Isometric embedding of ${\mathrm{\ell }}_1$ into Lipschitz-free spaces and ${\mathrm{\ell }}_{\mathrm{\infty }}$ into their duals

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of ${\mathrm{\ell }}_{\mathrm{\infty }}$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to ${\mathrm{\ell }}_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between “isometric embedding of~${\mathrm{\ell }}_{\mathrm{\infty }}$ into the dual” and “containing as good copy of~${\mathrm{\ell }}_1$ as possible” in a general Banach space.