EMBEDDING $\ell_1$ INTO THE PROJECTIVE TENSOR PRODUCT OF BANACH SPACES

Abstract

Let $X$ and $Y$ be Banach spaces such that $X$ has an unconditional basis. Then $X \hat{\otimes} Y$, the projective tensor product of $X$ and $Y$, contains no copy of $\ell_1$ if and only if both $X$ and $Y$ contain no copy of $\ell_1$ and each continuous linear operator from $X$ to $Y^\ast$ is compact.