A note on Tsirelson type ideals

Boban Veličković

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 3, page 259-268
  • ISSN: 0016-2736

Abstract

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Using Tsirelson’s well-known example of a Banach space which does not contain a copy of c 0 or l p , for p ≥ 1, we construct a simple Borel ideal I T such that the Borel cardinalities of the quotient spaces P ( ℕ ) / I T and P ( ℕ ) / I 0 are incomparable, where I 0 is the summable ideal of all sets A ⊆ ℕ such that ∑ n ∈ A 1 / ( n + 1 ) < ∞ . This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

How to cite

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Veličković, Boban. "A note on Tsirelson type ideals." Fundamenta Mathematicae 159.3 (1999): 259-268. <http://eudml.org/doc/212333>.

@article{Veličković1999,
abstract = {Using Tsirelson’s well-known example of a Banach space which does not contain a copy of $c_0$ or $l_p$, for p ≥ 1, we construct a simple Borel ideal $I_T$ such that the Borel cardinalities of the quotient spaces $P(ℕ)/I_T$ and $P(ℕ)/I_0$ are incomparable, where $I_0$ is the summable ideal of all sets A ⊆ ℕ such that $∑ _\{n ∈ A\}1/(n+1) < ∞$. This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.},
author = {Veličković, Boban},
journal = {Fundamenta Mathematicae},
keywords = {trichotomy conjecture; dichotomy conjecture; Borel equivalence relations; Polish space; Tsirelson's Banach space; Borel ideal; Borel cardinalities; quotient spaces},
language = {eng},
number = {3},
pages = {259-268},
title = {A note on Tsirelson type ideals},
url = {http://eudml.org/doc/212333},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Veličković, Boban
TI - A note on Tsirelson type ideals
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 3
SP - 259
EP - 268
AB - Using Tsirelson’s well-known example of a Banach space which does not contain a copy of $c_0$ or $l_p$, for p ≥ 1, we construct a simple Borel ideal $I_T$ such that the Borel cardinalities of the quotient spaces $P(ℕ)/I_T$ and $P(ℕ)/I_0$ are incomparable, where $I_0$ is the summable ideal of all sets A ⊆ ℕ such that $∑ _{n ∈ A}1/(n+1) < ∞$. This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.
LA - eng
KW - trichotomy conjecture; dichotomy conjecture; Borel equivalence relations; Polish space; Tsirelson's Banach space; Borel ideal; Borel cardinalities; quotient spaces
UR - http://eudml.org/doc/212333
ER -

References

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  1. [CS] P. G. Casazza and T. J. Shura, Tsirelson's Space, Lecture Notes in Math. 1363, Springer, 1989. Zbl0709.46008
  2. [FJ] T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no l p , Compositio Math. 29 (1974), 179-190. Zbl0301.46013
  3. [Hj1] G. Hjorth, Actions of S ∞ , manuscript. 
  4. [Hj2] G. Hjorth, Actions by the classical Banach spaces, manuscript. 
  5. [JN] S. Jalali-Naini, The monotone subsets of Cantor space, filters and descriptive set theory, doctoral dissertation, Oxford, 1976. 
  6. [Ke1] A. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  7. [Ke2] A. Kechris, Rigidity properties of Borel ideals on the integers, preprint. 
  8. [KL] A. Kechris and A. Louveau, The structure of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), 215-242. Zbl0865.03039
  9. [LV] A. Louveau and B. Veličković, A note on Borel equivalence relations, Proc. Amer. Math. Soc. 120 (1994), 255-259. Zbl0794.04002
  10. [Mat] A. R. D. Mathias, A remark on rare filters, in: Infinite and Finite Sets, A. Hajnal et al. (eds.), Colloq. Math. Soc. János Bolyai 10, Vol. III, North-Holland, 1975, 1095-1097. 
  11. [Ma1] K. Mazur, A modification of Louveau and Veličković construction for F σ -ideals, preprint. 
  12. [Ma2] K. Mazur, Towards a dichotomy for F σ -ideals, preprint. 
  13. [OS] E. Odell and T. Schlumprecht, Distortion and stabilized structure in Banach spaces; new geometric phenomena for Banach and Hilbert spaces, in: Proc. Internat. Congress of Mathematicians, Zürich, Birkhäuser, 1995, 955-965. Zbl0868.46010
  14. [So] S. Solecki, Analytic ideals, Bull. Symbolic Logic 2 (1996), 339-348. Zbl0862.04002
  15. [Ta] M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), 13-43. Zbl0435.46023
  16. [Ve] B. Veličković, Definable automorphisms of P(ω)/fin, Proc. Amer. Math. Soc. 96 (1986), 130-135. Zbl0614.03049

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