Gibbs states for AF algebras

Abstract

We consider a special class of C*-systems containing asymptotically Abelian binary shifts and shifts of Temperley–Lieb algebras. We study Gibbs states for these systems corresponding to potentials with finite range interaction, and obtain the same results as the well-known Araki’s results for a one-dimensional quantum lattice. In particular, it is proved that a Gibbs state in the infinite volume is a translation invariant KMS state having the exponential uniform clustering property. Entropic properties of the Gibbs states are also discussed. This allows us, in particular, to construct new examples of quantum K-systems.