This is an announcement for the paper “$c_{0} \widehat{\otimes}_πc_{0}\widehat{\otimes}_πc_{0}$ is not isomorphic to a subspace of $c_{0} \widehat{\otimes}_πc_{0}$ ” by R.M. Causeyhttps://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M, E. Galegohttps://arxiv.org/search/math?searchtype=author&query=Galego%2C+E, C. Samuelhttps://arxiv.org/search/math?searchtype=author&query=Samuel%2C+C.
Abstract: In the present paper we prove that the $3$-fold projective tensor product of $c_0$, $c_{0} \widehat{\otimes}_\pi c_{0}\widehat{\otimes}_\pi c_{0}$, is not isomorphic to a subspace of $c_{0} \widehat{\otimes}_\pi c_{0}$. In particular, this settles the long-standing open problem of whether $c_{0} \widehat{\otimes}_\pi c_{0}$ is isomorphic to $c_{0} \widehat{\otimes}_\pi c_{0}\widehat{\otimes}_\pi c_{0}$. 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.