Skip to main content
SpringerLink
Log in

Trivial automorphisms

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω.

We can also assure that the dominating number, σ, is equal to ℵ1 and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\). Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial.

Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. L. Alperin, J. Covington and and D. Macpherson, Automorphisms of quotients of symmetric groups, Ordered groups and infinite permutation groups, Applied Mathematics, Vol. 354, Kluwer Academic Publishers, Dordrecht, 1996, pp. 231–247.

    Chapter  Google Scholar 

  2. T. Bartoszynski and H. Judah, Set theory: on the Structure of the Real Line, A.K. Peters, Welleslet, MA, 1995.

    MATH  Google Scholar 

  3. I. Ben Yaacov, A. Berenstein, C.W. Henson and A. Usvyatsov, Model theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Vol. II (Z. Chatzidakis et al., eds.), London Mathematical Society Lecture Notes Series, no. 350, Cambridge University Press, 2008, pp. 315–427.

  4. C. C. Chang and H. J. Keisler, Model Theory, third edn., Studies in Logic and the Foundations of Mathematics, Vol. 73, North-Holland, Amsterdam, 1990.

    MATH  Google Scholar 

  5. S. Coskey and I. Farah, Automorphisms of corona algebras and group cohomology, Transactions of the American Mathematical Society, to appear.

  6. A. Dow, A non-trivial copy of βℕ\ℕ, Proceedings of the American Mathematical Society, to appear.

  7. I. Farah, Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers, Memoirs of the American Mathematical Society, Vol. 148, no. 702, 2000.

  8. I. Farah, Liftings of homomorphisms between quotient structures and Ulam stability, in Logic Colloquium’ 98, Lecture Notes in Logic, Vol. 13, A. K. Peters, 2000, pp. 173–196.

  9. I. Farah, How many Boolean algebras P(ℕ)/I are there?, Illinois Journal of Mathematics 46 (2003), 999–1033.

    MathSciNet  Google Scholar 

  10. I. Farah, Luzin gaps, Transactions of the American Mathematical Society 356 (2004), 2197–2239.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Farah, Rigidity conjectures, in Logic Colloquium 2000, Lecture Notes on Logic, Vol. 19, The Association for Symbolic Logic, Urbana, IL, 2005, pp. 252–271.

    Google Scholar 

  12. I. Farah, All automorphisms of the Calkin algebra are inner, Annals of Mathematics 173 (2011), 619–661.

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Farah, Absoluteness, truth, and quotients, in Infinity and Truth, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Vol. 25, World Scientific Publ., 2013, pp. 1–24.

  14. I. Farah and B. Hart, Countable saturation of corona algebras, Comptes Rendus Mathematique, Comptes Rendus Mathématiques l’Acadmie des Sciences Canada 35 (2013), 35–56.

    MathSciNet  MATH  Google Scholar 

  15. Q. Feng, M. Magidor and W.H. Woodin, Universally Baire sets of reals, in Set Theory of the Continuum (H. Judah, W. Just, and W. H. Woodin, eds.), Springer-Verlag, Berlin, 1992, pp. 203–242.

    Chapter  Google Scholar 

  16. S. Ghasemi, Automorphisms of FDD-algebras, York University, 2012, preprint. arXiv:1310.1353.

  17. W. Just, Repercussions on a problem of Erdős and Ulam about density ideals, Canadian Journal of Mathematics 42 (1990), 902–914.

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Just, A modification of Shelah’s oracle chain condition with applications, Transactions of the American Mathematical Society 329 (1992), 325–341.

    MathSciNet  MATH  Google Scholar 

  19. W. Just, A weak version of AT from OCA, MSRI Publications 26 (1992), 281–291.

    MathSciNet  Google Scholar 

  20. W. Just and A. Krawczyk, On certain Boolean algebras P(ω)/I, Transactions of the American Mathematical Society 285 (1984), 411–429.

    MathSciNet  MATH  Google Scholar 

  21. A. Kanamori, The Higher Infinite: Large Cardinals in Set Theory from their Beginnings, Perspectives in Mathematical Logic, Springer, Berlin-Heidelberg-New York, 1995.

    Google Scholar 

  22. V. Kanovei and M. Reeken, On Ulam’s problem concerning the stability of approximate homomorphisms, Tr. Mat. Inst. Steklova 231 (2000), 249–283.

    MathSciNet  Google Scholar 

  23. V. Kanovei and M. Reeken, New Radon-Nikodym ideals, Mathematika 47 (2002), 219–227.

    Article  MathSciNet  Google Scholar 

  24. J. Kellner, Non-elementary proper forcing, Rendiconti del Seminario Matematico. Università Politecnico Torino (2012), to appear.

  25. K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  26. P. B. Larson, The Stationary Tower, University Lecture Series, Vol. 32, American Mathematical Society, Providence, RI, 2004, Notes on a course by W. H. Woodin.

    MATH  Google Scholar 

  27. P. B. Larson, Forcing over models of determinacy, in Handbook of Set Theory, Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 2121–2177.

    Google Scholar 

  28. D. A. Martin and R. M. Solovay, A basis theorem for Σ 13 sets of reals, Annals of Mathematics 89 (1969), 138–159.

    Article  MathSciNet  MATH  Google Scholar 

  29. N. C. Phillips and N. Weaver, The Calkin algebra has outer automorphisms, Duke Mathematical Journal 139 (2007), 185–202.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Roslanowski and S. Shelah, Norms on possibilities i: forcing with trees and creatures, Memoirs of the American Mathematical Society 141 (1999), xii+167pp., math.LO/9807172.

  31. W. Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Mathematics Journal 23 (1956), 409–419.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Shelah, Proper Forcing, Lecture Notes in Mathematics, Vol. 940, Springer-Verlag, Berlin-New York, xxix+496 pp, 1982.

    MATH  Google Scholar 

  33. S. Shelah, Proper and improper forcing, Perspectives in Mathematical Logic, Springer, 1998.

  34. S. Shelah, Properness without elementaricity, Journal of Applied Analysis 10 (2004), 168–289, math.LO/9712283.

    Article  Google Scholar 

  35. S. Shelah and J. Steprāns, PFA implies all automorphisms are trivial, Proceedings of the American Mathematical Society 104 (1988), 1220–1225.

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Shelah and J. Steprāns, Martin’s axiom is consistent with the existence of nowhere trivial automorphisms, Proceedings of the American Mathematical Society 130 (2002), 2097–2106.

    Article  MathSciNet  MATH  Google Scholar 

  37. S. Shelah and J. Steprans, Nontrivial automorphisms of \(P({\Bbb N})/[{\Bbb N}] < {\aleph _0}\) from variants of small dominating number, available at http://www.math.yorku.ca/~steprans/Research/menu.shtml

  38. J. Steprāns, The autohomeomorphism group of the Čech-Stone compactification of the integers, Transactions of the American Mathematical Society 355 (2003), 4223–4240.

    Article  MathSciNet  MATH  Google Scholar 

  39. B. Veličković, OCA and automorphisms of P(ω)/Fin, Topology and its Applications 49 (1992), 1–13.

    Article  Google Scholar 

  40. W. H. Woodin, Beyond Σ 21 absoluteness, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 515–524.

    Google Scholar 

  41. W. H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, revised ed., de Gruyter Series in Logic and its Applications, Vol. 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010.

    Book  MATH  Google Scholar 

  42. J. Zapletal, Descriptive Set Theory and Definable Forcing, Memoirs of the American Mathematical Society 167 (2004), viii+141pp.

  43. J. Zapletal, Forcing Idealized, Cambridge University Press, 2008.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3

    Ilijas Farah

  2. Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia

    Ilijas Farah

  3. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel

    Saharon Shelah

  4. Department of Mathematics, Hill Center-Busch Campus, Rutgers The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

Authors
  1. Ilijas Farah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saharon Shelah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilijas Farah.

Additional information

The first author was partially supported by NSERC.

Second author’s research was supported by the United States-Israel Binational Science Foundation (Grant no. 2010405), and by the National Science Foundation (Grant no. DMS 1101597). No. 987 on Shelah’s list of publications.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farah, I., Shelah, S. Trivial automorphisms. Isr. J. Math. 201, 701–728 (2014). https://doi.org/10.1007/s11856-014-1048-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-014-1048-5

Keywords

Search

Navigation