AI 中文总结
研究包含$c_0$的$\ell_\infty$理想的格罗滕迪克性质相关问题，通过有限可加测度等给出刻画，证明存在解析理想使$c_{0,\mathcal I}$是格罗滕迪克空间，同时表明多数博雷尔理想不具有该性质，且格罗滕迪克性质在相关序中不向下封闭。
AI 中文摘要
我们回答了文献中关于包含$c_0$的巴拿赫格$\ell_\infty$的理想的格罗滕迪克性质的几个问题。任何这样的理想都可表示为$c_{0,\mathcal I}$，其中$\mathcal I$是自然数上的理想。我们根据$\mathcal P(\omega)$上的有限可加测度和$\mathcal I$的元素给出了$c_{0,\mathcal I}$是格罗滕迪克空间的一个刻画。利用此刻画，我们证明存在解析理想$\mathcal I$使得$c_{0,\mathcal I}$是格罗滕迪克空间。反之，我们表明对于任何几乎不交族$\mathcal A$，$c_{0,\mathcal I(\mathcal A)}$和$C(K_\mathcal A)$不是格罗滕迪克空间，并且对于文献中大多数博雷尔理想，$c_{0,\mathcal I}$不是格罗滕迪克空间。特别地，不具有格罗滕迪克性质的理想族在鲁丁 - 基斯勒序中是共尾的，所以格罗滕迪克性质在卡特托夫序中不是向下封闭的。继续[文献引用]中的工作，我们还给出了关于由理想$\mathcal I$生成的$\mathcal P(\omega)$的布尔子代数的尼科迪姆性质的类似结果。
英文摘要
We answer several questions in the literature concerning the Grothendieck property of ideals of the Banach lattice $\ell_\infty$ that contain $c_0$. Any such ideal can be represented as a space $c_{0,\mathcal I}$ for $\mathcal I$ an ideal over the natural numbers. We provide a characterization of when $c_{0,\mathcal I}$ is a Grothendieck space in terms of finitely additive measures over $\mathcal P(ω)$ and elements of $\mathcal I$. Using this characterization we show that there are analytic ideals $\mathcal I$ such that $c_{0,\mathcal I}$ is Grothendieck. In the opposite direction we show that for any AD family $\mathcal A$, $c_{0,\mathcal I(\mathcal A)}$ and $C(K_\mathcal A)$ are not Grothendieck spaces, and that for most of the Borel ideals present in the literature, $c_{0,\mathcal I}$ is not Grothendieck. In particular, the family of ideals that do not have the Grothendieck property is cofinal in the Rudin-Keisler order, so the Grothendieck property is not downward closed in the Katětov order. Continuing the work in \cite{Sobota-Zuchowski, Zuchowski}, we also provide similar results on the Nikodym property of the Boolean subalgebras of $\mathcal P(ω)$ generated by the ideal $\mathcal I$.
