On a question of Niels Gr\o nbaek

Abstract

Let F(X) denote the norm closure of the space of all finite rank operators on a Banach space X. We show that there are Banach spaces X for which the product map $a\otimes b\to ab$ does not define a surjective map from the projective tensor product $F(X) \hat\otimes F(X)$ onto F(X).