Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures

TL;DR

This paper explores new representations of Cuntz-Krieger algebras using semibranching function systems linked to stationary Bratteli diagrams, establishing conditions for their equivalence and classifying certain monic representations.

Contribution

It introduces a framework connecting semibranching function systems, Bratteli diagrams, and Cuntz-Krieger algebra representations, including classification results and graph-based isomorphism conditions.

Findings

Equivalent measures produce equivalent representations

Isomorphic graphs generate isomorphic semibranching systems

Classification of monic representations up to unitary equivalence

Abstract

We study a new class of representations of the Cuntz-Krieger algebras $\mathcal O_A$ constructed by semibranching function systems, naturally related to stationary Bratteli diagrams. The notion of isomorphic semibranching function systems is defined and studied. We show that any isomorphism of such systems implies the equivalence of the corresponding representations of Cuntz-Krieger algebra $\mathcal O_A$. In particular, we show that equivalent measures generate equivalent representations of $\mathcal O_A$. We use Markov measures which are defined on the path space of stationary Bratteli diagrams to construct isomorphic representations of $\mathcal O_A$. To do this, we associate a (strongly) directed graph to a stationary (simple) Bratteli diagram, and show that isomorphic graphs generate isomorphic semibranching function systems. We also consider a class of monic representations of the Cuntz-Krieger algebras, and we classify them up to unitary equivalence. Several examples that illustrate the results are included in the paper.