Factoriality and type classification of $\mathsf{\text{k}}$-graph von Neumann algebras

Let $\text{Fth}$ be a single vertex $\mathsf{\text{k}}$-graph, and ${\pi }_{\omega }({\text{O}}_{\theta })\mathrm{"}$ be the von Neumann algebra induced from the GNS representation of a distinguished state $\omega$ of its $\mathsf{\text{k}}$-graph C*-algebra ${\text{O}}_{\theta }$. In this paper, we prove the factoriality of ${\pi }_{\omega }({\text{O}}_{\theta })\mathrm{"}$ and further determine its type, when either $\text{Fth}$ has the little pull-back property, or the intrinsic group of $\text{Fth}$ has rank $0$. The key step to achieve this is to show that the fixed point algebra of the modular action corresponding to $\omega$ has a unique tracial state.