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

Abstract

Let $G$ be a compactly generated locally compact group and $(π, \mathcal H)$ 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 $\mathcal H.$ In particular, the affine isometric action given by a harmonic cocycle $b$ is irreducible (in the sense that $\mathcal H$ contains no non-empty, proper closed invariant affine subspace) if the linear span of $b(G)$ is dense in $\mathcal H.$ 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 necessary and sufficient condition for a factorial representation $π$ without almost invariant vectors to admit an irreducible affine action with $π$ as linear part.