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

Carla Farsi, Elizabeth Gillaspy, Sooran Kang, Judith A. Packer

Research output: Contribution to journalArticlepeer-review

Abstract

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, 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, M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

Original languageEnglish
Article number19767
Pages (from-to)241-270
Number of pages30
JournalJournal of Mathematical Analysis and Applications
Volume434
Issue number1
DOIs
StatePublished - Feb 1 2016

Keywords

  • Cantor-type fractal subspaces of [0, 1]
  • Cuntz-Krieger C-algebras of k-graphs
  • Separable representations
  • Λ-semibranching function systems

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