arXiv 1012.57132010-12-28math.GN

一类非Mackey的可度量化局部拟凸群

A class of metrizable locally quasi-convex groups which are not Mackey

Dikran Dikranjan, Elena Martín Peinador, Vaja Tarieladze

备注：: 20 pages

摘要：

一个来自MAP拓扑阿贝尔群的类$\mathcal G$的拓扑群$(G,μ)$在$\mathcal G$中称为{ \it Mackey群}，如果它具有以下性质：如果$ν$是$G$中的群拓扑，使得$(G,ν)\in \mathcal G$和$(G,ν)$具有相同的连续特征标，比如说$(G,ν)^{\wedge}=(G,μ)^{\wedge}$，那么$ν\le μ$。如果$\rm{LCS}$是Hausdorff拓扑阿贝尔群的一个类，且该类允许$\mathbb R$上的局部凸拓扑向量空间的结构，则每个可度量化的$(G,μ) \in \rm{LCS}$都是$\rm{LCS}$中的Mackey群.对于局部拟凸Hausdorff拓扑交换群类$\rm{LQC}$，1999年证明了每个{ \bf complete}可度量化$(G,μ)\in \rm{LQC}$是$\rm{LQC}$（\cite{CMPT}）中的Mackey群。正如我们在本文中证明的那样，完整性不能在类$\rm{LQC}$中被\NB丢弃。事实上，我们提供了一个大的可度量化的预紧\NB（非紧）群，{ \bf不是} Mackey群在\rm{LQC}（定理\ref{basth}）。这些例子是从$c_0(X)$形式的群构造的，其元素是拓扑阿贝尔群$X$的空序列，其拓扑是一致拓扑。首先证明了对于紧可度量化群$X\ne\{0\}$，拓扑群$c_0(X)$是非紧完全可度量化局部拟凸群，它具有{ \bf可数}拓扑对偶当且仅当$X$连通.然后证明了对于连通紧可度量化群$X\ne\{0\}$，具有由乘积$X^{\N}$导出的乘积拓扑的群$c_0(X)$是可度量化预紧的，但不是LQC中的Mackey群.

英文摘要：

A topological group $(G,μ)$ from a class $\mathcal G$ of MAP topological abelian groups will be called a {\it Mackey group} in $\mathcal G$ if it has the following property: if $ν$ is a group topology in $G$ such that $(G,ν)\in \mathcal G$ and $(G,ν)$ has the same continuous characters, say $(G,ν)^{\wedge}=(G,μ)^{\wedge}$, then $ν\le μ$. If $\rm{LCS}$ is the class of Hausdorff topological abelian groups which admit a structure of a locally convex topological vector space over $\mathbb R$, it is well-known that every metrizable $(G,μ) \in \rm{LCS}$ is a Mackey group in $\rm{LCS}$. For the class $\rm{LQC}$ of locally quasi-convex Hausdorff topological abelian groups, it was proved in 1999 that every {\bf complete} metrizable $(G,μ)\in \rm{LQC}$ is a Mackey group in $\rm{LQC}$ (\cite{CMPT}). The completeness cannot be \NB dropped within the class $\rm{LQC}$ as we prove in this paper. In fact, we provide a large family of metrizable precompact \NB(noncompact) groups which {\bf are not} Mackey groups in \rm{LQC} (Theorem \ref{basth}). Those examples are constructed from groups of the form $c_0(X)$, whose elements are the null sequences of a topological abelian group $X$, and whose topology is the uniform topology. We first show that for a compact metrizable group $X\ne\{0\}$ the topological group $c_0(X)$ is a non-compact complete metrizable locally quasi-convex group, which has {\bf countable} topological dual iff $X$ is connected. Then we prove that for a connected compact metrizable group $X\ne\{0\}$ the group $c_0(X)$ endowed with the product topology induced from the product $X^{\N}$ is metrizable precompact but not a Mackey group in LQC.

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