\(L^p\) -Cuntz Algebras and Spectrum of Weighted Composition Operators

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 \(X=\{1,\ldots ,n\}^{\mathbb N}\) , 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 radius given by the formula: \(\displaystyle r(aT)=\max _{\mu \in \operatorname {\mathrm {Erg}}(X,\varphi )} \exp \left ( \int _X \ln (|a|\varrho ^{\frac {1}{p}})d\mu + \frac {h_\varphi (\mu )}{p} \right ) \) where \(\varrho \) is a potential associated to the transfer operator and \(h_\varphi (\mu )\) is Kolmogorov-Sinai entropy. This generalizes classical results for \(p=2\) .