$L^p$-Cuntz algebras and spectrum of weighted composition operators

Abstract: Let $p\in [1,\infty)$. We define an $L^p$-operator algebra crossed product by a transfer operator for the topological Bernoulli shift $\varphi$ on $X={1,...,n}^{\mathbb{N}}$, and we prove it is isometrically isomorphic to the $L^p$-analog $\mathcal{O}n^p$ of the Cuntz algebra introduced by Phillips. As an application, we prove that the spectrum of the associated `abstract weighted shift operators' $aT$, $a\in C(X)$, is a disk with a radius given by the formula $$ r(aT)=\max{\mu\in\text{Erg}(X,\varphi)} \exp\left( \int_X \ln(|a|\varrho^{\frac{1}{p}})d\mu + \frac{h_\varphi(\mu)}{p} \right) $$ where $\varrho$ is a potential associated to the transfer operator, and $h_\varphi(\mu)$ is Kolmogorov-Sinai entropy. This generalizes classical results for $p=2$.