整数的对称紧化与空间 $C_p(X)$ 的可分商空间

中文摘要 AI 辅助

所有整数的离散空间 $\omega$ 的紧化被称为对称的，如果它是通过将 $\omega$ 的某个其他紧化的两个副本的余部粘合在一起而得到的商空间。这是收敛序列（在某种意义上是最小对称紧化）和 $C_p$-理论中研究的 Arkhangel'ski\u0131--Bereznitski\u0131--Schachermayer 空间（在某种意义上是 $\omega$ 的最大对称紧化）两者的推广。我们研究 $\omega$ 的对称紧化及其与空间 $C_p(X)$ 的可分商问题以及空间 $X$ 上有限支撑 Borel 测度的 Josefson--Nissenzweig 序列（特别是具有有界大小的支撑）的存在性的关系。此外，我们将空间 $C_p(K)$（$K$ 紧）的可分商问题归结为 $K$ 是 $\omega$ 的完全非对称紧化的情况。我们的结果为 Banach 空间 $C(K)$ 的 Grothendieck 性质提供了一些新的启示。

英文摘要

A compactification of the discrete space $ω$ of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of $ω$. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of $ω$, and the Arkhangel'ski\uı--Bereznitski\uı--Schachermayer space studied in $C_p$-theory, which is in a sense the maximal symmetric compactification of $ω$. We investigate symmetric compactifications of $ω$ and their relations to the Separable Quotient Problem for spaces $C_p(X)$ and to the existence of Josefson--Nissenzweig sequences of finitely supported Borel measures on spaces $X$, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces $C_p(K)$, $K$ compact, to the case when $K$ is a totally asymmetric compactification of $ω$. Our results shed some new light on the Grothendieck property of Banach spaces $C(K)$.

发表机构

• Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）
• University of Vienna（维也纳大学）

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