  In the present paper the Ising model with competing binary ($J$) and binary
($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is
considered. The structure of Gibbs measures for the model considered is
studied. We completely describe the set of all periodic Gibbs measures for the
model with respect to any normal subgroup of finite index of a group
representation of the Cayley tree. Types of von Neumann algebras, generated by
GNS-representation associated with diagonal states corresponding to the
translation invariant Gibbs measures, are determined. It is proved that the
factors associated with minimal and maximal Gibbs states are isomorphic, and if
they are of type III$_\lambda$ then the factor associated with the unordered
phase of the model can be considered as a subfactors of these factors
respectively. Some concrete examples of factors are given too. \\[10mm] {\bf
Keywords:} Cayley tree, Ising model, competing interactions, Gibbs measure,
GNS-construction, Hamiltonian, von Neumann algebra.
