关于拓扑环中的无条件收敛级数
On unconditionally convergent series in topological rings
摘要：
定义一个拓扑环$R$为\emph{Hirsch}，如果对$R$中的任意无条件收敛级数$\sum_{n\inω} x_i$和$R$的加性恒等式$0$的任意邻域$U$，存在$0$的邻域$V\subseteq R$使得对任意有限集$F\subsetω$和任意序列$(a_n)_{n\in F}\in V^F$有$\sum_{n\in F} a_n x_n\in U$.我们在某些已知的拓扑环类中认识到赫希环。为此，我们引进并发展了actogroups的技术。我们证明，特别是，一个拓扑环$R$是Hirsch提供$R$是局部紧或$R$有一个基地在零组成的开放理想或$R$是一个封闭的子环的Banach环$C(K)$，其中$K$是一个紧凑的Hausdorff空间。这意味着Banach环$\ell_\infty$及其子环$c_0$和$c$是Hirsch环。证明了对每个$p\in[1,2]$，Banach环$\ell_p$是Hirsch环.另一方面，对于任何不同的数$p,q\in[1,\infty]$，交换Banach环$\ell_p\oplus i\ell_q$不是Hirsch环。也对任何$p\in (1,\infty)$，Banach环$\ell_p$的连续自同态的（非交换）Banach环$L(\ell_p)$不是Hirsch。我们不知道Banach环$\ell_p$是否是$p\in(2,\infty)$的Hirsch环。
英文摘要：
We define a topological ring $R$ to be \emph{Hirsch}, if for any unconditionally convergent series $\sum_{n\inω} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$ such that $\sum_{n\in F} a_n x_n\in U$ for any finite set $F\subsetω$ and any sequence $(a_n)_{n\in F}\in V^F$. We recognize Hirsch rings in certain known classes of topological rings. For this purpose we introduce and develop the technique of seminorms on actogroups. We prove, in particular, that a topological ring $R$ is Hirsch provided $R$ is locally compact or $R$ has a base at the zero consisting of open ideals or $R$ is a closed subring of the Banach ring $C(K)$, where $K$ is a compact Hausdorff space. This implies that the Banach ring $\ell_\infty$ and its subrings $c_0$ and $c$ are Hirsch. Also we prove that for every $p\in[1,2]$ the Banach ring $\ell_p$ is Hirsch. On the other hand, for any distinct numbers $p,q\in[1,\infty]$ the commutative Banach ring $\ell_p\oplus i\ell_q$ is not Hirsch. Also for any $p\in (1,\infty)$, the (noncommutative) Banach ring $L(\ell_p)$ of continuous endomorphisms of the Banach ring $\ell_p$ is not Hirsch. We do not know whether the Banach rings $\ell_p$ are Hirsch for $p\in(2,\infty)$.
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