Factoriality and type classification of \textsf{k}-graph von Neumann algebras

Let $\Fth$ be a single vertex \textsf{k}-graph, and $\pi_\omega(\O_\theta)"$ be the von Neumann algebra induced from the GNS representation of a distinguished state $\omega$ of its $\textsf{k}$-graph C*-algebra $\O_\theta$. In this paper, we prove the factoriality of $\pi_\omega(\O_\theta)"$ and further determine its type, when either $\Fth$ has the little pull-back property, or the intrinsic group of $\Fth$ 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.