The Bishop-Phelps-Bollobas property for certain Banach spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910356808400896 |
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| author | Bhattacharyya, Tirthankar Bhowmik, Mainak Dhara, Kousik |
| author_facet | Bhattacharyya, Tirthankar Bhowmik, Mainak Dhara, Kousik |
| contents | Let $X$ be a complex Banach space. We prove that if $L$ is an extremally disconnected compact Hausdorff topological space, then the pair $(X, C(L))$ satisfies the Bishop-Phelps-Bollobás property (BPBp for short). As a byproduct, we obtain the BPBp for the pair $(X, L^\infty(ν))$ for any measure $ν$. In particular, this settles an unresolved question regarding the BPBp for the pair $(L^\infty(μ), L^\infty(ν) )$ for any two measures $μ$ and $ν$. Finally, we show that $(X,H^\infty(Ω)$ has the BPBp when $Ω$ is a multi-connected planar domain bounded by finitely many disjoint analytic simple closed curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16135 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Bishop-Phelps-Bollobas property for certain Banach spaces Bhattacharyya, Tirthankar Bhowmik, Mainak Dhara, Kousik Functional Analysis Complex Variables 46B20, 30H05, 46J15, 47L20, 46J10 Let $X$ be a complex Banach space. We prove that if $L$ is an extremally disconnected compact Hausdorff topological space, then the pair $(X, C(L))$ satisfies the Bishop-Phelps-Bollobás property (BPBp for short). As a byproduct, we obtain the BPBp for the pair $(X, L^\infty(ν))$ for any measure $ν$. In particular, this settles an unresolved question regarding the BPBp for the pair $(L^\infty(μ), L^\infty(ν) )$ for any two measures $μ$ and $ν$. Finally, we show that $(X,H^\infty(Ω)$ has the BPBp when $Ω$ is a multi-connected planar domain bounded by finitely many disjoint analytic simple closed curves. |
| title | The Bishop-Phelps-Bollobas property for certain Banach spaces |
| topic | Functional Analysis Complex Variables 46B20, 30H05, 46J15, 47L20, 46J10 |
| url | https://arxiv.org/abs/2401.16135 |