Embeddings of $L^p$-operator algebras
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866909997081821184 |
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| author | Gardella, Eusebio Gundelach, Jan |
| author_facet | Gardella, Eusebio Gundelach, Jan |
| contents | We study embeddings of $L^p$-operator algebras arising from (twisted) étale groupoids, with particular emphasis on rigidity phenomena for $p\neq 2$. Our methods rely on a detailed analysis of core normalizers and their functorial behavior under algebra homomorphisms. Using the notion of actors between groupoids, we show that under natural hypotheses, embeddings between reduced $L^p$-groupoid algebras can be described entirely in terms of morphisms of the underlying groupoids. We further show that embeddings of $L^p$-groupoid algebras induce embeddings of the associated topological full groups. Our results provide new tools for studying embeddability questions in the $L^p$-setting, and are particularly helpful when ruling out the existence of embeddings.
As applications, we obtain strong rigidity results for (spatial) $L^p$-AF-embeddability, showing that, for $p\neq 2$, an $L^p$-groupoid algebra embeds into a spatial $L^p$-AF-algebra if and only if the underlying groupoid is AF. In particular, irrational rotation $L^p$-operator algebras do not embed into spatial $L^p$-AF-algebras. We apply these results to tensor products of $L^p$-Cuntz algebras and prove that, for $p\neq 2$, there is no unital contractive homomorphism from $\mathcal{O}_2^p \otimes_p \mathcal{O}_2^p$ into $\mathcal{O}_2^p$, showing that there is no $L^p$-analog of Kirchberg's $\mathcal{O}_2$-embedding theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Embeddings of $L^p$-operator algebras Gardella, Eusebio Gundelach, Jan Functional Analysis Operator Algebras We study embeddings of $L^p$-operator algebras arising from (twisted) étale groupoids, with particular emphasis on rigidity phenomena for $p\neq 2$. Our methods rely on a detailed analysis of core normalizers and their functorial behavior under algebra homomorphisms. Using the notion of actors between groupoids, we show that under natural hypotheses, embeddings between reduced $L^p$-groupoid algebras can be described entirely in terms of morphisms of the underlying groupoids. We further show that embeddings of $L^p$-groupoid algebras induce embeddings of the associated topological full groups. Our results provide new tools for studying embeddability questions in the $L^p$-setting, and are particularly helpful when ruling out the existence of embeddings. As applications, we obtain strong rigidity results for (spatial) $L^p$-AF-embeddability, showing that, for $p\neq 2$, an $L^p$-groupoid algebra embeds into a spatial $L^p$-AF-algebra if and only if the underlying groupoid is AF. In particular, irrational rotation $L^p$-operator algebras do not embed into spatial $L^p$-AF-algebras. We apply these results to tensor products of $L^p$-Cuntz algebras and prove that, for $p\neq 2$, there is no unital contractive homomorphism from $\mathcal{O}_2^p \otimes_p \mathcal{O}_2^p$ into $\mathcal{O}_2^p$, showing that there is no $L^p$-analog of Kirchberg's $\mathcal{O}_2$-embedding theorem. |
| title | Embeddings of $L^p$-operator algebras |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2601.15204 |