Uniformly bounded representations of discrete measured groupoid into finite Von Neumann algebras

TL;DR

This work extends classical unitary-conjugacy results for uniformly bounded representations to the setting of $t$-discrete ergodic groupoids valued in finite von Neumann algebras. The authors adapt the circumcenter approach in CAT$(0)$ spaces, following Boutonnet and Roydor, to construct an explicit similarity that converts any uniformly bounded measurable representation $\rho:\mathcal{G}\to \mathrm{GL}(\mathcal{M})$ into a unitary one. The construction hinges on forming circumcenters $\sigma(x)$ of the sets $B_x=\{\rho(g)^*\rho(g): g\in \mathcal{G}_x\}$, proving the cocycle relation $\rho(g)^*\sigma(t(g))\rho(g)=\sigma(s(g))$, and obtaining a measurable square root map $\psi$ so that $\rho'(g)=\psi(t(g))\rho(g)\psi(s(g))^{-1}$ is unitary. Consequently, every uniformly bounded measurable representation is similar to a unitary representation, generalizing prior results from groups and cocycles to the groupoid framework with a constructive proof method.

Abstract

Let $(\mathcal{G},ν)$ be a $t$-discrete ergodic groupoid. Consider a finite Von Neumann algebra $\mathcal{M}$ with separable predual. We prove that every uniformly bounded measurable representation $ρ:\mathcal{G} \rightarrow \mathrm{GL}(\mathcal{M})$ into the invertible elements of $\mathcal{M}$ is similar to a unitary representation.

Theorems & Definitions (1)

• Theorem 1