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

Let $\Fth$ be a single vertex \textsf{k}-graph, and $π_ω(Ø_θ)"$ be the von Neumann algebra induced from the GNS representation of a distinguished state $ω$ of its $\textsf{k}$-graph C*-algebra $Ø_θ$. In this paper, we prove the factoriality of $π_ω(Ø_θ)"$ 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 $ω$ has a unique tracial state.

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