers. I: Foundations, J. Reine Angew. Math. 495 (1998), 135-162.

    Google Scholar 

  34. Lück, W.: Dimension theory of arbitrary modules over finite von Neumann algebras and L 2-Betti numbers. II: Applications to Grothendieck groups, L 2-Euler characteristics and Burnside groups, J. Reine Angew. Math. 496 (1998), 213-236.

    Google Scholar 

  35. Lück, W.: L 2-invariants of regular coverings of compact manifolds and CW-complexes, to appear, In: R. J. Davermann and R. B. Sher (eds), Handbook of Geometry, Elsevier, Amsterdam, 1997.

    Google Scholar 

  36. Lück, W., Reich, H. and Schick, Th.: Novikov-Shubin invariants for arbitrary group actions and their positivity, Contemp. Math. 231 (1999), 159-176.

    Google Scholar 

  37. Mathai, V. and Carey, A. L.: L 2-analyticity and L 2-torsion invariants, Contemp. Math. 105 (1990), 91-118.

    Google Scholar 

  38. Mathai, V. and Shubin, M. A.: Twisted invariants on covering spaces and asymptotic L 2-Morse inequalities, Russ. J. of Math. Phys. 4 (1996), 499-527.

    Google Scholar 

  39. Mathai, V.: L 2-invariants on covering spaces, In: A. L. Carey and M. Murray (eds), Geometric Analysis and Lie Theory in Mathematics and Physics, Austral. Math. Soc. Lecture Ser. 11 (1998), 209-242.

    Google Scholar 

  40. Mathai, V.: Von Neumann invariants of Dirac operators, J. Funct. Anal. 152 (1998), 1-21.

    Google Scholar 

  41. Mathai, V. and Rothenberg, M.: On the homotopy invariance of L 2-torsion for covering spaces, Proc. Amer. Math. Soc. 126 (1998), 887-897.

    Google Scholar 

  42. Mishchenko, A. S.: Representations of compact groups on Hilbert modules over C*-algebras (Russ./Engl.), Trudy Mat. Inst. Steklov 166 (1984), 161-176 / Proc. Steklov Inst. Math. 166 (1986), 179-195.

    Google Scholar 

  43. Paschke, W. L.: Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468.

    Google Scholar 

  44. Raeburn, I. and Williams, D. P.: Morita Equivalence and Continuous Trace C*-algebras, Math. Surveys and Monogr. 60, Amer. Math. Soc., Providence, RI, 1998.

    Google Scholar 

  45. Reich, H.: Group von Neumann algebras and related algebras, Ph.D. Thesis, Universität Göttingen, Göttingen, 1998.

    Google Scholar 

  46. Rieffel, M. A.: Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96.

    Google Scholar 

  47. Schick, Th.: Analysis on a-Manifolds of Bounded Geometry, Hodge-De Rham Isomorphism and L 2 -Index Theorem, Ph.D. thesis, Univ. Mainz, Mainz, 1996, Shaker Verlag, Aachen, 1996.

    Google Scholar 

  48. Schick, Th.: Bounded geometry and L 2-index theorems, Preprint, Universität Münster, Münster, Germany, 1998.

    Google Scholar 

  49. Singer, I. M.: Some remarks on operator theory and index theory, In: K-theory and Operator Algebras, Proc. Conf., Univ. Georgia, Athens, GA, 1975, Lecture Notes in Math. 575, Springer, Berlin, 1977, pp. 128-138.

    Google Scholar 

  50. Takesaki, M.: Theory of Operator Algebras I, Springer, New York, 1979.

    Google Scholar 

  51. Wegge-Olsen, N. E.: K-theory and C*-algebras-a Friendly Approach, Oxford Univ. Press, Oxford, 1993.

    Google Scholar 

Download references

Authors
  1. Michael Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frank, M. Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants. Acta Applicandae Mathematicae 68, 227–242 (2001). https://doi.org/10.1023/A:1012029129487

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012029129487

Search

Navigation