Nonassociative $\mathrm{L}^p$-spaces and embeddings in noncommutative $\mathrm{L}^p$-spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916130403123200 |
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| author | Arhancet, Cédric |
| author_facet | Arhancet, Cédric |
| contents | We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $φ$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More precisely, we show that our nonassociative $\mathrm{L}^p$-spaces contain isometrically the $\mathrm{L}^p$-spaces of Iochum and that all tracial nonassociative $\mathrm{L}^p$-spaces from $\mathrm{JW}^*$-factors arise as positively contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04452 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonassociative $\mathrm{L}^p$-spaces and embeddings in noncommutative $\mathrm{L}^p$-spaces Arhancet, Cédric Operator Algebras Functional Analysis Quantum Algebra We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $φ$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More precisely, we show that our nonassociative $\mathrm{L}^p$-spaces contain isometrically the $\mathrm{L}^p$-spaces of Iochum and that all tracial nonassociative $\mathrm{L}^p$-spaces from $\mathrm{JW}^*$-factors arise as positively contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. |
| title | Nonassociative $\mathrm{L}^p$-spaces and embeddings in noncommutative $\mathrm{L}^p$-spaces |
| topic | Operator Algebras Functional Analysis Quantum Algebra |
| url | https://arxiv.org/abs/2307.04452 |