论文标题
关于分离地图在非共同$ l^p $空间上的分解
On factorization of separating maps on noncommutative $L^p$-spaces
论文作者
论文摘要
对于任何半决赛von neumann代数$ {\ Mathcal M} $和任何$ 1 \ leq p <\ infty $，我们引入了一种natutal $ s^1 $ s^1 $ valued nont-valued nont-valued $ l^p $ -space $ -Space $ -Space $ l^p（{\ Mathcal m};我们说，如果$ s^p（{\ nathcal n}）$是$ s^1 $ bounded（resp。$ s^1 $ - contractive），如果$ t \ otime imime i_ {s^1} $扩展到formimiese n forts n forsime { i_ {s^1} $来自$ l^p（{\ Mathcal m}; s^1）$ to $ l^p（{\ Mathcal n}; s^1）$。我们表明，任何完全积极的地图都是$ s^1 $ bunded，带有$ \ vert t \ overline {\ otimes} i_ {s^1} \ vert = \ vert = \ vert t \ vert $。我们将上述工具用作研究分离的映射$ t \ colon l^p（{\ Mathcal m}）\ to l^p（{\ Mathcal n}）$，该$接纳了直接的Yeadon类型分解，也就是说，即存在$ w^*$ - 连续$*$ - 连续$*$ - homomorphism $ j \ colon m} \ to {\ Mathcal n} $，{\ Mathcal n} $ in {\ Mathcal n} $的部分等轴测$ W \和一个阳性运算符$ b $与$ {\ Mathcal n} $相关的$ {\ Mathcal n} $，以便$ w^*w = j（1）= s（b）$，$ b $，$ b $ for $ j $ $ j $（$ j $）（和$ j $ t（）） {\ Mathcal M} \ CAP l^P（{\ Mathcal M}）$。给定一个分离的等距$ t \ colon l^p（{\ Mathcal m}）\ to l^p（{\ Mathcal n}）$，我们表明$ t $是$ s^1 $ - 合同，并且仅当它承认直接Yeadon类型分解时。我们进一步表明，如果$ p \ not = 2 $，则以上时，仅当$ t $完全承包时。
For any semifinite von Neumann algebra ${\mathcal M}$ and any $1\leq p<\infty$, we introduce a natutal $S^1$-valued noncommutative $L^p$-space $L^p({\mathcal M};S^1)$. We say that a bounded map $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is $S^1$-bounded (resp. $S^1$-contractive) if $T\otimes I_{S^1}$ extends to a bounded (resp. contractive) map $T\overline{\otimes} I_{S^1}$ from $ L^p({\mathcal M};S^1)$ into $L^p({\mathcal N};S^1)$. We show that any completely positive map is $S^1$-bounded, with $\Vert T\overline{\otimes} I_{S^1}\Vert =\Vert T\Vert$. We use the above as a tool to investigate the separating maps $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ which admit a direct Yeadon type factorization, that is, maps for which there exist a $w^*$-continuous $*$-homomorphism $J\colon{\mathcal M}\to{\mathcal N}$, a partial isometry $w\in{\mathcal N}$ and a positive operator $B$ affiliated with ${\mathcal N}$ such that $w^*w=J(1)=s(B)$, $B$ commutes with the range of $J$, and $T(x)=wBJ(x)$ for any $x\in {\mathcal M}\cap L^p({\mathcal M})$. Given a separating isometry $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$, we show that $T$ is $S^1$-contractive if and only if it admits a direct Yeadon type factorization. We further show that if $p\not=2$, the above holds true if and only if $T$ is completely contractive.
