On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras

In the present paper a model with competing ternary (J)₂ and binary (J)₁ interactions with spin values ± 1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J₁, J₂ (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation- invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the (unordered phase is a factor of type III_λ, where λ = exp {-2βJ₂}, β > 0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type III_δ, otherwise they have type III₁.