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

TL;DR

This paper characterizes harmonic cocycles in group cohomology and explores their role in irreducible affine isometric actions using von Neumann algebra techniques, providing criteria for irreducibility based on the cocycles' span.

Contribution

It introduces a characterization of harmonic cocycles within their cohomology classes and links their properties to irreducible affine actions via operator algebra methods.

Findings

Harmonic cocycles span minimal closed subspaces of the Hilbert space.

Irreducibility of affine actions corresponds to the density of the cocycle's linear span.

Operator algebra techniques give criteria for irreducible actions in factorial representations.

Abstract

Let $G$ be a compactly generated locally compact group and $(\pi, \mathcal H)$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $\pi$ which are harmonic (with respect to a suitable probability measure on $G$) represent classes in the first reduced cohomology $\bar{H}^1(G,\pi).$ 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 $\pi$ moreover has no almost invariant vectors. Our approach exploits the natural structure of the space of harmonic $1$-cocycles with coefficients in $\pi$ as a Hilbert module over the von Neumann algebra $\pi(G)',$ which is the commutant of $\pi(G)$. Using operator algebras techniques, such as the von Neumann dimension, we give a necessary and sufficient condition for a factorial representation $\pi$ without almost invariant vectors to admit an irreducible affine action with $\pi$ as linear part.