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Renormings of ℓ1 and C 0 and Fixed Point Properties

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Handbook of Metric Fixed Point Theory

Abstract

As has been noted in previous chapters, there are many geometric conditions on a Banach space strong enough to imply that the Banach space has the fixed point property. Geometric conditions such as uniform rotundity, uniform smoothness, or normal structure together with reflexivity are sufficient to imply the fixed point property. Each of these conditions also implies (or assumes in the last case) that the Banach space is reflexive.

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References

  1. D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (3) (1981), 423–424.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Alspach, private communication.

    Google Scholar 

  3. E. Behrends, On Banach spaces X for which every infinite-dimensional closed subspace contains an isometric copy of X, unpublished, 1984.

    Google Scholar 

  4. M. Besbes, Points fixes dans les espaces des opérateurs compact, preprint.

    Google Scholar 

  5. C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164.

    MathSciNet  MATH  Google Scholar 

  6. S. Chen, Geometry of Orlicz spaces, Dissertationes Math. (Rozprawy Mat.) 356(1996).

    Google Scholar 

  7. N.L. Carothers, S.J. Dilworth and C.J. Lennard, On a localization of the UKK property and the fixed point property in L,,,,1, Interaction between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994 ), Lecture Notes in Pure and Applied Math. 175, Dekker, New York, 1996, 111–124.

    Google Scholar 

  8. B.J. Cole, T.W. Gamelin and W.B. Johnson, Analytic disks in fibers over the unit ball of a Banach space, Michigan Math. J. 39 (1992), 551–569.

    MathSciNet  MATH  Google Scholar 

  9. M. M. Day, R. C. James, S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Can. J. Math. 23 (6) (1971), 1051–1059.

    Google Scholar 

  10. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, Springer-Verlag, New York-Berlin, 1984.

    Google Scholar 

  11. S.J. Dilworth, Maria Girardi and J. Hagler, Dual Banach spaces which contain an isometric copy of Lí, Bull. Polish Acad. Sci. Math. 48 (1) (2000), 1–12.

    MathSciNet  MATH  Google Scholar 

  12. P.G. Dodds, T.K. Dodds, P.N. Dowling, C.J. Lennard, N. J. Randrianantoanina, and F. A. Sukochev, Subspaces of preduals of von Neumann algebras, preprint.

    Google Scholar 

  13. P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 125 (1997), 167–174.

    Article  MathSciNet  MATH  Google Scholar 

  14. P.N. Dowling and C.J. Lennard, Every nonreflexive subspace of Lí[0,1] fails the fixed point property, Proc. Amer. Math. Soc. 125 (1997), 443–446.

    Article  MathSciNet  MATH  Google Scholar 

  15. P.N. Dowling, C.J. Leonard and B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996), 653–662.

    Article  MathSciNet  MATH  Google Scholar 

  16. P.N. Dowling, C.J. Leonard and B. Turett, Asymptotically isometric copies of co in Banach spaces, J. Math. Anal. Appl. 219 (1998), 377–391.

    Article  MathSciNet  MATH  Google Scholar 

  17. P.N. Dowling, C.J. Lennard and B. Turett, Some fixed point results in e l and co, Nonlinear Analysis 39 (2000), 929–936.

    Article  MathSciNet  MATH  Google Scholar 

  18. P.N. Dowling, C.J. Leonard and B. Turett, The fixed point property for subsets of some classical Banach spaces, preprint.

    Google Scholar 

  19. P.N. Dowling and N. Randrianantoanina, Spaces of compact operators on a Hilbert space with the fixed point property, J. Funct. Anal. 168 (1999), 111–120.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. N. Dowling, Asymptotically isometric copies of co and renormings of Banach spacess, J. Math. Anal. Appl. 228 (1) (1998), 265–271.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. N. Dowling, The fixed point property for subsets of Li [0,1], Contemporary Math. 232 (1999), 131–137.

    Article  MathSciNet  Google Scholar 

  22. Patrick N. Dowling, Isometric copies of co and e°° in duals of Banach spaces, J. Math. Anal. Appl. 244 (1) (2000), 223–227.

    Article  MathSciNet  MATH  Google Scholar 

  23. V.P. Fonf and M.I. Kadec, Subspaces of eí with strictly convex norm, Math. Notes of the Academy of Sc., USSR 33 (1983), 213–215.

    MATH  Google Scholar 

  24. K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 1350–140.

    MathSciNet  Google Scholar 

  25. K. Goebel, W. A. Kirk, and R. L. Thiele, Uniformly lipschitzian families of transformations in Banach spaces, Canadian J. of Math 26 (1974), 1245–1256.

    Article  MATH  Google Scholar 

  26. K. Goebel and T. Kuczumow, Irregular convex sets with fixed-point property for nonexpansive mappings,Colloq. Math. 40 (2)(1978/79), 259–264.

    Google Scholar 

  27. J. Hagler, Embeddings of L í spaces into conjugate Banach spaces, University of California Berkeley, Ph.D. Thesis, 1972.

    Google Scholar 

  28. J. Hagler, Some more Banach spaces containing e í, Studia Math. 46 (1973), 35–42.

    MathSciNet  MATH  Google Scholar 

  29. R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550.

    Article  MathSciNet  MATH  Google Scholar 

  30. M.I. Kadec and A. Pelczynski, Bases, lacunary sequences and complemented subspaces in LP, Studia Math. 21 (1962), 161–176.

    MathSciNet  MATH  Google Scholar 

  31. M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordoff Ltd., Groningen, 1961.

    Google Scholar 

  32. C. Lennard, C„„ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1) (1990), 71–77.

    MathSciNet  Google Scholar 

  33. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: Sequence Spaces, Ergebnisse der Mathematik und Ihrer Grenzgebiete 92, Springer-Verlag, Berlin-Heidelberg-NewYork, 1977.

    Google Scholar 

  34. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II: Function Spaces, Ergebnisse der Mathematik und Ihrer Grenzgebiete 97, Springer-Verlag, Berlin-Heidelberg-NewYork, 1979.

    Google Scholar 

  35. B. Maurey, Points fixes des contractions de certain faiblement compacts de L 1,Seminaire d’Analyse Fonctionelle, Exposé VIII, École Polytechnique, Centre de Mathematiques, 19801981.

    Google Scholar 

  36. A. Pelczyrcski, On Banach spaces containing L1 [0,1], Studia Math. 30 (1968), 231–246.

    MathSciNet  Google Scholar 

  37. H. Pfitzner, A note on asymptotically isometric copies of eí and co, preprint.

    Google Scholar 

  38. H. Pfitzner, Perturbation of el-copies and measure convergence in preduals of von Neumann algebras, preprint.

    Google Scholar 

  39. H. Pfitzner, L-embedded Banach spaces and measure topology, preprint.

    Google Scholar 

  40. N. Randrianantoanina, Kadec-Pelczyriski decomposition for Haagerup LP-spaces, preprint.

    Google Scholar 

  41. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel-Dekker, New York, 1991.

    MATH  Google Scholar 

  42. P. M. Soardi, Existence of fixed points on nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25–29.

    Article  MathSciNet  MATH  Google Scholar 

  43. B. Sims and M. Smyth, On non-uniform conditions giving weak normal structure, Quaestiones Mathematicae 18 (1995), 9–19.

    Article  MathSciNet  MATH  Google Scholar 

  44. M. A. Smith and B. Turett, Some examples concerning normal and uniform normal structure in Banach spaces, J. Austral. Math. Soc. Ser. A, 48 (1990), 223–234.

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Turett, Rotundity of Orlicz spaces, Proc. Acad. Amsterdam A 79 (1976), 462–469.

    MathSciNet  MATH  Google Scholar 

  46. V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 1–33.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Miami University, Oxford, OH, 45056, USA

    P. N. Dowling

  2. Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    C. J. Lennard

  3. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    B. Turett

Authors
  1. P. N. Dowling
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  2. C. J. Lennard
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  3. B. Turett
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Editor information

Editors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, IA, USA

    William A. Kirk

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia

    Brailey Sims

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Dowling, P.N., Lennard, C.J., Turett, B. (2001). Renormings of ℓ1 and C 0 and Fixed Point Properties. In: Kirk, W.A., Sims, B. (eds) Handbook of Metric Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1748-9_9

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  • DOI: https://doi.org/10.1007/978-94-017-1748-9_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5733-4

  • Online ISBN: 978-94-017-1748-9

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