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

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

5 pages