TY - CHAP

T1 - Wavelets and spectral triples for fractal representations of Cuntz algebras

AU - Farsi, C.

AU - Gillaspy, E.

AU - Julien, A.

AU - Kang, S.

AU - Packer, J.

N1 - Funding Information:
This work was partially supported by a grant from the Simons Foundation (#316981 to Judith Packer).
Publisher Copyright:
© 2017 American Mathematical Society.

PY - 2017

Y1 - 2017

N2 - In this article we provide an identification between the wavelet decompositions of certain fractal representations of C∗ -algebras of directed graphs, as introduced by M. Marcolli and A. Paolucci (2011), and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of Cuntz C∗ -algebras OD. In particular, in this setting we use results of J. Pearson and J. Bellissard (2009), and A. Julien and J. Savinien (2011), to construct first the spectral triple and then the Laplace–Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace–Beltrami operators for the equal weight case for representations of OD, and in the uneven weight case for certain representations of O2, and show how the eigenspaces and wavelet subspaces at different levels (first constructed in C. Farsi, E. Gillaspy, S. Kang, and J. Packer) are related.

AB - In this article we provide an identification between the wavelet decompositions of certain fractal representations of C∗ -algebras of directed graphs, as introduced by M. Marcolli and A. Paolucci (2011), and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of Cuntz C∗ -algebras OD. In particular, in this setting we use results of J. Pearson and J. Bellissard (2009), and A. Julien and J. Savinien (2011), to construct first the spectral triple and then the Laplace–Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace–Beltrami operators for the equal weight case for representations of OD, and in the uneven weight case for certain representations of O2, and show how the eigenspaces and wavelet subspaces at different levels (first constructed in C. Farsi, E. Gillaspy, S. Kang, and J. Packer) are related.

KW - Laplace beltrami operators

KW - Spectral triples

KW - Ultrametric cantor set

KW - Wavelets

KW - Weighted bratteli diagrams

UR - http://www.scopus.com/inward/record.url?scp=85029562986&partnerID=8YFLogxK

U2 - 10.1090/conm/687/13795

DO - 10.1090/conm/687/13795

M3 - Chapter

AN - SCOPUS:85029562986

T3 - Contemporary Mathematics

SP - 103

EP - 133

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -