Codes and Expansions (CodEx) Seminar

Judith Packer (University of Colorado Boulder):
Wavelets coming from representations of graph \(C^*\)-algebras

A \(C^*\)-algebra corresponding to a finite directed graph can be described in a fairly straightforward way by using two finite sets of projections and partial isometries on a Hilbert space that satisfy relations determined by the directed graph. Here I describe a construction that has its origins in a 2011 paper of M. Marcolli and A. Paolucci, who constructed "wavelets" for Cuntz-Krieger \(C^*\)-algebras. We construct "wavelets" on Cantor sets whose generalized "translates" and "dilates" by the partial isometries and projections related to a finite, source-free directed graph can be used to form an orthonormal basis for an \(L^2\)-space of the Cantor set coming from the infinite path space for the graph. This method due to Marcolli and Paolucci uses multi-resolution analysis (MRA), and can be described with linear algebra, too. It is then noted how the wavelet subspaces of these MRAs can be also described using the language of spectral triples. This is based on joint work with C. Farsi, E. Gillaspy, A. Julien, and S. Kang.