An obstruction to \ell ^{p}-dimension

An lp-version of von Neumann dimension for Banach space representations of sofic groups II

“…The above shows that dim Σ,l p has many of the properties that the usual dimension in linear algebra and von Neumann dimension have and thus it makes sense to think of dim Σ,l p as a version of von Neumann dimension. We mention that in [20] Monod-Petersen show that if 2 < p < ∞, then any isomorphism invariant associated to Γ-invariant subspaces of l p (Γ) cannot satisfy all the properties that von Neumann dimension satisfies. In particular, they show that if Γ contains an infinite elementary amenable subgroup and 2 < p < ∞, then there exists closed Γ-invariant linear subspaces E n and F = {0} of l p (Γ) with E n ∩ F = {0} for all n, but…”