A New Crossnorm That Preserves Unconditional Bases in Banach Spaces

TL;DR

This paper introduces a novel tensor norm that preserves unconditional bases in Banach space tensor products, addressing a longstanding open problem by constructing a crossnorm with this property.

Contribution

The authors construct a new reasonable crossnorm that preserves unconditional bases in tensor products of Banach spaces, unlike previously known norms.

Findings

The new tensor norm preserves unconditional bases in tensor products.

It is generally non-uniform but still maintains unconditionality.

This construction addresses an open problem in Banach space theory.

Abstract

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.