论文标题
在拓扑环中无条件收敛的系列中
On unconditionally convergent series in topological rings
论文作者
论文摘要
如果对于任何无条件收敛的系列$ \ sum_ {n \inΩ} x_i $ in $ r $和任何附加$ 0 $ 0 $ 0 $ r $的附近$ r $ $ r $的$ 0 $ v \ sups $ v \ sups us_ x_n \ in U $对于任何有限的设置$ f \subsetΩ$和任何序列$（a_n）_ {n \ in v^f $ in f} \ in v^f $。我们认识到某些已知类别的拓扑环中的赫希环。为此，我们介绍并开发了对Actroups的符号技术的技术。我们特别证明，拓扑环$ r $是hirsch，前提是$ r $是本地紧凑的，或者$ r $在零的基础上，由开放的理想或$ r $组成，是Banach Ring $ c（k）$的封闭式，其中$ k $是一个紧凑的Hausdorff Space。这意味着Banach Ring $ \ ell_ \ infty $及其子环$ C_0 $和$ C $是Hirsch。我们还证明，在[1,2] $中的每一个$ p \ banach ring $ \ ell_p $是hirsch中。另一方面，对于任何不同的数字$ p，q \ in [1，\ infty] $ tomutative banach环$ \ ell_p \ oplus i \ ell_q $不是hirsch。同样，对于（1，\ infty）$中的任何$ p \，Banach Ring $ \ ell_p $的连续内态的（非共同）Banach Ring $ l（\ ell_p）$不是Hirsch。我们不知道Banach戒指$ \ ell_p $是否为$ p \ in（2，\ infty）$。
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 $p\in(2,\infty)$.
