The compact operators on $c_0$ as a Calkin algebra
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929267138363392 |
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| author | Motakis, Pavlos Puglisi, Daniele |
| author_facet | Motakis, Pavlos Puglisi, Daniele |
| contents | For a Banach space $X$, let $\mathcal{L}(X)$ denote the algebra of all bounded linear operators on $X$ and let $\mathcal{K}(X)$ denote the compact operator ideal in $\mathcal{L}(X)$. The quotient algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is called the Calkin algebra of $X$, and it is denoted $\mathcal{C}al(X)$. We prove that the unitization of $\mathcal{K}(c_0)$ is isomorphic as a Banach algebra to the Calkin algebra of some Banach space $\mathcal{Z}_{\mathcal{K}(c_0)}$. This Banach space is an Argyros-Haydon sum $(\oplus_{n=1}^\infty X_n)_\mathrm{AH}$ of a sequence of copies $X_n$ of a single Argyros-Haydon space $\mathfrak{X}_\mathrm{AH}$, and the external versus the internal Argyros-Haydon construction parameters are chosen from disjoint sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The compact operators on $c_0$ as a Calkin algebra Motakis, Pavlos Puglisi, Daniele Functional Analysis Operator Algebras 46B07, 46B25, 46B28 For a Banach space $X$, let $\mathcal{L}(X)$ denote the algebra of all bounded linear operators on $X$ and let $\mathcal{K}(X)$ denote the compact operator ideal in $\mathcal{L}(X)$. The quotient algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is called the Calkin algebra of $X$, and it is denoted $\mathcal{C}al(X)$. We prove that the unitization of $\mathcal{K}(c_0)$ is isomorphic as a Banach algebra to the Calkin algebra of some Banach space $\mathcal{Z}_{\mathcal{K}(c_0)}$. This Banach space is an Argyros-Haydon sum $(\oplus_{n=1}^\infty X_n)_\mathrm{AH}$ of a sequence of copies $X_n$ of a single Argyros-Haydon space $\mathfrak{X}_\mathrm{AH}$, and the external versus the internal Argyros-Haydon construction parameters are chosen from disjoint sets. |
| title | The compact operators on $c_0$ as a Calkin algebra |
| topic | Functional Analysis Operator Algebras 46B07, 46B25, 46B28 |
| url | https://arxiv.org/abs/2403.04137 |