Representations of étale groupoids on $L^p$-spaces

Abstract

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.

33 pages. v2: minor changes. v3: more minor changes. To appear in Advances in Math

References in corpus (6)

• Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces
• Quotients of Banach algebras acting on $L^p$-spaces
• Banach algebras generated by an invertible isometry of an $L^p$-space
• Simplicity of UHF and Cuntz algebras on $L^p$~spaces
• Simple reduced $L^p$ operator crossed products with unique trace
• Isomorphism, nonisomorphism, and amenability of L^p UHF algebras