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Surjective separating maps on noncommutative $L^p$-spaces

Abstract : Let $1\leq p<\infty$ and let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded map between noncommutative $L^p$-spaces. If $T$ is bijective and separating (i.e., for any $x,y\in L^p({\mathcal M})$ such that $x^*y=xy^*=0$, we have $T(x)^*T(y)=T(x)T(y)^*=0$), we prove the existence of decompositions ${\mathcal M}={\mathcal M}_1\mathop{\oplus}\limits^\infty{\mathcal M}_2$, ${\mathcal N}={\mathcal N}_1 \mathop{\oplus}\limits^\infty{\mathcal N}_2$ and maps $T_1\colon L^p({\mathcal M}_1)\to L^p({\mathcal N}_1)$, $T_2\colon L^p({\mathcal M}_2)\to L^p({\mathcal N}_2)$, such that $T=T_1+T_2$, $T_1$ has a direct Yeadon type factorisation and $T_2$ has an anti-direct Yeadon type factorisation. We further show that $T^{-1}$ is separating in this case. Next we prove that for any $1\leq p<\infty$ (resp. any $1\leq p\not=2<\infty$), a surjective separating map $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is $S^1$-bounded (resp. completely bounded) if and only if there exists a decomposition ${\mathcal M}={\mathcal M}_1 \mathop{\oplus}\limits^\infty{\mathcal M}_2$ such that $T|_{L^p({\tiny {\mathcal M}_1})}$ has a direct Yeadon type factorisation and ${\mathcal M}_2$ is subhomogeneous.
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https://hal.archives-ouvertes.fr/hal-03525871
Contributor : Christian Le Merdy Connect in order to contact the contributor
Submitted on : Friday, January 14, 2022 - 10:24:58 AM
Last modification on : Wednesday, January 19, 2022 - 4:38:44 PM
Long-term archiving on: : Friday, April 15, 2022 - 6:28:24 PM

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  • HAL Id : hal-03525871, version 1
  • ARXIV : 2009.05919

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Christian Le Merdy, Safoura Zadeh. Surjective separating maps on noncommutative $L^p$-spaces. Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, inPress. ⟨hal-03525871⟩

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