论文标题
投影张量产品的极端结构
Extremal structure of projective tensor products
论文作者
论文摘要
我们证明，给定两个Banach空间$ x $和$ y $以及有限的，封闭的凸组集合$ c \ subseteq x $和$ d \ subseteq y $，如果一个非零元素$ z \ in \ in \ in \ in \ incepline {\ mathrm {co}}（co}}}（c \ otimes d）\ otimies d）\ speteq x $ isse_; $ z = x_0 \ otimes y_0 $对于某些保留的极端点$ x_0 \ in c $和d $中的$ y_0 \，每当$ k（x，y^*）$分离$ x \ x \ wideHat {\ outimes}_πy$（特别是尤其是$ x $ y $ y $ y y $ ymimations commant commations commant commatect commate）的点。此外，我们证明，如果$ x_0 \在c $中，而d $中的$ y_0 \是弱暴露的点，那么$ x_0 \ otimes y_0 $在$ \ operline {\ mathrm {co}}}}（c \ otimimes d）$ x__0 $ x_0 \ y_0 $ x_0 \ otimes a y_0 $ x_0 \ otimes a y_0 potimes a potsation中均为$ x_0 \ y_0操作员。此外，我们在b_x $中找到了一个Banach Space $ x $同构为$ \ ell_2 $，而B_x $中的弱曝光点$ x_0 \，因此$ x_0 \ otimes x_0 $并不是$ x \ widehat的单位球的微弱曝光点，
We prove that, given two Banach spaces $X$ and $Y$ and bounded, closed convex sets $C\subseteq X$ and $D\subseteq Y$, if a nonzero element $z\in \overline{\mathrm{co}}(C\otimes D)\subseteq X\widehat{\otimes}_πY$ is a preserved extreme point then $z=x_0\otimes y_0$ for some preserved extreme points $x_0\in C$ and $y_0\in D$, whenever $K(X,Y^*)$ separates points of $X \widehat{\otimes}_πY$ (in particular, whenever $X$ or $Y$ has the compact approximation property). Moreover, we prove that if $x_0\in C$ and $y_0\in D$ are weak-strongly exposed points then $x_0\otimes y_0$ is weak-strongly exposed in $\overline{\mathrm{co}}(C\otimes D)$ whenever $x_0\otimes y_0$ has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space $X$ isomorphic to $\ell_2$ with a weak-strongly exposed point $x_0\in B_X$ such that $x_0\otimes x_0$ is not a weak-strongly exposed point of the unit ball of $X\widehat{\otimes}_πX$.
