Publication date:
1 October 2017
Source:Advances in Mathematics, Volume 318
Author(s): Eusebio Gardella, Martino Lupini
For $p\in (1,\infty )$, we study representations of étale groupoids on $L^p$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of étale groupoids on Hilbert spaces. We establish a correspondence between $L^p$-representations of an étale 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.