Abstract
Our paper aims to extend fusion frames to Hilbert C\(^{*}\)-modules. We introduce \(^*\)-fusion frames associated to weighted sequences of closed orthogonally complemented submodules, showcasing similarities to Hilbert space frames. Using Dragan S. Djordjevic’s distance, we define submodule angles and establish a new topology on the set of sequences of closed orthogonally complemented submodules. Relying on this topology, we obtain for our \(^*\)-fusion frames, some new perturbation results of topological and geometric character.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicable.
References
Aldroubi, A., Sun, Q., Tang, W.: p-Frames and shift invariant subspaces of \(L^{p}\). J. Fourier Anal. Appl. 7, 1–22 (2001)
Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)
Alijani, A., Dehghan, M.: \(*\)-frames in Hilbert \(C^*\)-modules. U.P.B. Sci. Bull. Ser. A (2011)
Arambašić, L.: On frames for countably generated Hilbert \(C^*\)-modules. Proc. Am. Math. Soc. 135, 469–478 (2007)
Bounader, N., Kabbaj, S.: \(*\)-g-frames in Hilbert \(C^*\)-modules. J. Math. Comput. Sci. 4(2), 246–256 (2014)
Casazza, P.G., Han, D., Larson, D.R.: Frames for Banach spaces. Contemp. Math. 247, 149–182 (1999)
Casazza, P.G., Kutyniok, G.: Frames of subspaces. In: Wavelets, Frames and Operator Theory (College Park, MD, 2003), pp. 87–113. Contemporary Mathematics. American Mathematical Society, Providence (2004)
Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25, 114–132 (2008)
Choquet, G., Feinstein, A.: Topology. Pure and Applied Mathematics, vol. 19. Academic Press, Cambridge (1966)
Christensen, O., Stoeva, D.T.: p-frames in separable Banach spaces. Adv. Comput. Math. 18, 117–126 (2003)
Daubechies, I., Grossman, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 275, 1271–1283 (1986)
Davidson, K.R.: \(C^*\)-Algebras by Example. Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)
Djordjevic̀, D.S.: Algebraic distance between submodules. Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathématiques) 44, 75–89 (2019)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Feichtinger, H.G., Gröchenig, K.: A unified approach to atomic decompositions via integrable group representations. In: Function Spaces and Applications, pp. 52–73. Springer, Berlin (1988)
Frank, M., Larson, D.R.: Frames in Hilbert \(C^*\)-modules and C*-algebras. J. Oper. Theory 48, 273–314 (2002)
Gabardo, J.P., Han, D.: Frames associated with measurable space. Adv. Comput. Math. 18, 127–147 (2003)
Gabor, D.: Theory of communication. J. Inst. Electr. Eng. 93, 429–457 (1946)
Khosravi, A., Khosravi, B.: Fusion frames and g-frames in Hilbert \(C^*\)-modules. Int. J. Wavelet Multiresolut. Inf. Process. 6, 433–446 (2008)
Lance, E.C.: Hilbert \(C^*\)-modules, a toolkit for operator algebraists, vol. 210. Cambridge University Press, Cambridge (1995)
Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert \(C^*\)-modules. J. Oper. Theory 71, 571–584 (2014)
Manuilov, V.M., Troitsky, E.V.: Hilbert C\(^*\)-Modules. Translations of Mathematical Monographs, vol. 226. American Mathematical Society, Providence (2005)
Murphy, G.J.: \(C^*\)-Algebras and Operator Theory. Academic Press, Cambridge (1990)
Shalit, O.M.: A First Course in Functional Analysis. CRC Press, Boca Raton (2017)
Young, R.M.: An Introduction to Nonharmonic Fourier Series, vol. 5, pp. 324–329 (1981)
Acknowledgements
The authors express their sincere gratitude to the anonymous referee for her / his helpful comments who has improved this work.
Author information
Authors and Affiliations
Contributions
The authors declare that they contributed equally to this work. Indeed, the author names are listed only in alphabetical order.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Communicated by Evgenij Troitsky.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Assila, N., Kabbaj, S. & Zoubeir, H. On \(^*\)-fusion frames for Hilbert \(C^*\)-modules. Adv. Oper. Theory 9, 35 (2024). https://doi.org/10.1007/s43036-024-00337-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-024-00337-6