HARMONIC COCYCLES, VON NEUMANN ALGEBRAS, AND IRREDUCIBLE AFFINE ISOMETRIC ACTIONSReport as inadecuate




HARMONIC COCYCLES, VON NEUMANN ALGEBRAS, AND IRREDUCIBLE AFFINE ISOMETRIC ACTIONS - Download this document for free, or read online. Document in PDF available to download.

1 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : 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 bG 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 πG , which is the commutant of πG. 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.

Keywords : Group cohomology Unitary Representations Von Neumann algebras Affine action





Author: Bachir Bekka -

Source: https://hal.archives-ouvertes.fr/



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