An obstruction to ${\mathrm{\ell }}^p$-dimension

For any group $G$ containing an infinite elementary amenable subgroup, and any $2<p<\mathrm{\infty }$, there exists closed invariant subspaces $E_i\nearrow {\mathrm{\ell }}^{p}G$ and $F\mathrm{\ne }0$ such that $E_i\cap F=0$ for all $i$. This is an obstacle to ${\mathrm{\ell }}^p$-dimension and gives a negative answer to a question of Gaboriau.