A New Crossnorm That Preserves Unconditional Bases in Banach Spaces

Let $\alpha$ be a tensor norm (i.e., a uniform reasonable crossnorm) on the class of all algebraic tensor products of Banach spaces $E \otimes F$. We say that $\alpha$ preserves unconditionality if, for every pair of Banach spaces $E$ and $F$ with unconditional Schauder bases (USBs), the completion $E \otimes_{\alpha} F$ also admits a USB. It is well known that none of Grothendieck's fourteen natural tensor norms satisfy this unconditionality-preserving condition. Moreover, the existence of a tensor norm $\alpha$ with this property remains an open question. In this paper, we construct for every such pair $(E,F)$ a new reasonable crossnorm $\alpha$. This norm has the surprising property that -- despite being generally non-uniform -- the space $E \otimes_{\alpha} F$ nevertheless admits a USB.