Skip to main content
SpringerLink
Log in

HilbertC *- andW *-modules and their morphisms

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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

References

  1. N. N. Luzin,Theory of Functions of a Real Variable [in Russian], Moscow (1948).

  2. V. M. Manuilov, “On eigenvalues of a perturbed Schrödinger operator in a magnetic field with irrational magnetic flow,”Funkts. Anal. Prilozh.,28, 57–60 (1994).

    MATH  MathSciNet  Google Scholar 

  3. A. Ya. Khelemskii,Banach and Polynormed Algebras. General Theory. Reperesentations. Homologies [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  4. C. Anantharaman-Delaroche and J.-F. Havet, “On approximate factorizations of completely positive maps,”J. Funct. Anal.,90, 411–428 (1990).

    MathSciNet  Google Scholar 

  5. S. Baaj and P. Julg, “Théorie bivariante de Kasparov et opérateurs non bornés dans lesC *-modules hilbertiens,”C. R. Acad. Sci. Paris, Sér. 1,296, 875–878 (1983).

    MathSciNet  Google Scholar 

  6. S. Baaj and G. Skandalis, “C *-algèbres de Hopf et théorie de Kasparov équivariante,”K-theory,2, 683–721 (1989).

    Article  MathSciNet  Google Scholar 

  7. M. Baillet, Y. Denizeau Y., and J.-F. Havet, “Indice d'une esperance conditionelle,”Compos. Math.,66, 199–236 (1988).

    MathSciNet  Google Scholar 

  8. A. Barut and R. R⩯czka,Theory of Group Representations and Applications, PWN-Polish Scientific Publishers, Warszawa (1977).

    Google Scholar 

  9. B. Blackadar,K-Theory for Operator Algebras, Springer-Verlag, New York (1986).

    Google Scholar 

  10. D. P. Blecher, “A new approach to HilbertC *-modules,”Math. Ann. 307, 253–290 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  11. D. P. Blecher, “A generalization of Hilbert modules,”J. Funct. Anal.,136, 365–421 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  12. O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics, I, Springer-Verlag, New York-Berlin-Heidelberg (1981).

    Google Scholar 

  13. L. G. Brown “Stable isomorphism of hereditary subalgebras ofC *-algebras,”Pacific J. Math.,71, 335–348 (1977).

    MATH  MathSciNet  Google Scholar 

  14. L. G. Brown, P. Green, and M. A. Rieffel, “Stable isomorphism and strong Morita equivalence ofC *-algebras,”Pacific J. Math.,71, 349–363 (1977).

    MathSciNet  Google Scholar 

  15. A. Connes,Noncommutative Geometry, Academic Press, San Diego (1994).

    Google Scholar 

  16. J. Cuntz, “A survey on some aspects of noncommutative geometry,”Jahresbericht der DMV,95, 60–84 (1993).

    MATH  MathSciNet  Google Scholar 

  17. J. Dixmier,Les C *-Algèbres et Leurs Représentations, Gauthier-Villars, Paris (1964).

    Google Scholar 

  18. J. Dixmier,Les Algèbres d'Operateurs dans l'Espace Hilbertien, Gauthier-Villars, Paris (1969).

    Google Scholar 

  19. J. Dixmier and A. Douady, “Champs continus d'espaces Hilbertiens,”Bull. Soc. Math. France,91, 227–284 (1963).

    MathSciNet  Google Scholar 

  20. M. J. Dupré and P. A. Fillmore, “Triviality theorems for Hilbert modules,” In:Topics in Modern Operator Theory. 5th Internat. Conf. on Operator Theory. Timisoara and Herculane (Romania), 1980, Birkhäuser Verlag, Basel-Boston-Stuttgart, (1981), pp. 71–79.

    Google Scholar 

  21. G. A. Elliott, K. Saitô, and J. D. M. Wright, “EmbeddingAW *-algebras as double commutants in type I algebras,”J. London, Math. Soc.,28, 376–384 (1983).

    MathSciNet  Google Scholar 

  22. M. Frank, “Self-duality andC *-reflexivity of HilbertC *-modules,”Z. Anal. Anw.,9, 165–176 (1990).

    Google Scholar 

  23. M. Frank,Geometrical aspects of Hilbert C *-modules, Københavns Universitet. Matematisk Institut. Preprint Series No. 22 (1993).

  24. M. Frank,Hilbert C *-modules and related subjects—a guided reference overview, Leipzig University. ZHS-NTZ preprint No. 13 (1996).

  25. M. Frank, V. M. Manuilov, and E. V. Troitsky, “On conditional expectations arising from group actions,”Z. Anal. Anw.,16, 831–850 (1997).

    MathSciNet  Google Scholar 

  26. M. Frank and E. V. Troitsky, “Lefschetz numbers and the geometry of operators inW *-modules,”Funkts. Anal. Prilozh.,30, No. 4, 45–57 (1996).

    Google Scholar 

  27. A. Irmatov, “On a new topology in the space of Fredholm operators,”Ann. Global Anal. Geom.,7, No. 2, 93–106 (1989).

    MATH  MathSciNet  Google Scholar 

  28. B. E. Johnson, “Centralisers and operators reduced by maximal ideals,”J. London Math. Soc.,43, 231–233 (1968).

    MATH  MathSciNet  Google Scholar 

  29. R. V. Kadison and J. R. Ringrose,Fundamentals of the Theory of Operator Algebras, Vol. 1, 2, Grad. Stud. Math. Amer. Math. Soc. (1997).

    Google Scholar 

  30. I. Kaplansky, “Modules over operator algebras,”Amer. J. Math.,75, 839–858 (1953).

    MATH  MathSciNet  Google Scholar 

  31. M. Karoubi,K-Theory. An Introduction, Springer-Verlag, Berlin-Heidelberg-New York (1978).

    Google Scholar 

  32. G. G. Kasparov, “Topological invariants of elliptic operators, I:K-homology,”Izv. Akad. Nauk SSSR, Ser. Mat.,39, 796–838 (1975).

    MATH  MathSciNet  Google Scholar 

  33. G. G. Kasparov,Izv. Akad. Nauk SSSR, Ser. Mat.,44, 571–636 (1980).

    MATH  MathSciNet  Google Scholar 

  34. G. G. Kasparov, “HilbertC *-modules: Theorems of Stinespring and Voiculescu,”J. Oper. Theory,4, 133–150, (1980).

    MATH  MathSciNet  Google Scholar 

  35. G. G. Kasparov, “Novikov's conjecture on higher signatures: the operatorK-theory approach,”Contemp. Math.,145, 79–99 (1993).

    MATH  MathSciNet  Google Scholar 

  36. A. Kumjian, “On equivariant sheaf cohomology and elementaryC *-bundles,”J. Oper. Theory,20, 207–240 (1988).

    MATH  MathSciNet  Google Scholar 

  37. E. C. Lance, “On nuclearC *-algebras,”J. Funct. Anal.,12, 157–176 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  38. E. C. Lance, “HilbertC *-modules—a toolkit for operator algebraists,” In:London Math. Soc. Lect. Note, Vol. 210, Cambridge University Press, England (1995).

    Google Scholar 

  39. N. P. Landsman, “Rieffel induction as generalized quantum Marsden-Weinstein reduction,”J. Geom. Phys.,15, 285–319 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  40. H. Lin, “Bounded module maps and pure completely positive maps,”J. Oper. Theory,26, 121–138 (1991).

    MATH  Google Scholar 

  41. H. Lin, “Injective HilbertC *-modules,”Pacif. J. Math.,154, 131–164 (1992).

    MATH  Google Scholar 

  42. B. Magajna, “HilbertC *-modules in which all closed submodules are complemented,”Proc. Amer. Math. Soc.,125, 849–852 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  43. V. M. Manuilov, “Diagonalization of compact operators in Hilbert modules over, finiteW *-algebras,”Ann. Global Anal. Geom.,13, 207–226 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  44. J. A. Mingo, “On the contractibility of the general linear group of Hilbert space over aC *-algebra,”J. Integral Equations Operator Theory,5, 888–891 (1982).

    MATH  MathSciNet  Google Scholar 

  45. J. Mingo and W. Phillips, “Equivariant triviality theorems for HilbertC *-modules,”Proc. Amer. Math. Soc.,91, 225–230 (1984).

    MathSciNet  Google Scholar 

  46. A. S. Mishchenko, “Banach algebras, pseudodifferential operators and their applications toK-theory,”Usp. Mat. Nauk,34, No. 6, 67–79 (1979).

    MATH  MathSciNet  Google Scholar 

  47. A. S. Mishchenko, “Representations of compact groups on Hilbert modules overC *-algebras,”Tr. Mat. Inst. Akad Nauk SSSR,166, 161–176 (1984).

    MATH  MathSciNet  Google Scholar 

  48. A. S. Mishchenko and A. T. Fomenko, “The index of elliptic operators overC *-algebras,”Izv. Akad. Nauk SSSR. Ser. Mat.,43, 831–859 (1979).

    MathSciNet  Google Scholar 

  49. G. D. Mostow, “Cohomology of topological groups and solvmanifolds,”Ann. Math.,73, 20–48 (1961).

    MATH  MathSciNet  Google Scholar 

  50. G. J. Murphy,C *-Algebras and Operator Theory, Academic Press, San Diego, (1990).

    Google Scholar 

  51. F. J. Murray and J. von Neumann, “On rings of operators,”Ann. Math.,37, 116–229 (1936).

    Google Scholar 

  52. W. L. Paschke, “Inner product modules overB *-algebras,”Trans. Amer. Math. Soc.,182, 443–468 (1973).

    MATH  MathSciNet  Google Scholar 

  53. W. L. Paschke, “The doubleB-dual of an inner product module over aC *-algebra,”Can. J. Math.,26, 1272–1280 (1974).

    MATH  MathSciNet  Google Scholar 

  54. W. L. Paschke, “Integrable group actions on von Neumann algebras,”Math. Scand.,40, 234–248 (1977).

    MATH  MathSciNet  Google Scholar 

  55. G. K. Pedersen,C *-Algebras and Their Automorphism Groups, Academic Press, London-New York-San Francisco (1979).

    Google Scholar 

  56. M. A. Rieffel, “Induced representations ofC *-algebras,”Adv. Math.,13, 176–257 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  57. M. A. Rieffel, “Morita equivalence forC *-algebras andW *-algebras,”J. Pure, Appl. Alg.,5, 51–96 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  58. M. A. Rieffel, “Strong Morita equivalence of certain transformation groupC *-algebras,”Math. Ann.,222, 7–22 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  59. W. Rudin,Functional Analysis, McGraw-Hill, New York (1973).

    Google Scholar 

  60. S. Sakai,C *-Algebras and W *-Algebras, Springer-Verlag, Berlin-New York (1971).

    Google Scholar 

  61. Yu. P. Solovyov and E. V. Troitsky,C *-Algebras and Elliptic Operators in Differential Topology [in Russian], Factorial, Moscow (1996).

    Google Scholar 

  62. M. Takesaki,Theory of Operator Algebras, 1, Springer-Verlag, New York-Heidelberg-Berlin (1979).

    Google Scholar 

  63. V. A. Trofimov, “Reflexivity of Hilbert modules over the algebra of compact operators with adjoint identity,”Vestn. Mosk. Univ. Ser. I: Mat.-Mekh., No. 5, 60–64 (1986).

    MATH  MathSciNet  Google Scholar 

  64. V. A. Trofimov, “Reflexive and self-dual Hilbert modules over someC *-algebras,”Usp. Mat. Nauk,42, No. 2, 247–248 (1987).

    MATH  MathSciNet  Google Scholar 

  65. E. V. Troitsky, “Contractibility of the full general linear group of the Hilbert C*-module,l 2(A),”Funkts. Anal. Prilozh.,20, No. 4, 58–64 (1986).

    Google Scholar 

  66. E. V. Troitsky, “The eqivariant index of elliptic operators overC *-algebras,”Izv. Akad. Nauk SSSR, Ser. Mat.,50, No. 4, 849–865 (1986).

    MathSciNet  Google Scholar 

  67. E. V. Troitsky, “Orthogonal complements and endomorphisms of Hilbert modules andC *-elliptic complexes,” In:Novikov Conjectures, Index Theorems and Rigidity, Part 2, London Math. Soc. Lect. Notes Series. Vol. 227 (1995), pp. 309–331.

  68. E. V. Troitsky, “Geometry and topology of operators on HilbertC *-modules,”this issue.

  69. Y. Watatani,Index for C *-subalgebras (Mem. Amer. Math. Soc., Vol. 424 Providence, AMS (1990).

  70. S. L. Woronowicz, “Unbounded elements affiliated withC *-algebras and noncompact quantum groups,”Commun. Math. Phys.,136, 399–432 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  71. S. L. Woronowicz and K. Napiórkowski, “Operator theory in theC *-algebra framework,”Rep. Math. Phys., 353–371 (1992).

  72. N. E. Wegge-Olsen,K-Theory and C *-Algebras, Oxford University Press (1993).

Download references

Authors
  1. V. M. Manuilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. V. Troitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 53, Functional Analysis-6, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Manuilov, V.M., Troitsky, E.V. HilbertC *- andW *-modules and their morphisms. J Math Sci 98, 137–201 (2000). https://doi.org/10.1007/BF02355447

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02355447

Search

Navigation