Representations of étale groupoids on L^p -spaces

Abstract

For p∈(1,∞), 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_(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.

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