2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866911855325216768 |
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| author | Arhancet, Cédric Raynaud, Yves |
| author_facet | Arhancet, Cédric Raynaud, Yves |
| contents | We prove the first theorem on projections on general noncommutative $\mathrm{L}^p$-spaces associated with non-type I von Neumann algebras where $1 \leqslant p < \infty$. This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative $\mathrm{L}^p$-space is completely order isometrically isomorphic to some noncommutative $\mathrm{L}^p$-space. This result is sharp and is even new for Schatten spaces $S^p$. Our approach relies on non-tracial Haagerup's noncommutative $\mathrm{L}^p$-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_03128 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | 2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces Arhancet, Cédric Raynaud, Yves Operator Algebras Functional Analysis We prove the first theorem on projections on general noncommutative $\mathrm{L}^p$-spaces associated with non-type I von Neumann algebras where $1 \leqslant p < \infty$. This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative $\mathrm{L}^p$-space is completely order isometrically isomorphic to some noncommutative $\mathrm{L}^p$-space. This result is sharp and is even new for Schatten spaces $S^p$. Our approach relies on non-tracial Haagerup's noncommutative $\mathrm{L}^p$-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman. |
| title | 2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/1912.03128 |