Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866914176358678528 |
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| author | Tylli, Hans-Olav Wirzenius, Henrik |
| author_facet | Tylli, Hans-Olav Wirzenius, Henrik |
| contents | We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_0$ into $\mathcal K(Z)/\mathcal A(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space $c_0(Γ)$ into $\mathcal K(Z_{FJ})/\mathcal A(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1811_09402 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property Tylli, Hans-Olav Wirzenius, Henrik Functional Analysis 46B28, 47L10 We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_0$ into $\mathcal K(Z)/\mathcal A(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space $c_0(Γ)$ into $\mathcal K(Z_{FJ})/\mathcal A(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel. |
| title | Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property |
| topic | Functional Analysis 46B28, 47L10 |
| url | https://arxiv.org/abs/1811.09402 |