New York Journal of Mathematics Volume 23 (2017) 1205-1218
 Volume 23 Volume Index

Bachir Bekka

Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions

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 Published: September 11, 2017 Keywords: Isometric group actions on Hilbert spaces, reduced first cohomology, von Neumann algebras, von Neumann dimension, first ℓ2-Betti number Subject: 22D10, 22D25, 22E41

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 \bar{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 nonempty, proper closed invariant affine subspace) if and only if the linear span of b(G) is dense in H. 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 necessary and sufficient condition for a factorial representation π without almost invariant vectors to admit an irreducible affine action with π as linear part.

Acknowledgements

The author acknowledges the partial support of the French Agence Nationale de la Recherche (ANR) through the projects Labex Lebesgue (ANR-11-LABX-0020-01) and GAMME (ANR-14-CE25-0004).

Author information

IRMAR, UMR-CNRS 6625 Université de Rennes 1, Campus Beaulieu, F-35042 Rennes Cedex, France
bachir.bekka@univ-rennes1.fr