A dynamical approach to von Neumann dimension

Pages: 967 - 987, Volume 26, Issue 3, March 2010

doi:10.3934/dcds.2010.26.967       Abstract        Full Text (280.7K)       Related Articles

Antoine Gournay - Max Planck Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany (email)

Abstract: Let $\Gamma$ be an amenable group and $V$ be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of l$^2(\Gamma;V)$ (with respect to $\Gamma$) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces $Y$ of l$^p(\Gamma;V)$ a real positive number dim_lp Y (which is the von Neumann dimension when $p=2$). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective $\Gamma$-equivariant linear map of finite-type from l$^p(\Gamma;V) \to $l$^p(\Gamma; V')$ if $\dim V > \dim V'$. A generalization of the Ornstein-Weiss lemma is developed along the way.

Keywords:  Von Neumann dimension, mean dimension, Alexandrov width, Ornstein-Weiss lemma, classification of l$^p$ spaces on discrete amenable groups.
Mathematics Subject Classification:  Primary: 37A35, 43A65, 70G60; Secondary: 37C45, 41A46, 43A15, 46E30.

Received: May 2009;      Revised: September 2009;      Published: December 2009.