The Szemeredi 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 Folner sequence satisfying the Tempelman condition. For a smaller class of groups (which include Z^q and R^q) this is then used to show that an asymptotically abelian ergodic W*-dynamical system either has the "Szemeredi 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.

v1: 39 pages. Incorporates large portions of the unpublished preprints math/0512059 and math/0611309. v2: Several corrections made, switched to amsart, 30 pages. v3: Small corrections made

References in corpus (1)

• Noncommutative recurrence over locally compact Hausdorff groups

Cited by in corpus (5)

• Relatively independent joinings and subsystems of W*-dynamical systems
• Relative ergodic properties of C*-dynamical systems
• Ergodic theory for quantum semigroups
• Almost weak polynomial stability of operators
• Ergodicity and mixing of W*-dynamical systems in terms of joinings