论文标题
Josefson - Nissenzweig定理和$ω$的过滤器
The Josefson--Nissenzweig theorem and filters on $ω$
论文作者
论文摘要
For a free filter $F$ on $ω$, endow the space $N_F=ω\cup\{p_F\}$, where $p_F\not\inω$, with the topology in which every element of $ω$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$.表格$ n_f $的空间构成了最简单的非污点Tychonoff空间的类。本文的目的是在著名的约瑟夫森（Josefson-Nissenzweig theorem）的背景下研究它们。我们证明，例如，对于过滤器$ f $，空间$ n_f $带有序列$ \langleμ_n\ colon n \ colon n \inΩ\inΩ\ rangle $有限支持的签名措施，以至于每一个有限的持续不动数$ n_的$μ_n（f）\ to $ n_ $ f^*\ le_k \ MATHCAL {z} $，即，双重理想$ f^*$是低于渐近密度理想$ \ Mathcal {z} $的katětov。 Consequently, we get that if $F^*\le_K\mathcal{Z}$, then: (1) if $X$ is a Tychonoff space and $N_F$ is homeomorphic to a subspace of $X$, then the space $C_p^*(X)$ of bounded continuous real-valued functions on $X$ contains a complemented copy of the space $c_0$ endowed with the pointwise拓扑，（2）如果$ k $是一个紧凑的豪斯多夫太空，而$ n_f $是$ k $的子空间同型，则$ k $连续实现的banach space $ c（k）$ a $ k $上的$ c（k）$不是薄荷空间。后一个结果概括了一个众所周知的事实，即如果紧凑的Hausdorff Space $ k $包含非平凡的收敛序列，那么Space $ C（K）$就不是Grothendieck。
For a free filter $F$ on $ω$, endow the space $N_F=ω\cup\{p_F\}$, where $p_F\not\inω$, with the topology in which every element of $ω$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. Spaces of the form $N_F$ constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson--Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter $F$, the space $N_F$ carries a sequence $\langleμ_n\colon n\inω\rangle$ of normalized finitely supported signed measures such that $μ_n(f)\to 0$ for every bounded continuous real-valued function $f$ on $N_F$ if and only if $F^*\le_K\mathcal{Z}$, that is, the dual ideal $F^*$ is Katětov below the asymptotic density ideal $\mathcal{Z}$. Consequently, we get that if $F^*\le_K\mathcal{Z}$, then: (1) if $X$ is a Tychonoff space and $N_F$ is homeomorphic to a subspace of $X$, then the space $C_p^*(X)$ of bounded continuous real-valued functions on $X$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology, (2) if $K$ is a compact Hausdorff space and $N_F$ is homeomorphic to a subspace of $K$, then the Banach space $C(K)$ of continuous real-valued functions on $K$ is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space $K$ contains a non-trivial convergent sequence, then the space $C(K)$ is not Grothendieck.
