# Wavelets and Graph C ∗ -Algebras

@article{Farsi2017WaveletsAG, title={Wavelets and Graph C ∗ -Algebras}, author={Carla Farsi and Elizabeth A. Gillaspy and Sooran Kang and Judith A. Packer}, journal={arXiv: Operator Algebras}, year={2017}, pages={35-86} }

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…

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## References

SHOWING 1-10 OF 64 REFERENCES

### Wavelets on Graphs via Spectral Graph Theory

- Computer Science, MathematicsArXiv
- 2009

### HOMOLOGY FOR HIGHER-RANK GRAPHS AND TWISTED

- Mathematics
- 2011

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homol- ogy of a k-graph coincides with the…

### Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scaleN

- Mathematics
- 1996

AbstractIn this paper we show how wavelets originating from multiresolution analysis of scaleN give rise to certain representations of the Cuntz algebrasON, and conversely how the wavelets can be…

### On twisted higher-rank graph C*-algebras

- Mathematics
- 2011

We define the categorical cohomology of a k-graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This…

### A family of 2-graphs arising from two-dimensional subshifts

- MathematicsErgodic Theory and Dynamical Systems
- 2009

Abstract Higher-rank graphs (or k-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz–Krieger C*-algebras of Robertson and Steger. Here we consider a…

### KMS states on the C*-algebras of reducible graphs

- Mathematics
- 2014

We consider the dynamics on the C*-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani proved that if the vertex matrix of the…

### Spectral graph theory

- MathematicsZeta and 𝐿-functions in Number Theory and Combinatorics
- 2019

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall…

### Higher Rank Graph C-Algebras

- Mathematics
- 2000

Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of…

### THE ANALYTIC ALGEBRAS OF HIGHER RANK GRAPHS

- MathematicsMathematical Proceedings of the Royal Irish Academy
- 2006

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hubert space and creation operators that are partial…