A dynamical approach to von Neumann dimension

TL;DR

This paper introduces a dynamical approach to defining and analyzing a generalized von Neumann dimension for G-invariant subspaces of l^p(G;V), extending classical concepts beyond Hilbert spaces and exploring their properties.

Contribution

It develops a new dynamical framework for von Neumann dimension applicable to l^p spaces, generalizes the Ornstein-Weiss lemma, and investigates properties of this dimension in relation to linear maps.

Findings

Defined a real-valued dimension for l^p(G;V) subspaces that generalizes von Neumann dimension.

Proved that no finite-type G-equivariant injective linear map exists when the dimension exceeds the target.

Extended the Ornstein-Weiss lemma within this dynamical context.

Abstract

Let G 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(G;V) (with respect to G) 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(G;V) a real positive number dim_{l^p} 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 G-equivariant linear map of finite-type from l^p(G;V) -> l^p(G; V') if dim V > dim V'. A generalization of the Ornstein-Weiss lemma is developed along the way.