Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 35-67.

The Szemeredi property in ergodic W*-dynamical systems

Authors: Conrad Beyers (1), Rocco Duvenhage (2), and Anton Stroh (3)
Author institution: (1) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa
(2) Department of Mathematics and Applied Mathematics, (Current address: Department of Physics), University of Pretoria, Pretoria, 0002, South Africa
(3) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

Summary: 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 F\o lner sequence satisfying the Tempelman condition. For a smaller class of groups (which include $\mathbb{Z}^{q}$\ and $\mathbb{R}^{q}$) this is then used to show that an asymptotically abelian ergodic W*-dynamical system either has the `Szemer\'{e}di 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.

Keywords: $C^*$- and {\rm W}*-dynamical systems, weak mixing, compact dynamical systems, Szemer\'{e}di property.

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