HARMONIC COCYCLES, VON NEUMANN ALGEBRAS, AND IRREDUCIBLE AFFINE ISOMETRIC ACTIONS
Résumé
Let G be a compactly generated locally compact group and (π, H) a unitary representation of G. The 1-cocycles with coefficients in π which are harmonic (with respect to a suitable probability measure on G) represent classes in the first reduced cohomology H 1- (G, π). We show that harmonic 1-cocycles are characterized inside their reduced cohomology class by the fact that they span a minimal closed subspace of H. In particular, the affine isometric action given by a harmonic cocycle b is irreducible (in the sense that H contains no non-empty, proper closed invariant affine subspace) if the linear span of b(G) is dense in H. The converse statement is true, if π moreover has no almost invariant vectors. Our approach exploits the natural structure of the space of harmonic 1-cocycles with coefficients in π as a Hilbert module over the von Neumann algebra π(G) , which is the commutant of π(G). Using operator algebras techniques, such as the von Neumann dimension, we give a a necessary and sufficient condition for a factorial representation π of G without almost invariant vectors to admit an irreducible affine action with π as linear part.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)