On injective tensor powers of $\ell_1$

Abstract: 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 $c_0.$ 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 $C(K) \widehat{\otimes}\pi C(K)$.