The Szemerédi Property in Ergodic W*-dynamical Systems

Abstract. We study weak mixing of all orders for asymptotically abelian weakly mixing state preserving C*-dynamical systems, where the dynamics is given by the action of an abelian second countable locally compact group which contains a Følner sequence satisfying the Tempelman condition. For a smaller class of groups (which include Z and R) this is then used to show that an asymptotically abelian ergodic W*-dynamical system either has the “Szemerédi property” or contains a nontrivial subsystem (a “compact factor”) that does. A van der Corput lemma for Hilbert space valued functions on the group is one of our main technical tools.