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

Abstract

Abstract. We study a new class of representations of the Cuntz-Krieger algebras OA 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 OA. In particular, we show that equivalent measures generate equivalent representations of OA. We use Markov measures which are defined on the path space of stationary Bratteli diagrams to construct isomorphic representations of OA. 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.

Monic representations of finite higher-rank graphs

“…This condition is crucial to making sense of the representation of C * (Λ) associated to the Λ-semibranching function system (see Theorem 2.10 below). As established in Theorem 2.22 of [5], in order to obtain a representation of a 1-graph algebra C * (Λ) from a semibranching function system, one must also assume that the semibranching function system satisfies condition (C-K).…”

“…[17,18,40,27,28,26]), representations of Cuntz-Krieger algebras have been linked to fractals and Cantor sets [48,35,25,26] and to the endomorphism group of a Hilbert space [8,39]. Indeed, the astonishing goal of identifying both discrete and continuous series of representations of Cuntz (and to some extent Cuntz-Krieger) C * -algebras, was accomplished in [19,20,5], building on the pioneering results of [7].…”

“…In this paper, we refine the Λ-semibranching function systems into a crucial technical tool for studying monic representations, namely the Λ-projective systems of Definition 3.1. Monic representations also have strong connections to Markov measures [19,5] and Nelson's universal representation of an abelian algebra [42]. Indeed, studying monic representations enables us to convert questions about the representation theory of higher-rank graphs into measure-theoretic questions (see Theorems 3.11 and 3.13 below).…”

Dirichlet Forms and Ultrametric Cantor Sets Associated to Higher-Rank Graphs

“…The measure M is often called the Perron-Frobenius measure on Λ ∞ . Now we describe Bratteli diagrams and k-Bratteli diagrams introduced in [2,9] as follows. 3 Since y ∈ Λ ∞ , we have y : Ω k → Λ by definition.…”

“…Definition 2.3. [2,9] A Bratteli diagram denoted by B is a directed graph with a vertex set B 0 = n∈N V n , and an edge set B 1 = ∞ n=1 E n , where E n consists of edges whose source vertex lies in V n+1 and whose range vertex lies in V n . A finite path η = η 1 · · · η ℓ is a finite sequence of edges with r(η n ) = s(η n+1 ).…”