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REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES

Published online by Cambridge University Press:  22 December 2015

PIOTR BORODULIN–NADZIEJA ,
BARNABÁS FARKAS  and
GRZEGORZ PLEBANEK
PIOTR BORODULIN–NADZIEJA
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW POLANDE-mail: pborod@math.uni.wroc.pl
BARNABÁS FARKAS
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW POLANDE-mail: barnabasfarkas@gmail.com Barnabás Farkas later at: KURT GÖDEL RESEARCH CENTER VIENNA AUSTRIA
GRZEGORZ PLEBANEK
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW POLANDE-mail: grzes@math.uni.wroc.pl
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Abstract

We investigate ideals of the form {A ⊆ ω: Σn∈Axn is unconditionally convergent} where (xn)n∈ω 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 nonpathological analytic P-ideal.

We focus on the family of ideals representable in c0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable in c0, and that a tall Fσ P-ideal is representable in c0 iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in ℓ1 but not in ℝ.

Finally, we summarize some open problems of this topic.

Keywords

03E0503E1546B15idealsummable idealdensity idealanalytic P-idealsubmeasurenonpathological submeasureBanach spacePolish groupunconditional convergenceclassical sequence spacenull settrace idealsummable-like idealdensity-like ideal
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1268 - 1289
Copyright
Copyright © The Association for Symbolic Logic 2015 

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