Results in Mathematics

, Volume 63, Issue 1–2, pp 241–250 | Cite as

Dilation of Dual Frame Pairs in Hilbert C*-Modules

  • Deguang Han
  • Wu Jing
  • David Larson
  • Pengtong Li
  • Ram N. Mohapatra


We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.

Mathematics Subject Classification (2010)

Primary 46L08 Secondary 42C15 46H25 


Frames Riesz bases dilation dual frame pairs frame vectors unitary groups Hilbert C*-modules 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Casazza P., Han D., Larson D.: Frames in Banach spaces. Contemp. Math. 247, 149–181 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Christensen O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2002)Google Scholar
  3. 3.
    Duffin R., Schaeffer A.: A class of nonhamonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Frank M., Larson D.: Modular frames for Hilbert C*-modules and symmetric approximation of frames. Proc. SPIE 4119, 325–336 (2000)CrossRefGoogle Scholar
  5. 5.
    Frank M., Larson D.: Frames in Hilbert C*-modules and C*-algebras. J. Oper. Theory 48, 273–314 (2002)MathSciNetMATHGoogle Scholar
  6. 6.
    Han D., Jing W., Larson D., Mohapatra R.: Riesz bases and their dual modular frames in Hilbert C*-modules. J. Math. Anal. Appl. 343, 246–256 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Han D., Jing W., Mohapatra R.: Perturbation of frames and Riesz bases in Hilbert C*-modules. Linear Algebra Appl. 431, 746–759 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Han D., Larson D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147(697), 1–94 (2000)MathSciNetGoogle Scholar
  9. 9.
    Jing, W.: Frames in Hilbert C*-modules, Ph. D. Thesis, University of Central Florida, Orlando (2006)Google Scholar
  10. 10.
    Jing W., Han D., Mohapatra R.: Structured Parseval frames in Hilbert C*-modules. Contemp. Math. 414, 275–287 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kasparov G.: Hilbert C*-modules: the theorem of Stinespring and Voiculescu. J. Oper. Theory 4, 133–150 (1980)MathSciNetMATHGoogle Scholar
  12. 12.
    Lance, E.: Hilbert C*-modules—a Toolkit for Operator Algebraists. London Mathematical Society lecture Note Series, vol. 210, Cambridge University Press, Cambridge (1995)Google Scholar
  13. 13.
    Li H.: A Hilbert C*-module admitting no frames. Bull. Lond. Math. Soc. 43, 388–394 (2010)CrossRefGoogle Scholar
  14. 14.
    Manuilov V., Troisky E.: Hilbert C*-Modules. American Mathematical Society, Providence (2005)Google Scholar
  15. 15.
    Packer J., Rieffel M.: Wavelet filter functions, the matrix completion problem, and projective modules over \({C({\mathbb T}^n)}\). J. Fourier Anal. Appl. 9, 101–116 (2003)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Packer J., Rieffel M.: Projective multi-resolution analyses for \({L^2({\mathbb R}^2)}\). J. Fourier Anal. Appl. 10, 439–464 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Paley R., Wiener N.: Fourier Transforms in the Complex Domains, AMS Colloquium Publications, vol. 19. American Mathematical Society, Providence (1987)Google Scholar
  18. 18.
    Paschke W.: Inner product modules over B*-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)MathSciNetMATHGoogle Scholar
  19. 19.
    Raeburn I., Thompson S.: Countably generated Hilbert modules, the Kasparov stabilization theorem, and frames in Hilbert modules. Proc. Am. Math. Soc. 131, 1557–1564 (2003)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Wegge-Olsen N.: K-Theory and C*-Algebras—a Friendly. Approach Oxford University Press, Oxford (1993)Google Scholar
  21. 21.
    Wood P.: Wavelets and Hilbert modules. J. Fourier Anal. Appl. 10, 573–598 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Deguang Han
    • 1
  • Wu Jing
    • 2
  • David Larson
    • 3
  • Pengtong Li
    • 4
  • Ram N. Mohapatra
    • 1
  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA
  2. 2.Department of Mathematics and Computer ScienceFayetteville State UniversityFayettevilleUSA
  3. 3.Department of MathematicTexas A&M UniversityCollege StationUSA
  4. 4.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China

Personalised recommendations