The ergodicity of Orlicz sequence spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915621930795008 |
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| author | de Rancourt, Noé Kurka, Ondřej |
| author_facet | de Rancourt, Noé Kurka, Ondřej |
| contents | We prove that non-Hilbertian separable Orlicz sequence spaces are ergodic, i.e., the equivalence relation $\mathbb{E}_0$ Borel reduces to the isomorphism relation between subspaces of every such space. This is done by exhibiting non-Hilbertian asymptotically Hilbertian subspaces in those spaces, and appealing to a result by Anisca. In particular, each non-Hilbertian Orlicz sequence space contains continuum many pairwise non-isomorphic subspaces.
As a consequence, we prove that the twisted Hilbert spaces $\ell_2(ϕ)$ constructed by Kalton and Peck are either Hilbertian, or ergodic. This applies in particular to the Kalton--Peck space $Z_2$ and all twisted Hilbert spaces generated by complex interpolation between Orlicz sequence spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17756 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The ergodicity of Orlicz sequence spaces de Rancourt, Noé Kurka, Ondřej Functional Analysis 46B45 (Primary) 46B03, 46B06, 46B20, 03E15, 46M18, 46B70 (Secondary) We prove that non-Hilbertian separable Orlicz sequence spaces are ergodic, i.e., the equivalence relation $\mathbb{E}_0$ Borel reduces to the isomorphism relation between subspaces of every such space. This is done by exhibiting non-Hilbertian asymptotically Hilbertian subspaces in those spaces, and appealing to a result by Anisca. In particular, each non-Hilbertian Orlicz sequence space contains continuum many pairwise non-isomorphic subspaces. As a consequence, we prove that the twisted Hilbert spaces $\ell_2(ϕ)$ constructed by Kalton and Peck are either Hilbertian, or ergodic. This applies in particular to the Kalton--Peck space $Z_2$ and all twisted Hilbert spaces generated by complex interpolation between Orlicz sequence spaces. |
| title | The ergodicity of Orlicz sequence spaces |
| topic | Functional Analysis 46B45 (Primary) 46B03, 46B06, 46B20, 03E15, 46M18, 46B70 (Secondary) |
| url | https://arxiv.org/abs/2501.17756 |