Skip to main content
SpringerLink
Log in
Book cover

Positivity and Noncommutative Analysis pp 405–423Cite as

Birkhäuser

On the Lipschitz Decomposition Problem in Ordered Banach Spaces and Its Connections to Other Branches of Mathematics

  • Chapter
  • First Online:

Part of the book series: Trends in Mathematics ((TM))

Abstract

Consider the following still-open problem: for any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅+ : X → C and ⋅− : X → C satisfying x = x + − x − for every x ∈ X?

We discuss the connections of this problem to a large number of other branches of mathematics: set-valued analysis, selection theorems, the non-linear geometry of Banach spaces, Ramsey theory, Lipschitz function spaces, duality theory, and tensor products of Banach spaces. We give numerous equivalent reformulations of the problem, and, through known examples, provide circumstantial evidence that the above question could be answered in the negative.

Dedicated to the occasion of Ben de Pagter’s 65th birthday

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    For our current purpose a closed cone inside a Banach space is sufficient. See the more general definitions: [10, Definitions 2.2 and 2.3].

  2. 2.

    Although very seldomly used in analysis, this is purely an observation about real numbers: let \(a,b\in \mathbb {R}\) be distinct real numbers and let K, ε > 0 satisfy \(0<(K-\varepsilon )<\left |a-b\right |\leq K.\) If \(c\in \mathbb {R}\) is such that \(\left |a-c\right |\leq 2^{-1}K\) and \(\left |b-c\right |\leq 2^{-1}K\) , then \(\left |c-2^{-1}(a+b)\right |\leq 2^{-1}\varepsilon \).

  3. 3.

    Let \((x,x)\in \overline {({\mathbf {B}}_{X\oplus _{\infty }X}+C\oplus _{\infty }(-C))}^{\left \Vert \cdot \right \Vert _{\infty }}\cap \Xi _{\infty }\). Then there exist sequences \(((a_{n},b_{n}))\subseteq {\mathbf {B}}_{X\oplus _{\infty }X}\) and ((c n, −d n)) ⊆ C ⊕∞(−C) so that (a n, b n) + (c n, −d n) → (x, x) as n →∞. For every \(n\in \mathbb {N}\), let p n := (a n + c n) − (b n − d n) and consider the sequence \(S:=\left ((a_{n},b_{n}+p_{n})+(c_{n},-d_{n})\right )=((a_{n}+c_{n},a_{n}+c_{n}))\subseteq \Xi _{\infty }\). This sequence S converges to (x, x) and, since p n → 0 as n →∞, for every ε > 0 the tail of S eventually lies in \(((1+\varepsilon /\alpha ){\mathbf {B}}_{X\oplus _{\infty }X}+C\oplus _{\infty }(-C))\cap \Xi _{\infty }\subseteq (1+\varepsilon /\alpha )\alpha \Xi _{1}\), and hence \(\left \Vert (x,x)\right \Vert _{\infty }\leq (\alpha +\varepsilon ).\) But this holds for every ε > 0, so \(\left \Vert (x,x)\right \Vert _{\infty }\leq \alpha \) and therefore (x, x) ∈ α Ξ1. Because the wk-closure and \(\left \Vert \cdot \right \Vert _{\infty }\)-closure of convex sets coincide, we obtain \(\overline {({\mathbf {B}}_{X\oplus _{\infty }X}+C\oplus _{\infty }(-C))}^{\mathrm{wk}}\cap \Xi _{\infty }\subseteq \alpha \Xi _{1}.\)

  4. 4.

    Of some relevance here is Kalton and Godefroy’s result [15, Theorem 5.3] showing that bounded approximation properties transfer between X and F(X).

References

  1. I. Aharoni, J. Lindenstrauss, Uniform equivalence between Banach spaces. Bull. Am. Math. Soc. 84(2), 281–284 (1978)

    Article  MathSciNet  Google Scholar 

  2. F. Albiac, N.J. Kalton, Topics in Banach Space Theory, 2nd edn. (Springer, Berlin, 2016)

    Book  Google Scholar 

  3. C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis, 3rd edn. (Springer, Berlin, 2006)

    MATH  Google Scholar 

  4. C.D. Aliprantis, R. Tourky, Cones and Duality (American Mathematical Society, Providence, 2007)

    Book  Google Scholar 

  5. T. Andô, On fundamental properties of a Banach space with a cone. Pac. J. Math. 12(4), 1163–1169 (1962)

    Article  MathSciNet  Google Scholar 

  6. L. Asimow, A.J. Ellis, Convexity Theory and Its Applications in Functional Analysis (Academic, London, 1980)

    MATH  Google Scholar 

  7. C.J.K. Batty, D.W. Robinson, Positive one-parameter semigroups on ordered Banach spaces. Acta Appl. Math. 2(3–4), 221–296 (1984)

    Article  MathSciNet  Google Scholar 

  8. Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis (American Mathematical Society, Providence, 2000)

    MATH  Google Scholar 

  9. E.B. Davies, The structure and ideal theory of the predual of a Banach lattice. Trans. Am. Math. Soc. 131, 544–555 (1968)

    Article  MathSciNet  Google Scholar 

  10. M. de Jeu, M. Messerschmidt, A strong open mapping theorem for surjections from cones onto Banach spaces. Adv. Math. 259, 43–66 (2014)

    Article  MathSciNet  Google Scholar 

  11. K. de Leeuw, Banach spaces of Lipschitz functions. Stud. Math. 21, 55–66 (1961/1962)

    Article  MathSciNet  Google Scholar 

  12. A. Defant, F. Floret, Tensor Norms and Operator Ideals (North-Holland, Amsterdam, 1993)

    MATH  Google Scholar 

  13. A.J. Ellis, The duality of partially ordered normed linear spaces. J. Lond. Math. Soc. 39, 730–744 (1964)

    Article  MathSciNet  Google Scholar 

  14. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler, Banach Space Theory (Springer, New York, 2011)

    Book  Google Scholar 

  15. G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces. Stud. Math. 159(1), 121–141 (2003)

    Article  MathSciNet  Google Scholar 

  16. J. Grosberg, M. Krein, Sur la décomposition des fonctionnelles en composantes positives. C. R. (Dokl.) Acad. Sci. URSS (N.S.) 25, 723–726 (1939)

    Google Scholar 

  17. J.B. Guerrero, G. López-Pérez, A. Rueda Zoca, Octahedrality in Lipschitz-free Banach spaces. Proc. Roy. Soc. Edinburgh Sect. A 148(3), 447–460 (2018)

    Article  MathSciNet  Google Scholar 

  18. P. Hájek, R.J. Smith, Some duality relations in the theory of tensor products. Expo. Math. 30(3), 239–249 (2012)

    Article  MathSciNet  Google Scholar 

  19. G. Jameson, Ordered Linear Spaces (Springer, Berlin, 1970)

    Book  Google Scholar 

  20. W.B. Johnson, J. Lindenstrauss, G. Schechtman, Banach spaces determined by their uniform structures. Geom. Funct. Anal. 6(3), 430–470 (1996)

    Article  MathSciNet  Google Scholar 

  21. N.J. Kalton, Spaces of Lipschitz and Hölder functions and their applications. Collect. Math. 55(2), 171–217 (2004)

    MathSciNet  MATH  Google Scholar 

  22. N.J. Kalton, Lipschitz and uniform embeddings into ℓ ∞. Fundam. Math. 212(1), 53–69 (2011)

    Article  MathSciNet  Google Scholar 

  23. V.L. Klee, Boundedness and continuity of linear functionals. Duke Math. J. 22(2), 263–269 (1955)

    Article  MathSciNet  Google Scholar 

  24. J. Lindenstrauss, Uniform embeddings, homeomorphisms and quotient maps between Banach spaces (a short survey). Topology Appl. 85(1–3), 265–279 (1998). 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996)

    Google Scholar 

  25. M. Messerschmidt, Geometric duality theory of cones in dual pairs of vector spaces. J. Funct. Anal. 269(7), 2018–2044 (2015)

    Article  MathSciNet  Google Scholar 

  26. M. Messerschmidt, A pointwise Lipschitz selection theorem. Set-Valued Var. Anal. (2017). https://doi.org/10.1007/s11228-017-0455-2

    Article  MathSciNet  Google Scholar 

  27. M. Messerschmidt, Strong Klee–Andô theorems through an open mapping theorem for cone-valued multi-functions. J. Funct. Anal. 275(12), 3325–3337 (2018)

    Article  MathSciNet  Google Scholar 

  28. K.F. Ng, The duality of partially ordered Banach spaces. Proc. Lond. Math. Soc. (3) 19, 269–288 (1969)

    Article  MathSciNet  Google Scholar 

  29. K.F. Ng, On a theorem of Dixmier. Math. Scand. 29(1971), 279–280 (1972)

    MathSciNet  MATH  Google Scholar 

  30. K.F. Ng, C.K. Law, Monotonic norms in ordered Banach spaces. J. Austral. Math. Soc. Ser. A 45(2), 217–219 (1988)

    Article  MathSciNet  Google Scholar 

  31. V.G. Pestov, Free Banach spaces and representations of topological groups. Funktsional. Anal. i Prilozhen. 20(1), 81–82 (1986)

    Article  MathSciNet  Google Scholar 

  32. M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces. Isr. J. Math. 48(2–3), 139–147 (1984)

    Article  MathSciNet  Google Scholar 

  33. R.A. Ryan, Introduction to Tensor Products of Banach Spaces (Springer, London, 2002)

    Book  Google Scholar 

  34. H.H. Schaefer, Banach Lattices and Positive Operators (Springer, Berlin, 1974)

    Book  Google Scholar 

  35. N. Weaver, Lipschitz Algebras (World Scientific, River Edge, 1999)

    Book  Google Scholar 

  36. N. Weaver, On the unique predual problem for Lipschitz spaces. Math. Proc. Camb. Philos. Soc. 165(3), 467–473 (2018)

    Article  MathSciNet  Google Scholar 

  37. R. Whitley, Projecting m onto c 0. Am. Math. Mon. 73(3), 285–286 (1966)

    Article  Google Scholar 

  38. G. Wittstock, Ordered normed tensor products, in Foundations of Quantum Mechanics and Ordered Linear Spaces. Lecture Notes in Physics, vol. 29 (Springer, Berlin, 1974), pp. 67–84

    Google Scholar 

  39. D. Yost, There can be no Lipschitz version of Michael’s selection theorem, in Proceedings of the Analysis Conference, Singapore 1986. North Holland Mathematics Studies, vol. 150 (North-Holland, Amsterdam, 1988), pp. 295–299

    Google Scholar 

Download references

Acknowledgements

The author would like to express his thanks to the MathOverflow community, especially to Bill Johnson for bringing work described in Sect. 5 to the author’s attention.

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria, South Africa

    Miek Messerschmidt

Authors
  1. Miek Messerschmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miek Messerschmidt .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Mississippi, Oxford, MS, USA

    Gerard Buskes

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands; Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Marcel de Jeu

  3. College of Science and Engineering, Flinders University, Adelaide, Australia

    Peter Dodds

  4. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Anton Schep

  5. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Fedor Sukochev

  6. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Jan van Neerven

  7. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, UK

    Anthony Wickstead

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Messerschmidt, M. (2019). On the Lipschitz Decomposition Problem in Ordered Banach Spaces and Its Connections to Other Branches of Mathematics. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_22

Download citation

Publish with us

Policies and ethics

Search

Navigation