Further properties of $\ell^p$ dimension

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 $\srl{\cup Y_n} = \ell^p(\Gamma;V)$, then there exist a $k$ such that $Y_k \cap Y \neq \{0\}$.