Further properties of ${\ell }^p$ dimension

Abstract

This article establishes more properties of the ${\ell }^p$ dimension introduced in a previous article. Given an amenable group $\Gamma$ acting by translation on ${\ell }^p(\Gamma )$, this is just a number, associated to the (usually infinite dimensional) subspaces $Y$ of ${\ell }^p(G)$ which are invariant under the action of $G$, satisfying dimension-like properties. As a consequence, for $p\in [1,2]$, if $Y$ is a closed non-trivial $\Gamma$-invariant subspace of ${\ell }^p(\Gamma ;V)$ and let $Y_n$ is an increasing sequence of closed $\Gamma$-invariant subspace such that $\srl{\cup Y_n} = \ell^p(Γ;V)$, then there exist a $k$ such that $Y_k\cap Y\ne \{0\}$.

arXiv comment22 pages