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C*-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states

Zahra Afsar, Nathan Brownlowe, Jacqui Ramagge, Michael F. Whittaker

Abstract

We introduce the notion of a self-similar action of a groupoid $G$ on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodules, and thereby obtain corresponding universal Nica--Toeplitz and Cuntz--Pimsner algebras. We consider natural actions of the real numbers on both algebras and study the KMS states of the associated dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica--Toeplitz algebra in terms of traces on the full $C^*$-algebra of $G$. We prove that if the graph is $G$-aperiodic and the action satisfies a finite-state condition, then there is a unique KMS state on the Cuntz--Pimsner algebra.

C*-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states

Abstract

We introduce the notion of a self-similar action of a groupoid on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodules, and thereby obtain corresponding universal Nica--Toeplitz and Cuntz--Pimsner algebras. We consider natural actions of the real numbers on both algebras and study the KMS states of the associated dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica--Toeplitz algebra in terms of traces on the full -algebra of . We prove that if the graph is -aperiodic and the action satisfies a finite-state condition, then there is a unique KMS state on the Cuntz--Pimsner algebra.

Paper Structure

This paper contains 21 sections, 39 theorems, 274 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids
  4. $k$-graphs
  5. Product systems of Hilbert bimodules
  6. $C^*$-algebras of product systems of Hilbert bimodules
  7. Self-similar groupoid actions on $k$-graphs
  8. Coloured-graph automata
  9. $k$-coloured graphs and their $k$-graphs
  10. Automata associated to coloured graphs
  11. Product systems from self-similar actions
  12. $C^*$-algebras of self-similar groupoid actions on $k$-graphs
  13. An algebraic characterisation of KMS states of $\mathcal{T}(G,\Lambda)$
  14. KMS states of $\mathcal{T}(G,\Lambda)$ for large inverse temperatures
  15. The $G$-periodicity group
  16. ...and 6 more sections

Key Result

Lemma 3.2

Let $\Lambda$ be a $k$-graph. Then $\operatorname{PIso}(\Lambda)$ is a groupoid, with unit space $\Lambda^0$; range map $\operatorname{cod}$ and source map $\operatorname{dom}$; and composition and inverse given by composition and inverse of functions, respectively.

Figures (2)

  • Figure 1: The 2-graph $\Lambda$ of Example \ref{['ex: single vertex']}.
  • Figure 2: The 2-graph $\Lambda$ of Example \ref{['ex: bascilica2']}.

Theorems & Definitions (103)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Example 3.5
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • ...and 93 more