On injective tensor powers of ${\mathrm{\ell }}_1$

In this paper we prove that the $3$-fold injective tensor product $\ell_1 \widehat{\otimes}\varepsilon \ell_1 \widehat{\otimes}\varepsilon \ell_1 $ is not isomorphic to any subspace of $\ell_1 \widehat{\otimes}_\varepsilon \ell_1$. This result provides a new solution to a problem of Diestel on the projective tensor products of $c_0\mathrm{.}$ Moreover, this result implies that for any infinite countable compact space $K,$ the $3$-fold projective tensor product $C(K){\widehat{\otimes }}_{\pi }C(K){\widehat{\otimes }}_{\pi }C(K)$ is not isomorphic to any quotient of $C(K){\widehat{\otimes }}_{\pi }C(K)$.