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

TL;DR

This paper systematically analyzes irreducibility of representations of $k$-graph $C^*$-algebras arising from $\Lambda$-semibranching systems, providing conditions, examples, and exploring their relation to atomic representations in a general setting.

Contribution

It offers the first comprehensive analysis of $\Lambda$-semibranching representations for row-finite source-free $k$-graphs, including necessary and sufficient conditions for irreducibility.

Findings

Established criteria for irreducibility of $\Lambda$-semibranching representations.

Connected irreducible representations with purely atomic representations.

Provided examples demonstrating the optimality of the results.

Abstract

The representations of a $k$-graph $C^*$-algebra $C^*(\Lambda)$ which arise from $\Lambda$-semibranching function systems are closely linked to the dynamics of the $k$-graph $\Lambda$. 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 $\Lambda$-semibranching representations and purely atomic representations of $C^*(\Lambda)$. Throughout the paper, we work in the setting of row-finite source-free $k$-graphs; this paper constitutes the first analysis of $\Lambda$-semibranching representations at this level of generality.