On the extension of positive maps to Haagerup non-commutative $L^p$-spaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929697724563456 |
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| author | Merdy, Christian Le Zadeh, Safoura |
| author_facet | Merdy, Christian Le Zadeh, Safoura |
| contents | Let $M$ be a von Neumann algebra, let $φ$ be a normal faithful state on $M$ and let $L^p(M,φ)$ be the associated Haagerup non-commutative $L^p$-spaces, for $1\leq p\leq\infty$. Let $D\in L^1(M,φ)$ be the density of $φ$. Given a positive map $T\colon M\to M$ such that $φ\circ T\leq C_1φ$ for some $C_1\geq 0$, we study the boundedness of the $L^p$-extension $T_{p,θ}\colon D^{\frac{1-θ}{p}} M D^{\fracθ{p}}\to L^p(M,φ)$ which maps $D^{\frac{1-θ}{p}} x D^{\fracθ{p}}$ to $D^{\frac{1-θ}{p}} T(x) D^{\fracθ{p}}$ for all $x\in M$. Haagerup-Junge-Xu showed that $T_{p,\frac12}$ is always bounded and left open the question whether $T_{p,θ}$ is bounded for $θ\not=\frac12$. We show that for any $1\leq p<2$ and any $θ\in [0,2^{-1}(1-\sqrt{p-1})]\cup[2^{-1}(1+\sqrt{p-1}), 1]$, there exists a completely positive $T$ such that $T_{p,θ}$ is unbounded. We also show that if $T$ is $2$-positive, then $T_{p,θ}$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $θ\in[1-p/2,p/2]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the extension of positive maps to Haagerup non-commutative $L^p$-spaces Merdy, Christian Le Zadeh, Safoura Operator Algebras 46L51 Let $M$ be a von Neumann algebra, let $φ$ be a normal faithful state on $M$ and let $L^p(M,φ)$ be the associated Haagerup non-commutative $L^p$-spaces, for $1\leq p\leq\infty$. Let $D\in L^1(M,φ)$ be the density of $φ$. Given a positive map $T\colon M\to M$ such that $φ\circ T\leq C_1φ$ for some $C_1\geq 0$, we study the boundedness of the $L^p$-extension $T_{p,θ}\colon D^{\frac{1-θ}{p}} M D^{\fracθ{p}}\to L^p(M,φ)$ which maps $D^{\frac{1-θ}{p}} x D^{\fracθ{p}}$ to $D^{\frac{1-θ}{p}} T(x) D^{\fracθ{p}}$ for all $x\in M$. Haagerup-Junge-Xu showed that $T_{p,\frac12}$ is always bounded and left open the question whether $T_{p,θ}$ is bounded for $θ\not=\frac12$. We show that for any $1\leq p<2$ and any $θ\in [0,2^{-1}(1-\sqrt{p-1})]\cup[2^{-1}(1+\sqrt{p-1}), 1]$, there exists a completely positive $T$ such that $T_{p,θ}$ is unbounded. We also show that if $T$ is $2$-positive, then $T_{p,θ}$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $θ\in[1-p/2,p/2]$. |
| title | On the extension of positive maps to Haagerup non-commutative $L^p$-spaces |
| topic | Operator Algebras 46L51 |
| url | https://arxiv.org/abs/2404.04400 |