Topological properties of the group of the null sequences valued in an Abelian topological group

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• 作者：S. Gabriyelyan ^{saak@math.bgu.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：22A10 ; 22A35 ; 43A05 ; 43A40 ; 54H11
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 July 2016
• 年：2016
• 卷：207
• 期：Complete
• 页码：136-155
• 全文大小：533 K

文摘

For a Hausdorff Abelian topological group X  , we denote by F₀(X)${\mathfrak{F}}_0(X)$ the group of all X-valued null sequences endowed with the uniform topology. We prove that if X is an (E  )-space (respectively, a strictly angelic space or a Š-space), then so is F₀(X)${\mathfrak{F}}_0(X)$. We essentially simplify and clarify the theory of properties respected by the Bohr functor on Abelian topological groups, denoted below by X↦X⁺$X\mapsto X^{+}$. We prove that for a complete maximally almost periodic group X, the group X   shares with X⁺$X^{+}$ the same functionally bounded sets iff it shares the same compact sets and X⁺$X^{+}$ is a μ-space. We show that for a locally compact Abelian (LCA) group X the following are equivalent: 1) X   is totally disconnected, 2) F₀(X)${\mathfrak{F}}_0(X)$ is a Schwartz group, 3) F₀(X)${\mathfrak{F}}_0(X)$ respects compactness, 4) F₀(X)${\mathfrak{F}}_0(X)$ has the Schur property. So, if a LCA group X   is not totally disconnected, the group F₀(X)${\mathfrak{F}}_0(X)$ is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group X   the group F₀(X)${\mathfrak{F}}_0(X)$ is monothetic and every real-valued uniformly continuous function on F₀(X)${\mathfrak{F}}_0(X)$ is bounded.