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

Abstract

The representations of a k-graph C * -algebra C * (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C * (Λ). Throughout the paper, we work in the setting of rowfinite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.

Unifying interval maps and branching systems with applications to relative graph C*-algebras

“…Applying the theory of representations associated with branching systems, we obtain, by [10,Thm. 5.3], that ρ is purely atomic and, by [10,Thm. 5.5], that ρ is monic.…”

“…Building on the work in [13] for graph C * -algebras, our study leads to a characterization of those representations that are faithful (in terms of the underlying branching systems). These results have interest in their own right, as the current streamline of research on the faithfulness of representations of ultragraph algebras and higher rank graph algebras from branching systems shows, see [10,11,12,14,15]. As we mentioned above, we will connect (and apply) the theory of branching systems with the open dynamics of interval maps.…”

“…Our study indicates a unifying perspective among the subjects of interval maps, (relative) branching systems, and relative graph algebras, whenever the underlying graphs are finite. This is built starting from previous work on the construction of branching systems and the associated representations of graph algebras by the second author [3,10,15,16,18], and by the three other authors in relating open dynamical systems and representations of relative graph algebras (underlying the transition matrices) on the orbit spaces [5,6,7,8].…”