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

Abstract

In this paper we prove that the $3$-fold injective tensor product ${\ell }_1{\hat{\otimes }}_{\epsilon }{\ell }_1{\hat{\otimes }}_{\epsilon }{\ell }_1$ is not isomorphic to any subspace of ${\ell }_1{\hat{\otimes }}_{\epsilon }{\ell }_1$. 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 $C(K){\hat{\otimes }}_{\pi }C(K){\hat{\otimes }}_{\pi }C(K)$ is not isomorphic to any quotient of $C(K){\hat{\otimes }}_{\pi }C(K)$.