Lp spaces of the von Neumann algebra of a measured groupoid

The Hausdorff-Young inequality is well known for the Fourier transform in R-n. More recently, Hausdorff-Young inequalities were established for unimodular groups (Kunze, 1958) and non-unimodular groups (Terp, 1980). A version was also given for X x X by Russo (1977), where X denotes a measured space. In this Note, we first study the L-P-spaces of the von Neumann algebra of a groupoid, and propose identifications of some of them as function spaces. Using interpolation, we then give a Hausdorff-Young inequality for groupoids. To cite this article: P Boivin, C. R. Acad. Sci. Paris, Ser. I 346 (2008). (C) 2008 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.