The Dirac operator for the Ruelle-Koopman pair on L^p-spaces: an interplay between Connes distance and symbolic dynamics

Abstract

Denote by $\mathbf{\mu }$ the maximal entropy measure for the shift \(σ\) acting on $\Omega =\{0,1{\}}^{\mathbb{N}}$, by $\ruelle$ the associated Ruelle operator and by $\koopman = \ruelle^{\dagger}$ the Koopman operator, both acting on $\lp{2}(\bmμ)$. Using a diagonal representation $\pi$, the Ruelle-Koopman pair can be used for defining a dynamical Dirac operator $\mathcal{D},$ as in \cite{BL}. $\mathcal{D}$ plays the role of a derivative. In \cite{lpspec}, the notion of a spectral triple was generalized to \(\lp{p}\)-operator algebras; in consonance, here, we generalize results for $\mathcal{D}$ to results for a Dirac operator ${\mathcal{D}}_p$ , and the associated Connes distance $d_p$, to this new \(\lp{p}\) context, \(p \geq 1\). Given the states $\eta ,\xi$: $d_{p}(η, ξ) \defn \sup \{ \,|η(a) - ξ(a) | where a \in \mathcal{A} and \norm{\left[\mathcal{D}_p,π(a)\right]} \leq 1\}$. The operator $M_f$ acts on $L^p(\mu ).$ We explore the relationship of ${\mathcal{D}}_p$ with dynamics, in particular with $f\circ \sigma -f$, the discrete-time derivative of a continuous $f:\Omega \to \mathbb{R}$. Take $p,p^{\prime }>0$ satisfying $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. We show for any continuous function $f$: $\norm{\left[ \dirac_p, π(\mult_f) \right]} = | \sqrt[λ]{\ruelle \abs{f \circ σ- f}^λ} |_{\infty}$, where $\lambda =\max \{p,p^{\prime }\}$. Furthermore, we show $\norm{\left[ \mathcal{D}_p, π(\koopman^{n} \mathcal{L}^{n})]\right]}=1$ for all \(n \geq 1\). We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.