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Wavelets and Graph C ∗-Algebras

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Excursions in Harmonic Analysis, Volume 5

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Here we give an overview on the connection between wavelet theory and representation theory for graph C ∗-algebras, including the higher-rank graph C ∗-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In Farsi et al. (J Math Anal Appl 425:241–270, 2015), we introduced the “cubical wavelets” associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of Hammond et al. (Appl Comput Harmon Anal 30:129–150, 2011) to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.

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Notes

  1. 1.

    Recall that an isometry in \(B(\mathcal{H})\) is an operator T such that T ∗ T = I; a partial isometry S satisfies S = SS ∗ S. A projection in \(B(\mathcal{H})\) is an operator that is both self-adjoint and idempotent.

  2. 2.

    We think of \(\mathbb{N}^{k}\) as a category with one object, namely 0, and with composition of morphisms given by addition.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Colorado at Boulder, Boulder, CO, 80309-0395, USA

    Carla Farsi & Judith Packer

  2. Mathematisches Institut der Universität Münster, Einsteinstrasse 62, Münster, 48149, Germany

    Elizabeth Gillaspy

  3. Department of Mathematics, Sungkyunkwan University, Seobu-ro 2066, Jangan-gu, Suwon, 16419, Republic of Korea

    Sooran Kang

Authors
  1. Carla Farsi
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  2. Elizabeth Gillaspy
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  3. Sooran Kang
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  4. Judith Packer
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Corresponding author

Correspondence to Carla Farsi .

Editor information

Editors and Affiliations

  1. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Radu Balan

  2. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    John J. Benedetto

  3. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Wojciech Czaja

  4. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Matthew Dellatorre

  5. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Kasso A. Okoudjou

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Farsi, C., Gillaspy, E., Kang, S., Packer, J. (2017). Wavelets and Graph C ∗-Algebras. In: Balan, R., Benedetto, J., Czaja, W., Dellatorre, M., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 5. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-54711-4_3

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