Remarks on the Daugavet Property for Complex Banach Spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913363028606976 |
|---|---|
| author | Lee, Han Ju Tag, Hyung-Joon |
| author_facet | Lee, Han Ju Tag, Hyung-Joon |
| contents | In this article, we study the Daugavet property and the diametral diameter two properties in complex Banach spaces. The characterizations for both Daugavet and $Δ$-points are revisited in the context of complex Banach spaces. We also provide relationships between some variants of alternative convexity and smoothness, nonsquareness, and the Daugavet property. As a consequence, every strongly locally uniformly alternatively convex or smooth (sluacs) Banach space does not contain $Δ$-points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property (convex-DLD2P) and the polynomial Daugavet property in the vector-valued function space $A(K, X)$. From an explicit computation of the polynomial Daugavetian index of $A(K, X)$, we show that the space $A(K, X)$ has the polynomial Daugavet property if and only if either the base algebra $A$ or the range space $X$ has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, the Daugavet property, the diameteral diameter two properties, and the property ($\mathcal{D}$) are equivalent for infinite-dimensional uniform algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11153 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Remarks on the Daugavet Property for Complex Banach Spaces Lee, Han Ju Tag, Hyung-Joon Functional Analysis In this article, we study the Daugavet property and the diametral diameter two properties in complex Banach spaces. The characterizations for both Daugavet and $Δ$-points are revisited in the context of complex Banach spaces. We also provide relationships between some variants of alternative convexity and smoothness, nonsquareness, and the Daugavet property. As a consequence, every strongly locally uniformly alternatively convex or smooth (sluacs) Banach space does not contain $Δ$-points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property (convex-DLD2P) and the polynomial Daugavet property in the vector-valued function space $A(K, X)$. From an explicit computation of the polynomial Daugavetian index of $A(K, X)$, we show that the space $A(K, X)$ has the polynomial Daugavet property if and only if either the base algebra $A$ or the range space $X$ has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, the Daugavet property, the diameteral diameter two properties, and the property ($\mathcal{D}$) are equivalent for infinite-dimensional uniform algebras. |
| title | Remarks on the Daugavet Property for Complex Banach Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2302.11153 |