$L^p$-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 $\phi$ 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 $$r(aT)=\underset{\mu \in \text{Erg}(X,\phi )}{\max }\exp \left({\int }_X\ln (|a|{\varrho }^{\frac{1}{p}})d\mu +\frac{h_{\phi }(\mu )}{p}\right)$$ where $\varrho$ is a potential associated to the transfer operator, and $h_{\phi }(\mu )$ is Kolmogorov-Sinai entropy. This generalizes classical results for $p=2$.

9 pages