On factors associated with quantum Markov states corresponding to nearest neighbor models on a Cayley tree

In this paper we consider nearest neighbour models where the spin takes values in the set Phi = {eta(1), eta(2), ..., eta(q)} and is assigned to the vertices of the Cayley tree Gamma(k). The Hamiltonian is defined by some given lambda-function. We find a condition for the function lambda to determine the type of the von Neumann algebra generated by the GNS-construction associated with the quantum Markov state corresponding to the unordered phase of the lambda-model. Also we give some physical applications of the obtained result.