A noncommutative Ruelle's Theorem for a normalized potential taking values on positivity-improving operators

Abstract

Let \(\mathcal{A}\) be a finite-dimensional real (or complex) C*-algebra, \(\Omega_{A}\) an aperiodic subshift of finite type, and \(\mathcal{C}(\Omega_{A}; \mathcal{A})\) the set of continuous functions from \(\Omega_{A}\) to \(\mathcal{A}\). The shift $\sigma$ provides dynamics. Given a real Lipschitz potential $\varphi\in \mathcal{C}(\Omega_{A}; \mathfrak{L}(\mathcal{A}))$, where $\mathfrak{L}(\mathcal{A})$ is the set of linear operators acting on a real $\mathcal{A}$, we introduce a noncommutative analogue of Ruelle's operator, which acts on \(\mathcal{C}(\Omega_{A}; \mathcal{A})\). Assuming the positivity-improving hypothesis, we prove a version of Ruelle's Theorem whenever the operator is normalized. An eigenstate (a linear functional) invariant for the action of the noncommutative Ruelle's operator will play the role of the Gibbs probability of Thermodynamic Formalism; to be called a Gibbs eigenstate. We introduce the concept of entropy for a Gibbs eigenstate (obtained from a certain family of potentials $\varphi$) - generalizing the classical one. In our setting, there is currently no direct relationship with cocycles and Lyapunov exponents. We present examples illustrating the novelty of the cases that can be considered, ranging from topics related to quantum channels to Pauli matrices. Interesting cases: $\mathcal{A}=M_{N \times N}(\mathbb{R})$ and $\mathcal{A}= \mathbb{R}^{N}$.