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REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES

  • PIOTR BORODULIN–NADZIEJA (a1), BARNABÁS FARKAS (a2) (a3) and GRZEGORZ PLEBANEK (a4)
Abstract

We investigate ideals of the form {Aω: ΣnAxn is unconditionally convergent} where (xn)nω 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 1 but not in ℝ.

Finally, we summarize some open problems of this topic.

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References
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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