论文标题
Josefson- Nissenzweig定理，Grothendieck属性，并有限支持紧凑的空间
The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces
论文作者
论文摘要
著名的Josefson-Nissenzweig定理意味着对于Banach空间$ c（k）$连续的真实价值功能，在无限的紧凑型空间上$ k $上存在一系列radon测量$ \langleμ_n\langleμ_n\ colon \ colon \ colon \ colon \ n \ n \ in $ k $ k $ k $ k in $ k y y y y y n in n and n and cons of zero coment to convers to cons to cons to cons to convermint converminge conversgint $ \ big \ |μ_n\ big \ | = 1 $，每$ n \inΩ$。我们称之为一系列度量\ textit {a josefson-nissenzweig序列}。在本文中，我们研究了空间$ k $承认约瑟夫森·尼森兹韦格（Josefson-Nissenzweig）的措施序列的情况，以使其每个元素都具有有限的支持。我们在其他人中证明，$ k $在且仅当$ c（k）$没有限制于Space $ \ ell_1（k）$的功能的grothendieck属性时，就承认了这样的Josefson-Nissenzweig序列。我们还研究了一般tychonoff空间上有限支持的约瑟夫森 - 尼森兹韦格序列的杂项分析和拓扑特性。 我们证明，紧凑空间的各种特性可以保证存在有限支持的Josefson-Nissenzweig序列。这样的属性是，例如，基于简单的扩展，可以将紧凑空间表示为紧凑型空间的逆系统的极限。该结果的直接结果是，许多经典的Efimov空间的一致示例，即，空间是对著名的Efimov问题的反例，都承认了这样的措施序列。 同样，我们表明，如果$ k $和$ l $是无限紧凑的空间，那么他们的产品$ k \ times l $总是承认有限支持的Josefson-Nissenzweig序列。作为推论，我们获得了一个建设性的证明，即空间$ c_p（k \ times l）$包含带有点端拓扑的空间$ C_0 $的补充副本 - 这概括了cembranos和freniche的结果。 最后，我们直接证明了Josefson-Nissenzweig定理Banach Spaces $ C（K）$。
The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langleμ_n\colon\ n\inω\rangle$ on $K$ which is weakly* convergent to the zero measure on $K$ and such that $\big\|μ_n\big\|=1$ for every $n\inω$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space $K$ admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others that $K$ admits such a Josefson-Nissenzweig sequence if and only if $C(K)$ does not have the Grothendieck property restricted to functionals from the space $\ell_1(K)$. We also investigate miscellaneous analytic and topological properties of finitely supported Josefson-Nissenzweig sequences on general Tychonoff spaces. We prove that various properties of compact spaces guarantee the existence of finitely supported Josefson-Nissenzweig sequences. One such property is, e.g., that a compact space can be represented as the limit of an inverse system of compact spaces based on simple extensions. An immediate consequence of this result is that many classical consistent examples of Efimov spaces, i.e. spaces being counterexamples to the famous Efimov problem, admit such sequences of measures. Similarly, we show that if $K$ and $L$ are infinite compact spaces, then their product $K\times L$ always admits a finitely supported Josefson--Nissenzweig sequence. As a corollary we obtain a constructive proof that the space $C_p(K\times L)$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology--this generalizes results of Cembranos and Freniche. Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for the case of Banach spaces $C(K)$.
