 

Bachir Bekka
Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions view print


Published: 
September 11, 2017 
Keywords: 
Isometric group actions on Hilbert spaces, reduced first cohomology, von Neumann algebras, von Neumann dimension, first ℓ^{2}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 1cocycles 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 1cocycles
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 1cocycles 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 (ANR11LABX002001) and GAMME (ANR14CE250004).


Author information
IRMAR, UMRCNRS 6625 Université de Rennes 1, Campus Beaulieu, F35042 Rennes Cedex, France
bachir.bekka@univrennes1.fr

