$c_0{\hat{\otimes }}_{\pi }{c}_0{\hat{\otimes }}_{\pi }{c}_0$ is not isomorphic to a subspace of $c_0{\hat{\otimes }}_{\pi }{c}_0$

Abstract

In the present paper we prove that the $3$-fold projective tensor product of $c_0$, $c_0{\hat{\otimes }}_{\pi }{c}_0{\hat{\otimes }}_{\pi }{c}_0$, is not isomorphic to a subspace of $c_0{\hat{\otimes }}_{\pi }{c}_0$. In particular, this settles the long-standing open problem of whether $c_0{\hat{\otimes }}_{\pi }{c}_0$ is isomorphic to $c_0{\hat{\otimes }}_{\pi }{c}_0{\hat{\otimes }}_{\pi }{c}_0$. The origin of this problem goes back to Joe Diestel who mentioned it in a private communication to the authors of paper "Unexpected subspaces of tensor products" published in 2006.