Representations of étale groupoids on Lp-spaces

For p∈(1,∞), we study representations of étale groupoids on Lp-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 Lp-representations of an étale groupoid G, contractive Lp-representations of Cc(G), and tight regular Lp-representations of any countable inverse semigroup of open slices of G that is a basis for the topology of G. We define analogs Fp(G) and Fredp(G) of the full and reduced groupoid C*-algebras using representations on Lp-spaces. As a consequence of our main result, we deduce that every contractive representation of Fp(G) or Fredp(G) is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group Lp-operator algebras, the analogs of Cuntz algebras on Lp-spaces, and the analogs of AF-algebras on Lp-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid Lp-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras.