Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Casazza P., Han D., Larson D.: Frames in Banach spaces. Contemp. Math. 247, 149–181 (1999)
Christensen O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2002)
Duffin R., Schaeffer A.: A class of nonhamonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Frank M., Larson D.: Modular frames for Hilbert C*-modules and symmetric approximation of frames. Proc. SPIE 4119, 325–336 (2000)
Frank M., Larson D.: Frames in Hilbert C*-modules and C*-algebras. J. Oper. Theory 48, 273–314 (2002)
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)
Han D., Jing W., Mohapatra R.: Perturbation of frames and Riesz bases in Hilbert C*-modules. Linear Algebra Appl. 431, 746–759 (2009)
Han D., Larson D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147(697), 1–94 (2000)
Jing, W.: Frames in Hilbert C*-modules, Ph. D. Thesis, University of Central Florida, Orlando (2006)
Jing W., Han D., Mohapatra R.: Structured Parseval frames in Hilbert C*-modules. Contemp. Math. 414, 275–287 (2006)
Kasparov G.: Hilbert C*-modules: the theorem of Stinespring and Voiculescu. J. Oper. Theory 4, 133–150 (1980)
Lance, E.: Hilbert C*-modules—a Toolkit for Operator Algebraists. London Mathematical Society lecture Note Series, vol. 210, Cambridge University Press, Cambridge (1995)
Li H.: A Hilbert C*-module admitting no frames. Bull. Lond. Math. Soc. 43, 388–394 (2010)
Manuilov V., Troisky E.: Hilbert C*-Modules. American Mathematical Society, Providence (2005)
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)
Packer J., Rieffel M.: Projective multi-resolution analyses for \({L^2({\mathbb R}^2)}\). J. Fourier Anal. Appl. 10, 439–464 (2004)
Paley R., Wiener N.: Fourier Transforms in the Complex Domains, AMS Colloquium Publications, vol. 19. American Mathematical Society, Providence (1987)
Paschke W.: Inner product modules over B*-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)
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)
Wegge-Olsen N.: K-Theory and C*-Algebras—a Friendly. Approach Oxford University Press, Oxford (1993)
Wood P.: Wavelets and Hilbert modules. J. Fourier Anal. Appl. 10, 573–598 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Han, D., Jing, W., Larson, D. et al. Dilation of Dual Frame Pairs in Hilbert C*-Modules. Results. Math. 63, 241–250 (2013). https://doi.org/10.1007/s00025-011-0195-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-011-0195-9