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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.11809 |
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| _version_ | 1866908708178493440 |
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| author | Azouzi, Youssef |
| author_facet | Azouzi, Youssef |
| contents | Let E be a Dedekind complete Riesz space with weak unit e, equipped with a conditional expectation operator T. We prove that the spaces Lp(T), with their natural vector-valued norms, are strongly complete, extending the p=2 case of Kuo, Kalauch, and Watson. This resolves a question that has remained open for several years. We begin by studying a general type of convergence and its unbounded modification, unifying and generalizing order, norm, and absolute weak convergence while providing simpler proofs. As an application, we consider vector-valued norms and their unbounded variants, generalizing strong convergence in Lp-spaces and convergence in probability. This framework establishes the completeness of Lp(T) and of the universal completion E^{u}, reinforcing the uo-completeness of universally complete vector lattices. Finally we apply our main theorem to obtain a new result in ergodicity for conditional preserving systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong completeness of Lp-type vector lattices Azouzi, Youssef Functional Analysis 60B12, 60F15 Let E be a Dedekind complete Riesz space with weak unit e, equipped with a conditional expectation operator T. We prove that the spaces Lp(T), with their natural vector-valued norms, are strongly complete, extending the p=2 case of Kuo, Kalauch, and Watson. This resolves a question that has remained open for several years. We begin by studying a general type of convergence and its unbounded modification, unifying and generalizing order, norm, and absolute weak convergence while providing simpler proofs. As an application, we consider vector-valued norms and their unbounded variants, generalizing strong convergence in Lp-spaces and convergence in probability. This framework establishes the completeness of Lp(T) and of the universal completion E^{u}, reinforcing the uo-completeness of universally complete vector lattices. Finally we apply our main theorem to obtain a new result in ergodicity for conditional preserving systems. |
| title | Strong completeness of Lp-type vector lattices |
| topic | Functional Analysis 60B12, 60F15 |
| url | https://arxiv.org/abs/2512.11809 |