Representations of etale groupoids on LP-spaces

For p is an element of (1, infinity), 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-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-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. (C) 2017 Elsevier Inc. All rights reserved.