Journal of Symbolic Logic

Borel reducibility and Hölder(α) embeddability between Banach spaces

Longyun Ding

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We investigate Borel reducibility between equivalence relations E(X;p)=X/ℓp(X)'s where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(Lr;p)'s and E(c₀;p)'s for r,p∈[1,+∞).

We also answer a problem presented by Kanovei in the affirmative by showing that C(ℝ⁺)/C₀(ℝ⁺) is Borel bireducible to ℝ/c₀.

Article information

J. Symbolic Logic, Volume 77, Issue 1 (2012), 224-244.

First available in Project Euclid: 20 January 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary 03E15, 46B20, 47H99


Ding, Longyun. Borel reducibility and Hölder(α) embeddability between Banach spaces. J. Symbolic Logic 77 (2012), no. 1, 224--244. doi:10.2178/jsl/1327068700.

Export citation


  • I. Aharoni Every separable metric space is Lipschitz equivalent to a subset of $c_0^+$, Israel Journal of Mathematics, vol. 19(1974), pp. 284–291.
  • F. Albiac and N. J. Kalton Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer-Verlag,2006.
  • H. Becker and A. S. Kechris The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, vol. 232, Cambridge University Press,1996.
  • Y. Benyamini and J. Lindenstrauss Geometric nonlinear functional analysis, Colloquium Publications, vol. 48, American Mathematical Society,2000.
  • J. Bourgain, V. Milman, and H. Wolfson On type of metric spaces, Transactions of the American Mathematical Society, vol. 294(1986), pp. 295–317.
  • R. Dougherty and G. Hjorth Reducibility and nonreducibility between $\ell^p$ equivalence relations, Transactions of the American Mathematical Society, vol. 351(1999), pp. 1835–1844.
  • I. Farah Basis problem for turbulent actions II: $c_0$-equalities, Proceedings of the London Mathematical Society, vol. 82(2001), no. 3, pp. 1–30.
  • S. Gao Equivalence relations and classical Banach spaces, Mathematical logic in Asia, proceedings of the 9th Asian logic conference, Novosibirsk, Russia, 2005 (S. S. Goncharov, R. Downey, and H. Ono, editors), World Scientific,2006, pp. 70–89.
  • –––– Invariant descriptive set theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 293, CRC Press,2008.
  • G. Hjorth Actions by the classical Banach spaces, Journal of Symbolic Logic, vol. 65(2000), pp. 392–420.
  • V. Kanovei Borel equivalence relations: Structure and classification, University Lecture Series, vol. 44, American Mathematical Society,2008.
  • J. Lindenstrauss and L. Tzafriri Classical Banach spaces, II: Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer-Verlag,1979.
  • B. Maurey and G. Pisier Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Mathematica, vol. 58(1976), pp. 45–90.
  • M. Mendel and A. Naor Euclidean quotients of finite metric spaces, Advances in Mathematics, vol. 189(2004), pp. 451–494.
  • –––– Metric cotype, Annals of Mathematics, vol. 168(2008), pp. 247–298.
  • G. Pisier Holomorphic semigroups and the geometry of Banach spaces, Annals of Mathematics, vol. 115(1982), pp. 375–392.
  • M. Talagarnd Approximating a helix in finiely many dimensions, Annales de l'Institute Henri Poincaré Probabilités et Statistiques, vol. 28(1992), pp. 355–363.
  • N. Tomczak-Jaegermann Banach–Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38, Longman Scientific & Technical,1989.