Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T19:34:33.006Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
BARNABÁS FARKAS
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW POLANDE-mail: [email protected] 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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Banach, Stefan, Théorie Des Opérations Linéaires, Chelsea, New York, 1955.Google Scholar
Diestel, Joseph, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984.CrossRefGoogle Scholar
Drewnowski, Lech and Paúl, Pedro J., The Nikodým property for ideals of sets defined by matrix summability methods. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.), vol. 94 (2000), no. 4, pp. 485–503. Perspectives in mathematical analysis (Spanish.)Google Scholar
Farah, Ilijas, Ideals induced by Tsirelson submeasures. Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 243–258.CrossRefGoogle Scholar
Farah, Ilijas, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702.CrossRefGoogle Scholar
Freedman, A. R. and Sember, J. J., Densities and summability. Pacific Journal of Mathematics, vol. 95 (1981), no. 2, pp. 293–305.CrossRefGoogle Scholar
Farkas, Barnabás and Soukup, Lajos, More on cardinal invariants of analytic P-ideals. Commentationes Mathematicae Universitatis Carolinae, vol. 50 (2009), no. 2, pp. 281–295.Google Scholar
Filipów, Rafał and Szuca, Piotr, Rearrangement of conditionally convergent series on a small set. Journal of Mathematical Analysis and Applications, vol. 362 (2010), no. 1, pp. 64–71.CrossRefGoogle Scholar
Heil, Christopher, A basis theory primer. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, expanded edition, 2011.Google Scholar
Hernández-Hernández, Fernando and Hrušák, Michael, Cardinal invariants of analytic P-ideals. Canadian Journal of Mathematics, vol. 59 (2007), no. 3, pp. 575–595.CrossRefGoogle Scholar
Hrušák, Michael, Combinatorics of filters and ideals, Set theory and its applications, vol. 533, American Mathematical Society, Providence, RI, 2011, pp. 29–69.CrossRefGoogle Scholar
Klee, V. L. Jr., Invariant metrics in groups (solution of a problem of Banach). Proceedings of the American Mathematical Society, vol. 3 (1952), pp. 484–487.CrossRefGoogle Scholar
Kwela, Adam and Sabok, Marcin, Topological representations, doi:10.1016/j.jmaa.2014.09.059, preprint.CrossRefGoogle Scholar
Meza-Alcántara, D., Ideals and filters on countable sets, PhD thesis, Universidad Nacional Autónoma México, Mexico City, 2009.Google Scholar
Mátrai, Tamás, More cofinal types, preprint.Google Scholar
Mazur, Krzysztof, F σ-ideals and ${\omega _1}\omega _1^*$ -gaps in the Boolean algebras P(ω)/I. Fundamenta Mathematicae, vol. 138 (1991), no. 2, pp. 103–111.CrossRefGoogle Scholar
Solecki, Sławomir, Analytic ideals. Bulletin of Symbolic Logic, vol. 2 (1996), no. 3, pp. 339–348.CrossRefGoogle Scholar
Solecki, Sławomir and Todorcevic, Stevo, Cofinal types of topological directed orders. Annales de l’institut Fourier (Grenoble), vol. 54 (2004), no. 6, pp. 1877–1911.CrossRefGoogle Scholar
Solecki, Sławomir and Todorcevic, Stevo, Avoiding families and Tukey functions on the nowhere-dense ideal. Journal of the Institute of Mathematics of Jussieu, vol. 10 (2011), no. 2, pp. 405–435.CrossRefGoogle Scholar
Veličković, Boban, A note on Tsirelson type ideals. Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 259–268.CrossRefGoogle Scholar