Representations of \'etale groupoids on $L^p$-spaces

Abstract

For $p\in (1,\infty)$, we study representations of \'etale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of \'etale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an \'etale groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight regular $L^{p}$-representations of any countable inverse semigroup of open slices of $G$ that is a basis for the topology of $G$. We define analogs $F^{p}(G)$ and $F_{\mathrm{red}}^{p}(G)$ of the full and reduced groupoid C*-algebras using representations on $L^{p}$-spaces. As a consequence of our main result, we deduce that every contractive representation of $F^{p}(G)$ or $F_{\mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on $L^{p}$-spaces, and the analogs of AF-algebras on $L^{p}$-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid $L^{p}$-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras.