Exotic closed subideals of algebras of bounded operators
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914653580296192 |
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| author | Tylli, Hans-Olav Wirzenius, Henrik |
| author_facet | Tylli, Hans-Olav Wirzenius, Henrik |
| contents | We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathscr F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the subideals in $\mathscr F$ are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras $\mathcal L(X)$ of bounded operators on $X$, where closed ideals $\mathcal I \neq \mathcal J$ are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators $\mathcal S(X)$ for classical spaces such as $X = L^p$ with $p \neq 2$, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exotic closed subideals of algebras of bounded operators Tylli, Hans-Olav Wirzenius, Henrik Functional Analysis 46H10, 46B28, 47L10 We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathscr F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the subideals in $\mathscr F$ are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras $\mathcal L(X)$ of bounded operators on $X$, where closed ideals $\mathcal I \neq \mathcal J$ are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators $\mathcal S(X)$ for classical spaces such as $X = L^p$ with $p \neq 2$, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. |
| title | Exotic closed subideals of algebras of bounded operators |
| topic | Functional Analysis 46H10, 46B28, 47L10 |
| url | https://arxiv.org/abs/2301.09425 |