Quasi-Product States and Factor Types for the One-Dimensional Hard-Core Model

We study a quasi-product state associated with the one-dimensional hard-core Gibbs measure. After coding the model by the topological Markov chain, we construct the standard path $AF$-algebra of admissible hard-core words and show that the stationary Markov measure induces on it a faithful diagonal state in the sense of Evans. We then analyze the von Neumann algebra generated by the corresponding GNS representation. The resulting algebra is a hyperfinite factor, and its type is determined by the single parameter \(\kappa=q/p^2,\) where \(\begin{pmatrix}p&q\\ 1&0\end{pmatrix}\) is the transition matrix of the Markov chain. More precisely, the factor is of type $\mathrm{II}_1$ when $\kappa =1$, and of type $\mathrm{III}_{\l}$ with \( \l=\min\{\kappa,\kappa^{-1}\}\) for $\k\neq 1$. We also specify the centralizer and the weight flow for the resulting factor.