Further properties of ℓp dimension

This article establishes more properties of the ℓp dimension introduced in a previous article. Given an amenable group Γ acting by translation on ℓp(Γ), a number satisfying dimension-like properties is associated to the (usually infinite dimensional) subspaces Y of ℓp(Γ) which are invariant under the action of Γ. This may be interpreted as a von Neumann dimension for ℓp spaces. As a consequence, for p∈[1,2], if Y is a closed non-trivial Γ-invariant subspace of ℓp(Γ) and let Yn be an increasing sequence of closed Γ-invariant subspace such that ⋃Yn¯=ℓp(Γ;V), then there exist a k such that Yk∩Y≠{0}.