We investigate ideals of the form $\{A \subseteq ω\colon \sum_{n\in A} x_n$ is unconditionally convergent $\}$, where $(x_n)_{n\inω}$ is a sequence in a Polish group or in a Banach space. If an ideal on $ω$ 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 $F_σ$ 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.
We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, resp. pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two different topological approaches. We investigate generic properties in these spaces and compare them with those in admissible topologies, confirming the suspicion of Godefroy and Saint-Raymon…
The well known ideal presentations of countably based domains were recently extended to (effective) quasi-Polish spaces. Continuing these investigations, we explore some classes of effective quasi-Polish spaces. In particular, we prove an effective version of the domain-characterization of quasi-Polish spaces, describe effective extensions of quasi-Polish topologies, discover natural numberings of classes of effective quasi-Polish spaces, estimate the complexity of the (effective) homeomorphism relation and of some classes of spaces w.r.t. these numberings, and investigate degree spectra of co…
In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah--Solecki. In particular, we obtain a characterization of such canonical approximations in terms of their Borel complexity class. As an application we provide a complete list of all the possible Borel complexity classes of Polishable subgroups of Polish groups or, equivalently, of the ranges of continuous group homomorphisms between Polish groups. We also provide a complete list of all the possible Borel complexity classes of the ranges…
For an analytic $P$-ideal $I$, $S_I$ is the Polish group of all the permutations of $\mathbb{N}$ whose support is in $I$, with Polish topology given by the corresponding submeasure on $I$. We show that if $\mbox{Fin} \subsetneq I$, then $S_I$ has ample generics. This implies that there exists a non non-archimedean Polish group with ample generics.
We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the "Schreier type" Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these…
We give a method of producing a Polish module over an arbitrary subring of $\mathbb Q$ from an ideal of subsets of $\mathbb N$ and a sequence in $\mathbb N$. The method allows us to construct two Polish $\mathbb Q$-vector spaces, $U$ and $V$, such that -- both $U$ and $V$ embed into $\mathbb R$ but -- $U$ does not embed into $V$ and $V$ does not embed into $U$, where by an embedding we understand a continuous $\mathbb Q$-linear injection. This construction answers a question of Frisch and Shinko. In fact, our method produces a large number of incomparable with respect to embeddings Polish $\ma…
设I是波兰空间X的子集的理想，包含所有的单元素，并拥有Borel基。假设I不满足ccc，我们考虑以下条件（B）、（M）和（D）。条件（B）说明存在一个大小为c的不相交族F子集P（X），它由不在I中的Borel集组成。条件（M）说明存在函数f：X-> X，对于X中的每个x，f^{-1}[{x}]记为I。设X是一个群，I是不变的，条件（D）说明存在一个博雷尔集B，它在I中，有一个完满集P子集X，它的族{B+x：x in P}是不相交的。本文的目的是研究链（D）=>（M）=>（B）=> not-ccc中的逆蕴涵是否成立。我们在Cantor群上构造了一个σ-理想，它满足（M）而不满足（D）（第2节）。这个sigma-理想的一个修改版本包含了整个空间（第3节）。对于“nicely”定义的理想建立了（M）从（B）导出的一些相容性结果（第4节）.我们证明了ccc和（M）都可能失败（定理1.3和4.2）。最后，研究了Polish群上不变理想的（M）的一些尖锐形式（第5节）。
We show some basic results on the characterization of quasi-Polish spaces in terms of spaces of ideals, with an emphasis on the connections with computable topology.
Given a compact Polish space $E$ and the hyperspace of its compact subsets $\mathcal{K}(E)$, we consider the class of $G_δ$ $σ$-ideals of compact subsets of $E$ that can be represented via a compact subset of $\mathcal{K}(E)$. If we extend such an ideal $I$ by considering $G_δ$ (or analytic) sets that are covered by countable unions of sets in $I$, we show that the extended collection can still be represented via some compact subset of $\mathcal{K}(E)$.
The following will be shown: Let $I$ be a $σ$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence relation on this Polish space with all equivalence classes Borel. If sharps of certain sets exist, then there is an $I^+$ Borel subset $C$ of $X$ such that $E \upharpoonright C$ is a Borel equivalence relation.
A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $μ$ and a Borel set $B$ containing $X$ such that $μ(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $σ$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}…
We study the lattice structure of the family of weakly compact subsets of the unit ball $B_X$ of a separable Banach space $X$, equipped with the inclusion relation (this structure is denoted by $\mathcal{K}(B_X)$) and also with the parametrized family of almost inclusion relations $K \subseteq L+εB_X$, where $ε>0$ (this structure is denoted by $\mathcal{AK}(B_X)$). Tukey equivalence between partially ordered sets and a suitable extension to deal with $\mathcal{AK}(B_X)$ are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely…
