Wavelets and graph C ^∗-algebras

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.