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 $Φ=\{\z_1,\z_2,...,\z_q\}$ and is assigned to the vertices of the Cayley tree ${\G}^k$. The Hamiltonian is defined by some given $λ$-function. We find a condition for the function $λ$ 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 $λ$-model. Also we give some physical applications of the obtained result.

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