Factorizations and minimality of the Calkin Algebra norm for $C(K)$-spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911379788660736 |
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| author | Acuaviva, Antonio |
| author_facet | Acuaviva, Antonio |
| contents | For a scattered, locally compact Hausdorff space $K$, we prove that the essential norm on the Calkin algebra \break $\mathscr{B}(C_0(K))/\mathscr{K}(C_0(K))$ is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on $c_0$ through non-compact operators $T: C_0(K) \to X$, where $X$ is any Banach space that does not contain a copy of $\ell_1$ or whose dual unit ball is weak$^*$ sequentially compact. It follows that, for every ordinal $α$, the algebras $\mathscr{B}(C[0,α]))$ and $\mathscr{B}(C[0,α]))/\mathscr{K}(C[0,α]))$ have an unique algebra norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Factorizations and minimality of the Calkin Algebra norm for $C(K)$-spaces Acuaviva, Antonio Functional Analysis 46H10, 47L10 (Primary) 46E15, 47B01, 47L20 (Secondary) For a scattered, locally compact Hausdorff space $K$, we prove that the essential norm on the Calkin algebra \break $\mathscr{B}(C_0(K))/\mathscr{K}(C_0(K))$ is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on $c_0$ through non-compact operators $T: C_0(K) \to X$, where $X$ is any Banach space that does not contain a copy of $\ell_1$ or whose dual unit ball is weak$^*$ sequentially compact. It follows that, for every ordinal $α$, the algebras $\mathscr{B}(C[0,α]))$ and $\mathscr{B}(C[0,α]))/\mathscr{K}(C[0,α]))$ have an unique algebra norm. |
| title | Factorizations and minimality of the Calkin Algebra norm for $C(K)$-spaces |
| topic | Functional Analysis 46H10, 47L10 (Primary) 46E15, 47B01, 47L20 (Secondary) |
| url | https://arxiv.org/abs/2408.11132 |