论文标题
grothendieck $ c（k）$ - 空间和约瑟夫森 - 尼森森温定理
Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem
论文作者
论文摘要
对于紧凑的空间$ k $，据说Banach Space $ C（K）$具有$ \ ell_1 $ -grothendieck属性，如果每个弱*收敛序列$ \ big \ big \ big \ big \langleμ_n\ colon \ colon \ colon \ n \inΩ弱收敛。因此，$ \ ell_1 $ -grothendieck属性是连续功能的Banach空间的标准Grothendieck属性的削弱。我们观察到，$ c（k）$具有$ \ ell_1 $ -grothendieck属性，并且仅在不存在任何一系列函数$ \ big \ big \ big \ big \langleμ_n\ colon \ n \ n \inmoΩ Josefson - Nissenzweig定理。我们构建一个可分离的紧凑型空间$ k $的示例，以便$ c（k）$具有$ \ ell_1 $ -grothendieck属性，但没有Grothendieck属性。我们还表明，对于许多经典的Efimov Spaces $ K $的一致示例，它们的Banach Spaces $ C（k）$没有$ \ ell_1 $ -grothendieck属性。
For a compact space $K$, the Banach space $C(K)$ is said to have the $\ell_1$-Grothendieck property if every weak* convergent sequence $\big\langleμ_n\colon\ n\inω\big\rangle$ of functionals on $C(K)$ such that $μ_n\in\ell_1(K)$ for every $n\inω$, is weakly convergent. Thus, the $\ell_1$-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that $C(K)$ has the $\ell_1$-Grothendieck property if and only if there does not exist any sequence of functionals $\big\langleμ_n\colon\ n\inω\big\rangle$ on $C(K)$, with $μ_n\in\ell_1(K)$ for every $n\inω$, satisfying the conclusion of the classical Josefson--Nissenzweig theorem. We construct an example of a separable compact space $K$ such that $C(K)$ has the $\ell_1$-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces $K$ their Banach spaces $C(K)$ do not have the $\ell_1$-Grothendieck property.
