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

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 row-finite source-free $k$-graphs; this paper constitutes the first analysis of $Λ$-semibranching representations at this level of generality.

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