论文标题
Banach空间张量产品的拓扑特性
Topological properties in tensor products of Banach spaces
论文作者
论文摘要
给定两个Banach Spaces $ x $和$ Y $，我们分析了何时投影量张量产品$ x \ wideHat {\ otimes}_πy$具有Corson的财产（C），或者是弱的Lindelöf（WLD）（wld），弱压实生成（WCG）空间的弱化（WCG）空间的子空间。例如，我们表明：（i）$ x \ wideHat {\ otimes}_πy$在且仅当$ x $和$ y $ is wld is wld is wld is wld is wld is wld，所有运营商从$ x $到$ y^*$，从$ y $到$ y $到$ x^*$具有分离范围； （ii）$ x \ wideHat {\ otimes}_πy$是wcg空间的子空间，如果在每个运营商$ x $ to x $ to y^$ compact的假设下都有相同的$ x $和$ y $的子空间； （iii）$ \ ell_p（γ）\ wideHat {\ otimes}_π\ ell_q（γ）$是Hilbert生成的空间的子空间，用于任何$ 1 <p，q <\ infty $，因此$ 1/p+1/q <1 $，以及任何Infinite Set $ unfinite set $ gum $γ$。我们还注意注射量张量产品$ x \ wideHat {\ otimes} _ \ varepsilon y $。在这种情况下，属性（C）的稳定性和被视为与所有常规的Borel概率测量在双球上具有可数的Maharam类型的条件密切相关。按照这种方式，我们概括了Plebanek和Sobota的结果，即如果$ k $是一个紧凑的空间，那么$ c（k \ times k）$具有属性（c），那么所有常规的borel概率指标$ k $都具有可计数的maharam类型。这种概括为Ruess和Werner的问题提供了一致的负面答案，该问题是关于$ W^*$ - 在注射量张量产品下的双单元球的天使。
Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_πY$ has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_πY$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_πY$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $Y^*$ is compact; (iii) $\ell_p(Γ)\widehat{\otimes}_π\ell_q(Γ)$ is subspace of a Hilbert generated space for any $1< p,q<\infty$ such that $1/p+1/q<1$ and for any infinite set $Γ$. We also pay attention to the injective tensor product $X\widehat{\otimes}_\varepsilon Y$. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if $K$ is a compact space such that $C(K\times K)$ has property (C), then all regular Borel probability measures on $K$ have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the $w^*$-angelicity of the dual unit ball under injective tensor products.
