A note on Tsirelson type ideals

Boban Veličković

Fundamenta Mathematicae (1999)

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

Abstract

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.