Monic representations of finite higher-rank graphs

Abstract

In this paper we define the notion of monic representation for the C * -algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative C * -algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the Λ-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

“…We now recall the definition of a Λ-projective system from [18]. Roughly speaking, a Λprojective system on (X, µ) consists of a Λ-semibranching function system plus some extra information (encoded in the functions f λ below).…”

“…For higher-rank graphs Λ, branching systems were introduced as Λsemibranching function systems by the first and second authors together with S. Kang and J. Packer in [22], where the associated representations were used to construct wavelets and to analyze the KMS states of the higher-rank graph C * -algebra C * (Λ). Subsequent work by Farsi, Gillaspy, Kang, and Packer together with P. Jorgensen [18] showed that a large class of representations of C * (Λ) -the so-called monic representations -all arise from Λsemibranching function systems, and these authors provide in [19] a more detailed analysis of the structure of Λ-semibranching function systems in the case when the associated measure space is atomic. The question of when a Λ-semibranching representation is faithful has been explored in [22,27,20].…”

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

“…Let µ be the probability measure on ∂B Λ given in (3). Let {ρ i : 1 ≤ i ≤ k} be the set of the spectral radii of all vertex matrices of Λ, and suppose that ρ i > 1 for all 1 ≤ i ≤ k. Let ∆ s , s ∈ R be the Laplace-Beltrami operator associated to the Dirichlet form Q s given in (8) and let λ s,γ be its eigenvalues given in (10), where γ ∈ F B Λ . If s < 2 + δ, then for γ ∈ F B Λ , the eigenvalue λ s,γ goes to −∞ as |γ| → ∞.…”

“…Also, note that the family of the spectral triples is indexed by the space of choice functions.) [9] A choice function for (∂B Λ , d w δ ) is a map φ : (3-1).) Here φ 1 (λ) and φ 2 (λ) are infinite paths in [λ] and the subscripts 1, 2 imply that the choice function gives a pair of (distinct) infinite paths in [λ] satisfying (3-2).…”

“…Higher-rank graphs and the associated C * -algebras have extensively been studied in the last two decades in classification theory, K-theory of C * (Λ) and the study of spectral triples and of KMS states. (See [3,7,9,11,14,20] and references therein.) The C * -theory of Cuntz algebras and Cuntz-Krieger algebras associated to graphs (and k-graphs) not only has played an important role in operator algebras and noncommutative geometry, but also has been used in many applications in engineering and physics: branching laws for endomorphisms, Markov measures and topological Markov chains, wavelets and multiresolution analysis, and iterated function systems (IFSs) and fractals.…”