A New Crossnorm That Preserves Unconditional Bases in Banach Spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915482467041280 |
|---|---|
| author | Karkri, Rafik Kabbaj, Samir |
| author_facet | Karkri, Rafik Kabbaj, Samir |
| contents | Let $α$ 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 $α$ preserves unconditionality if, for every pair of Banach spaces $E$ and $F$ with unconditional Schauder bases (USBs), the completion $E \otimes_α F$ 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 $α$ with this property remains an open question. In this paper, we construct for every such pair $(E,F)$ a new reasonable crossnorm $α$. This norm has the surprising property that -- despite being generally non-uniform -- the space $E \otimes_α F$ nevertheless admits a USB. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New Crossnorm That Preserves Unconditional Bases in Banach Spaces Karkri, Rafik Kabbaj, Samir Functional Analysis [2008] 46B04, 46B15, 46B25, 46B45, 46A32 Let $α$ 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 $α$ preserves unconditionality if, for every pair of Banach spaces $E$ and $F$ with unconditional Schauder bases (USBs), the completion $E \otimes_α F$ 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 $α$ with this property remains an open question. In this paper, we construct for every such pair $(E,F)$ a new reasonable crossnorm $α$. This norm has the surprising property that -- despite being generally non-uniform -- the space $E \otimes_α F$ nevertheless admits a USB. |
| title | A New Crossnorm That Preserves Unconditional Bases in Banach Spaces |
| topic | Functional Analysis [2008] 46B04, 46B15, 46B25, 46B45, 46A32 |
| url | https://arxiv.org/abs/2506.22522 |