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

TL;DR

This paper proves that any uniformly bounded measurable representation of a discrete ergodic groupoid into the invertible elements of a finite von Neumann algebra can be transformed into a unitary representation, extending classical results to groupoids.

Contribution

It establishes a similarity result for uniformly bounded representations of discrete ergodic groupoids into finite von Neumann algebras, generalizing known theorems from groups.

Findings

Every uniformly bounded measurable representation is similar to a unitary one.

The result applies to representations into the invertible elements of finite von Neumann algebras.

Extension of classical group representation theorems to the setting of groupoids.

Abstract

Let $(\mathcal{G},\nu)$ 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 $\rho:\mathcal{G} \rightarrow \mathrm{GL}(\mathcal{M})$ into the invertible elements of $\mathcal{M}$ is similar to a unitary representation.