L2-invariants and Rank Metric

  • Andreas Thom
Part of the Trends in Mathematics book series (TM)


We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L 2-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invariance of L 2-Betti numbers under orbit equivalence.


Cauchy Sequence Full Subcategory Projective Object Complete Module Local Isomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ati76]
    M.F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. In Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), pages 43–72. Astérisque, No. 32–33. Soc. Math. France, Paris, 1976.Google Scholar
  2. [CG86]
    Jeff Cheeger and Mikhael Gromov. L 2-cohomology and group cohomology. Topology, 25(2):189–215, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [CS05]
    Alain Connes and Dimitri Shlyakhtenko. L 2-homology for von Neumann algebras. J. Reine Angew. Math., 586:125–168, 2005.zbMATHMathSciNetGoogle Scholar
  4. [FM77a]
    Jacob Feldman and Calvin C. Moore. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc., 234(2):289–324, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [FM77b]
    Jacob Feldman and Calvin C. Moore. Ergodic equivalence relations, cohomology, and von Neumann algebras. II. Trans. Amer. Math. Soc., 234(2):325–359, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Gab02a]
    Damien Gaboriau. Invariants l 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci., (95):93–150, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [Gab02b]
    Damien Gaboriau. On orbit equivalence of measure preserving actions. In Rigidity in dynamics and geometry (Cambridge, 2000), pages 167–186. Springer, Berlin, 2002.Google Scholar
  8. [Lüc02]
    Wolfgang Lück. L 2-invariants: theory and applications to geometry and K-theory, volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2002.Google Scholar
  9. [Pas97]
    William L. Paschke. L 2-homology over traced *-algebras. Trans. Amer. Math. Soc., 349(6):2229–2251, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Ped79]
    Gert K. Pedersen. C*-algebras and their automorphism groups, volume 14 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1979.Google Scholar
  11. [Sau03]
    Roman Sauer. Power series over the group ring of a free group and applications to Novikov-Shubin invariants. In High-dimensional manifold topology, pages 449–468. World Sci. Publishing, River Edge, NJ, 2003.CrossRefGoogle Scholar
  12. [Sau05]
    Roman Sauer. L 2-Betti numbers of discrete measured groupoids. Internat. J. Algebra Comput., 15(5–6):1169–1188, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Tho06a]
    Andreas Thom. L 2-Betti numbers for sub-factors. to appear in the Journal of Operator algebras, arXiv:math.OA/0601408, 2006.Google Scholar
  14. [Tho06b]
    Andreas Thom. L 2-cohomology for von Neumann algebras. to appear in GAFA, arXiv:math.OA/0601447, 2006.Google Scholar
  15. [Voi05]
    Dan Voiculescu. Free probability and the von Neumann algebras of free groups. Reports on Mathematical Physics, 55(1):127–133, 2005.CrossRefMathSciNetGoogle Scholar
  16. [Wei94]
    Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Andreas Thom
    • 1
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenGermany

Personalised recommendations