Representations of Higher Rank Graph Algebras

Let $\text{Fth}$ be a $\text{Bk}$-graph on a single vertex. We show that every irreducible atomic $\ast$-representation is the minimal $\ast$-dilation of a group construction representation. It follows that every atomic representation decomposes as a direct sum or integral of such representations. We characterize periodicity of $\text{Fth}$ and identify a symmetry subgroup $H_{\theta }$ of ${\text{bZ}}^{\text{Bk}}$. If this has rank $s$, then $\text{ca}(\text{Fth})\cong \text{rC}({\text{bT}}^s)\otimes \text{fA}$ for some simple C*-algebra $\text{fA}$.