arXiv Analytics

Sign in

arXiv:1402.0441 [math.LO]AbstractReferencesReviewsResources

Representations of ideals in Polish groups and in Banach spaces

Piotr Borodulin-Nadzieja, Barnabas Farkas, Grzegorz Plebanek

Published 2014-02-03Version 1

We investigate ideals of the form $\{A \subseteq \omega\colon \sum_{n\in A} x_n$ 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.

Related articles: Most relevant | Search more
arXiv:1012.5051 [math.LO] (Published 2010-12-22)
A Boolean algebra and a Banach space obtained by push-out iteration
arXiv:1810.12855 [math.LO] (Published 2018-10-30)
Some Results on Polish Groups
arXiv:math/0409110 [math.LO] (Published 2004-09-07)
The number of translates of a closed nowhere dense set required to cover a Polish group