Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice

The Ising model on a Bethe lattice of order $k\geq2$ is considered. For maximum or minimum translation-invariant Gibbs states of this model, the relations between the von Neumann algebras generated by these states for the Gelfand–Neimark–Segal representation are found. These algebras can be of types $\mathrm{III}_\lambda$, $\lambda\in(0,1)$, and $\mathrm{III}_1$.