Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.07137 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909422183251968 |
|---|---|
| author | Bao, Qiyao Liu, Rui Shen, Jie |
| author_facet | Bao, Qiyao Liu, Rui Shen, Jie |
| contents | A Banach space is said to have the ball-covering property (BCP) if its unit sphere can be covered by countably many closed or open balls off the origin. Let $X$ be a Banach space with a shrinking $1$-unconditional basis. In this paper, by constructing an equivalent norm on $B(X)$, we prove that the quotient Banach algebra $B(X)/K(X)$ fails the BCP. In particular, the result implies that the Calkin algebra $B(H)/ K(H)$, $B(\ell^p)/K(\ell^p)$ ($1 \leq p <\infty$) and $B(c_0)/K(c_0)$ all fail the BCP. We also show that $B(L^p[0,1])$ has the uniform ball-covering property (UBCP) for $3/2< p < 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ball-covering property of non-commutative spaces of operators on Banach spaces Bao, Qiyao Liu, Rui Shen, Jie Functional Analysis 46B20, 46B15, 46B28 A Banach space is said to have the ball-covering property (BCP) if its unit sphere can be covered by countably many closed or open balls off the origin. Let $X$ be a Banach space with a shrinking $1$-unconditional basis. In this paper, by constructing an equivalent norm on $B(X)$, we prove that the quotient Banach algebra $B(X)/K(X)$ fails the BCP. In particular, the result implies that the Calkin algebra $B(H)/ K(H)$, $B(\ell^p)/K(\ell^p)$ ($1 \leq p <\infty$) and $B(c_0)/K(c_0)$ all fail the BCP. We also show that $B(L^p[0,1])$ has the uniform ball-covering property (UBCP) for $3/2< p < 3$. |
| title | The ball-covering property of non-commutative spaces of operators on Banach spaces |
| topic | Functional Analysis 46B20, 46B15, 46B28 |
| url | https://arxiv.org/abs/2412.07137 |