On a factor associated with the unordered phase of λ-model on a cayley tree

In this paper we consider nearest-neighbours models, where the spin takes values in the set Φ = {η1, η1, …, ηq} and is assigned to the vertices of the Cayley tree Γk. The Hamiltonian is defined by some given λ-function. We find a condition for the function λ to determine a type of von Neumann algebra generated by the GNS construction associated with the unordered phase of the λ-model. We give also some physical applications of the obtained result.