These are the lectures notes from a minicourse given at the Masterclass on “Ergodic theory and von Neumann algebras” at the University of Copenhagen in October 2013. In the Lectures 2, 3 and 4, we will follow S. Vaes’s exposition for his mini-course at the “ Intensive Month on Operator Algebra and Harmonic Analysis” at ICMAT in Madrid in May 2013. The exposition in itself is very close to the one in the articles by S. Popa and S. Vaes [PV11, PV12]. In the Lectures 5 and 6, we will follow the… Expand

We prove that for any free ergodic probability measure preserving action $\Gamma \actson (X,\mu)$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated… Expand

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$Fn↷(X,μ) of a free group on n generators $${\mathbb{F}_n, 2\leq n \leq… Expand