Representations of ideals in Polish groups and in Banach spaces

Abstract

We investigate ideals of the form $\{A\subseteq \omega :{\sum }_{n\in A}{x}_n$ is unconditionally convergent $\}$, where $(x_n{)}_{n\in \omega }$ is a sequence in a Polish group or in a Banach space. If an ideal on $\omega$ can be seen in this form for some sequence in $X$, then we say that it is representable in $X$. After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a non-pathological analytic P-ideal. We focus on the family of ideals representable in $c_0$. We prove that the trace of the null ideal, Farah's ideal, and Tsirelson ideals are not representable in $c_0$, and that a tall $F_{\sigma }$ P-ideal is representable in $c_0$ iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in ${\ell }_1$ but not in $\mathbb{R}$. Finally, we summarize some open problems of this topic.