Ergodic properties and KMS conditions on $C^{\ast }$-symbolic dynamical systems

Abstract

A $C^{\ast }$-symbolic dynamical system $(\mathcal{A},\rho ,\Sigma )$ consists of a unital $C^{\ast }$-algebra $\mathcal{A}$ and a finite family ${{\rho }_{\alpha }}_{\alpha \in \Sigma }$ of endomorphisms ${\rho }_{\alpha }$ of $\mathcal{A}$ indexed by symbols $\alpha$ of $\Sigma$ satisfying some conditions. The endomorphisms ${\rho }_{\alpha },\alpha \in \Sigma$ yield both a subshift ${\Lambda }_{\rho }$ and a $C^{\ast }$-algebra ${\mathcal{O}}_{\rho }$. We will study ergodic properties of the positive operator $lambd{a}_{\rho }={\sum }_{\alpha \in \Sigma }{\rho }_{\alpha }$ on $\mathcal{A}$. We will next introduce KMS conditions for continuous linear functionals on ${\mathcal{O}}_{\rho }$ under gauge action at inverse temperature taking its value in complex numbers. We will study relationships among the eigenvectors of $lambd{a}_{\rho }$ in ${\mathcal{A}}^{\ast }$, the continuous linear functionals on ${\mathcal{O}}_{\rho }$ satisfying KMS conditions and the invariant measures on the associated one-sided shifts. We will finally present several examples of continuous linear functionals satisfying KMS conditions.