This is an announcement for the paper "Representations of '{e}tale groupoids on $L^p$-spaces" by Eusebio Gardella and Martino Lupini.

Abstract: For $p\in (1,\infty)$, we study representations of '{e}tale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of '{e}tale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an '{e}tale 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.

Archive classification: math.OA

Mathematics Subject Classification: 47L10, 22A22 (Primary) 46H05 (Secondary)

Remarks: 52 pages

Submitted from: mlupini@yorku.ca

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1408.3752

or

http://arXiv.org/abs/1408.3752