Skip to main content
SpringerLink
Log in

Shift invariant preduals of ℓ 1(ℤ)

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

Abstract

The Banach space ℓ 1(ℤ) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ℓ 1(ℤ) weak*-continuous. This is equivalent to making the natural convolution multiplication on ℓ 1(ℤ) separately weak*-continuous and so turning ℓ 1(ℤ) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C 0(K) (for countable locally compact K) and ℓ 1(ℤ) is c 0(ℤ). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak*-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c 0. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ℤ. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c 0.

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. D. Alspach, R. Judd and E. Odell, The Szlenk index and local l 1-indices, Positivity, An International Journal devoted to the Theory and Applications of Positivity in Analysis 9 (2005), 1–44.

    MathSciNet  MATH  Google Scholar 

  2. S. A. Argyros and R. G. Haydon, A hereditarily indecomposable L ∞-space that solves the scalar-plus-compact problem, Acta Mathematica 206 (2011), 1–54.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Banach, Théorie des opérations linéaires, Subwencji Funduszu Kultury Narodowej, 1932.

  4. Y. Benyamini, Separable G spaces are isomorphic to C(K) spaces, Israel Journal of Mathematics 14 (1973), 287–293.

    Article  MathSciNet  MATH  Google Scholar 

  5. Y. Benyamini and J. Lindenstrauss, A predual of l 1 which is not isomorphic to a C(K) space, in Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel Journal of Mathematics 13 (1972), 246–254.

    MathSciNet  Google Scholar 

  6. J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1989.

    MATH  Google Scholar 

  7. C. Bessaga and A. Pełlczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Polska Akademia Nauk, Instytut Matematyczny, Studia Mathematica 19 (1960), 53–62.

    MATH  Google Scholar 

  8. J. Bourgain and F. Delbaen, A class of special L ∞ spaces, Acta Mathematica 145 (1980), 155–176.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Choi, Odd element of l1 group algebra of the integers, Posting on MathOverFlow.net, http://mathoverflow.net/questions/37305/odd-element-of-l1-group-algebra-of-the-integers/37336#37336, 2010.

  10. E. Christensen, F. Pop, A. M. Sinclair and R. R. Smith, Hochschild cohomology of factors with property Γ, Annals of Mathematics, Second Series 158 (2003), 635–659.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. G. Dales, F. Ghahramani and A. Ya. Helemskii, The amenability of measure algebras, Journal of the London Mathematical Society, Second Series 66 (2002), 213–226.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Daws, Dual Banach algebras: representations and injectivity, Studia Mathematica 178 (2007), 231–275.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Daws, H. L. Pham and S. White, Conditions implying the uniqueness of the weak*-topology on certain group algebras, Houston Journal of Mathematics 35 (2009), 253–276.

    MathSciNet  MATH  Google Scholar 

  14. M. Daws, H. L. Pham and S. White, Preduals of semigroup algebras, Semigroup Forum 80 (2010), 61–78.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Freeman, E. Odell and T. Schlumprecht, The universality of ℓ 1 as dual space, Mathematische Annalen 351 (2011), 149–186

    Article  MathSciNet  MATH  Google Scholar 

  16. S. L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific Journal of Mathematics 18 (1966), 121–137.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. E. Johnson, Separate continuity and measurability, Proceedings of the American Mathematical Society 20 (1969), 420–422.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Kaijser, On Banach modules. I, Mathematical Proceedings of the Cambridge Philosophical Society 90 (1981), 423–444.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Lancien, Dentability indices and locally uniformly convex renormings, The Rocky Mountain Journal of Mathematics 23 (1993), 635–647.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. T.-M. Lau and R. J. Loy, Weak amenability of Banach algebras on locally compact groups, Journal of Functional Analysis 145 (1997), 175–204.

    Article  MathSciNet  MATH  Google Scholar 

  21. D. J. Newman, Homomorphisms of l +, American Journal of Mathematics 91 (1969), 37–46.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Pełczyński, On the isomorphism of the spaces m and M, Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 6 (1958), 695–696.

    MATH  Google Scholar 

  23. H. P. Rosenthal, The Banach spaces C(K), in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1547–1602.

    Google Scholar 

  24. V. Runde, Amenability for dual Banach algebras, Studia Mathematica 148 (2001), 47–66.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Runde, Connes-amenability and normal, virtual diagonals for measure algebras, I, Journal of the London Mathematical Society, Second Series 67 (2003), 643–656.

    Article  MathSciNet  MATH  Google Scholar 

  26. W. A. F. Ruppert, On signed a-adic expansions and weakly almost periodic functions, Proceedings of the London Mathematical Society, Third Series 63 (1991), 620–656.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Sakai, C*-algebras and W*-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York, 1971.

    Book  Google Scholar 

  28. C. Samuel, Indice de Szlenk des C(K) (K espace topologique compact dénombrable), in Seminar on the Geometry of Banach Spaces, Vol. I, II (Paris, 1983), Publ. Math. Univ. Paris VII, Vol. 18, University Paris VII, Paris, 1984, pp. 81–91.

    Google Scholar 

  29. A. M. Sinclair and R. R. Smith, A survey of Hochschild cohomology for von Neumann algebras, in Operator Algebras, Quantization, and Noncommutative Geometry, Contemporary Mathematics, Vol. 365, American Mathematical Society, Providence, RI, 2004, pp. 383–400.

    Chapter  Google Scholar 

  30. W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Polska Akademia Nauk, Instytut Matematyczny, Studia Mathematica 30 (1968), 53–61.

    MathSciNet  MATH  Google Scholar 

  31. N. J. Young, Periodicity of functionals and representations of normed algebras on reflexive spaces, Proceedings of the Edinburgh Mathematical Society, Series II 20 (1976/77), 99–120.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Matthew Daws

  2. Brasenose College, Radcliffe Square, Oxford, OX1 4AJ, UK

    Richard Haydon

  3. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Thomas Schlumprecht

  4. School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, Scotland

    Stuart White

Authors
  1. Matthew Daws
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard Haydon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Schlumprecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stuart White
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Daws.

Additional information

Supported by NSF grants DMS0856148 and DMS0556013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Daws, M., Haydon, R., Schlumprecht, T. et al. Shift invariant preduals of ℓ 1(ℤ). Isr. J. Math. 192, 541–585 (2012). https://doi.org/10.1007/s11856-012-0040-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0040-1

Keywords

Search

Navigation