  In this paper we prove that $L_{p}$ does not admit an equivariant coarse
embedding into $\ell_p$ i.e there is no proper, affine, isometric action of
$L_{p}$, viewed as a group under addition with the standard metric $|| . ||_p$,
on $\ell_p$. This is done by showing that representations of $L_{p}$ into $
Isom(\ell_p)$ has to be trivial, which allows us to reduce the question to
bi-Lipschitz setting.
