TY - JOUR

T1 - Separable representations, KMS states, and wavelets for higher-rank graphs

AU - Farsi, Carla

AU - Gillaspy, Elizabeth

AU - Kang, Sooran

AU - Packer, Judith A.

N1 - Funding Information:
This work was partially supported by a grant from the Simons Foundation (# 316981 to Judith Packer). The third author would like to thank the Department of Mathematics of University of Colorado, Boulder for hosting her during the initial portion of this research.
Publisher Copyright:
© 2015.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C*(Λ) on certain separable Hilbert spaces of the form L2(X, μ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation of C*(Λ) on L2(Λ∞, M), where M is the Perron-Frobenius probability measure on the infinite path space Λ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C*(Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of C*(Λ) on L2(X, μ), where X is a fractal subspace of [0, 1] by embedding Λ∞ into [0, 1] as a fractal subset X of [0, 1]. Moreover, when the Radon-Nikodym derivatives of a Λ-semibranching function system are constant, we show that we can associate to it a KMS state for C*(Λ). Finally, we construct a wavelet system for L2(Λ∞, M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

AB - Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C*(Λ) on certain separable Hilbert spaces of the form L2(X, μ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation of C*(Λ) on L2(Λ∞, M), where M is the Perron-Frobenius probability measure on the infinite path space Λ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C*(Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of C*(Λ) on L2(X, μ), where X is a fractal subspace of [0, 1] by embedding Λ∞ into [0, 1] as a fractal subset X of [0, 1]. Moreover, when the Radon-Nikodym derivatives of a Λ-semibranching function system are constant, we show that we can associate to it a KMS state for C*(Λ). Finally, we construct a wavelet system for L2(Λ∞, M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

KW - Cantor-type fractal subspaces of [0, 1]

KW - Cuntz-Krieger C-algebras of k-graphs

KW - Separable representations

KW - Λ-semibranching function systems

UR - http://www.scopus.com/inward/record.url?scp=84943363764&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2015.09.003

DO - 10.1016/j.jmaa.2015.09.003

M3 - Article

AN - SCOPUS:84943363764

SN - 0022-247X

VL - 434

SP - 241

EP - 270

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

M1 - 19767

ER -