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Descriptive Topology and Functional Analysis pp 209–240Cite as

Some Non-linear Geometrical Properties of Banach Spaces

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Abstract

In this survey we report on very recent results about some non-linear geometrical properties of many classes of real and complex Banach spaces and uniform algebras, including the ball algebra \(\fancyscript{A}_u(B_X)\) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space \(X\). These geometrical properties are: Polynomial numerical index, Polynomial Daugavet property and Bishop-Phelp-Bollobás property for multilinear mappings.

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References

  1. Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. Graduate Texts in Mathematics, vol. 50. AMS, Providence (2002)

    Google Scholar 

  2. Abramovich, Y.A., Aliprantis, C.D.: Problems in Operator Theory. Graduate Texts in Mathematics, vol. 51. AMS, Providence (2002)

    Google Scholar 

  3. Abramovich, Y.A., Aliprantis, C.D., Burkinshaw, O.: The Daugavet equation in uniformly convex Banach spaces. J. Funct. Anal. 97, 215–230 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Acosta, M.D.: Denseness of norm attaining mappings. Rev. R. Acad. Cien. Ser. A. Mat. (RACSAM) 100, 9–30 (2006)

    Google Scholar 

  5. Acosta, M.D.: Boundaries for spaces of holomorphic functions on \(C(K)\). Publ. Res. I. Math. Sci. 42, 27–44 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Acosta, M.D., Aron, R.M., García, D., Maestre, M.: The Bishop-Phelps-Bollobás Theorem for operators. J. Funct. Anal. 254, 2780–2799 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Acosta, M.D., Becerra-Guerrero, J., García, D., Kim, S.K., Maestre, M.: The Bishop-Phelps-Bollobás property and its finite dimensional approach (preprint)

    Google Scholar 

  8. Acosta, M.D., Becerra-Guerrero, J., García, D., Maestre, M.: The Bishop-Phelps-Bollobás theorem for bilinear forms. Trans. Amer. Math. Soc. 365–11, 5911–5932 (2013)

    Article  Google Scholar 

  9. Acosta, M.D., Becerra-Guerrero, J., Choi, Y.S., García, D., Kim, S.K., Lee, H.J., Maestre, M.: The Bishop-Phelps-Bollobás property for bilinear forms and polynomials. J. Math. Soc. Japan 66, 957–979 (2014)

    Google Scholar 

  10. Acosta, M.D., García, D., Maestre, M.: A multilinear Lindenstrauss theorem. J. Funct. Anal. 235, 122–136 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Acosta, M.D., Kamińska, A.: Weak neighborhoods and the Daugavet property of the interpolation spaces \(L^1 + L^\infty \) and \(L^1 \cap L^\infty \). Indiana Univ. Math. J. 57, 77–96 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Acosta, M.D., Kim, S.G.: Numerical boundaries for some classical Banach spaces. J. Math. Anal. Appl. 350, 694–707 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Aksoy, A.G., Lewicki, G.: Limit theorems for the numerical index. J. Math. Anal. Appl. 398, 296–302 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Aron, R.M., Cascales, B., Kozhushkina, O.: The Bishop-Phelps-Bollobás theorem and Asplund operators. Proc. Amer. Math. Soc. 139(10), 3553–3560 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Aron, R.M., Choi, Y.S., García, D., Maestre, M.: The Bishop-Phelps-Bollobás Theorem for \({\fancyscript {L}}(L_1(\mu ), L_\infty [0,1])\). Adv. Math. 228, 617–628 (2011)

    Google Scholar 

  16. Aron, R.M., García, D., Maestre, M.: On norm attaining polynomials. Publ. Res. Inst. Math. Sci. 39, 165–172 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Aron, R.M., Globevnik, J.: Analytic functions on \(c_0\). Rev. Mat. Univ. Complt. Madrid 2, 27–33 (1989)

    MATH  MathSciNet  Google Scholar 

  18. Avilés, A., Kadets, V.M., Martín, M., Merí, J., Shepelska, V.: Slicely countably determined Banach spaces. C. R. Math. Acad. Sci. Paris. 347, 1277–1280 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Avilés, A., Kadets, V.M., Martín, M., Merí, J., Shepelska, V.: Slicely countably determined Banach spaces. Trans. Amer. Math. Soc. 362, 4871–4900 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bauer, F.L.: On the field of values subordinate to a norm. Numer. Math. 4, 103–111 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  21. Guerrero, Becerra: J., Rodríguez-Palacios, A.: Banach spaces with the Daugavet property, and the centralizer. J. Funct. Anal. 254, 2294–2302 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Bishop, E., Phelps, R.R.: A proof that every Banach space is subreflexive. Bull. Amer. Math. Soc. 67, 97–98 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  23. Bogdanowicz, W.: On the weak continuity of the polynomial functionals defined on the space \(c_0\). Bull. Acad. Polon. Sci. Cl. III. 5, 243–246 (in Russian) (1957)

    Google Scholar 

  24. Bollobás, B.: An extension to the theorem of Bishop and Phelps. Bull. London Math. Soc. 2, 181–182 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  25. Bonsall, F.F., Cain, B.E., Schneider, H.: The numerical range of a continuos mapping of a normed space. Aequationes Math. 2, 86–93 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  26. Bonsall, F.F., Duncan, J.: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras. London Mathematical Society Lecture Note Series 2, Cambridge (1971)

    Google Scholar 

  27. Bonsall, F.F., Duncan, J.: Numerical Ranges II. London Mathematical Society Lecture Note Series 10, Cambridge (1973)

    Google Scholar 

  28. Bourgain, J.: Dentability and the Bishop-Phelps property. Israel J. Math. 28, 265–271 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  29. Boyko, K., Kadets, V.M., Martín, M., Werner, D.: Numerical index of Banach spaces and duality. Math. Proc. Cambridge Phil. Soc. 142, 93–102 (2007)

    Article  MATH  Google Scholar 

  30. Boyko, K., Kadets, V.M., Martín, M., Werner, D.: Properties of lush spaces and applications to Banach spaces with numerical index \(1\). Studia Math. 190, 117–133 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Cascales, B., Guirao, A.J., Kadets, V.M.: A Bishop-Phelps-Bollobás type theorem for uniform algebras. Adv. Math. 240, 370–382 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Cheng, L., Dai, D., Dong, Y.: A sharp operator version of the Bishop-Phelps Theorem for operators from \(\ell _1\) to CL-spaces. Proc. Amer. Math. Soc. 141, 867–872 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Choi, Y.S.: Norm attaining bilinear forms on \(L^1[0,1]\). J. Math. Anal. Appl. 211, 295–300 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  34. Choi, Y.S., García, D., Kim, S.G., Maestre, M.: The polynomial numerical index of a Banach space. Proc. Edinb. Math. Soc. 49, 32–52 (2006)

    Article  Google Scholar 

  35. Choi, Y.S., García, D., Kim, S.G., Maestre, M.: Composition, numerical range and Aron-Berner extension. Math. Scand. 103, 97–110 (2008)

    MATH  MathSciNet  Google Scholar 

  36. Choi, Y.S., García, D., Kim, S.K., Maestre, M.: Some geometric properties on disk algebra. J. Math. Anal. Appl. 409, 147–157 (2014)

    Article  MathSciNet  Google Scholar 

  37. Choi, Y.S., García, D., Maestre, M., Martín, M.: The polynomical numerical index for some complex vector-valued function spaces. Q. J. Math. 59, 455–474 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  38. Choi, Y.S., García, D., Maestre, M., Martín, M.: The Daugavet equation for polynomials. Studia Math. 178, 63–82 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Choi, Y.S., Kim, S.G.: Norm or numerical radius attaining multilinear mappings and polynomials. J. London Math. Soc. 54, 135–147 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  40. Choi, Y.S., Kim, S.K.: The Bishop-Phelps-Bollobás theorem for operators from \(L_1(\mu )\) to Banach spaces with the Radon-Nikodým property. J. Funct. Anal. 261, 1446–1456 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Choi, Y.S., Kim, S.K.: The Bishop-Phelps-Bollobás property and lush spaces. J. Math. Anal. Appl. 390, 549–555 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  42. Choi, Y.S., Song, H.G.: The Bishop-Phelps-Bollobás theorem fails for bilinear forms on \(l_1 \times l_1\). J. Math. Anal. Appl. 360, 752–753 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  43. Dai, D.: The Bishop-Phelps-Bollobás Theorem for bilinear mappings (preprint)

    Google Scholar 

  44. Daugavet, I.K.: On a property of completely continuous operators in the space \(C\). Uspekhi Mat. Nauk 18, 157–158 (1963)

    MATH  MathSciNet  Google Scholar 

  45. Diestel, J.: Sequences and Series in Banach Space. Graduate Texts in Mathematics, vol. 92. Springer, New York (1984)

    Google Scholar 

  46. Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys 15. AMS, Providence (1977)

    Google Scholar 

  47. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics, Springer, London (1999)

    Book  MATH  Google Scholar 

  48. Duncan, J., McGregor, C.M., Pryce, J.D., White, A.J.: The numerical index of a normed space. J. London Math. Soc. 2, 481–488 (1970)

    MATH  MathSciNet  Google Scholar 

  49. Ed-dari, E.: On the numerical index of Banach spaces. Linear Algebra Appl. 403, 86–96 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  50. Faried, N., Ibrahim, F.A., Bakery, A.A.: The numerical index of the \(L_{(p)}\) space of dimension two. Int. Math. Forum 6, 2101–2108 (2011)

    MATH  MathSciNet  Google Scholar 

  51. Franchetti, C., Payá, R.: Banach spaces with strongly differentiable norm. Boll. U.M.I. 7, 45–70 (1993)

    Google Scholar 

  52. García, D., Grecu, B., Maestre, M., Martín, M., Merí, J.: Two-dimensional Banach spaces with Polynomial numerical index zero. Linear Algebra Appl. 430, 2488–2500 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  53. García, D., Grecu, B., Maestre, M., Martín, M., Merí, J.: Polynomial numerical indices of \(C(K)\) and \(L_1(\mu )\). Proc. Amer. Math. Soc. 142(4), 1229–1235 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  54. García, D., Lee, H.J., Maestre, M.: The Bishop-Phelps-Bollobás property for Hermitian forms on Hilbert spaces. Q. J. Math. 65, 201–209 (2014)

    Article  MathSciNet  Google Scholar 

  55. Harris, L.A.: The numerical range of holomorphic functions in Banach spaces. Amer. J. Math. 93, 1005–1019 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  56. Harris, L.A.: The numerical range of functions and best approximation. Proc. Camb. Phil. Soc. 76, 133–141 (1974)

    Article  MATH  Google Scholar 

  57. Kadets, V.M., Martín, M., Merí, J., Shepelska, V.: Lushness, numerical index one and duality. J. Math. Anal. Appl. 357, 15–24 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  58. Kadets, V.M., Martín, M., Merí, J., Werner, D.: Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces. Canad. J. Math. 65, 331–348 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  59. Kadets, V.M., Martín, M., Payá, R.: Recent progress and open questions on the numerical index of Banach spaces. Rev. R. Acad. Cien. Serie A Mat. (RACSAM) 100, 155–182 (2006)

    Google Scholar 

  60. Kadets, V.M., Popov, M.M.: The Daugavet property for narrow operators in rich subspaces of \(C[0,1]\) and \(L_1[0,1]\). St. Petersburg Math. J. 8, 571–584 (1997)

    MathSciNet  Google Scholar 

  61. Kadets, V.M., Shvidkoy, R.V., Sirotkin, G.G., Werner, D.: Banach spaces with the Daugavet property. Trans. Amer. Math. Soc. 352, 855–873 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  62. Kim, J., Lee, H.J.: Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices. J. Funct. Anal. 257, 931–947 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  63. Kim, S.G., Martín, M., Merí, J.: On the polynomial numerical index of the real spaces \(c_0\), \(\ell _1\) and \(\ell _{\infty }\). J. Math. Anal. Appl. 337, 98–06 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  64. Kim, S.K.: The Bishop-Phelps-Bollobás Theorem for operators from \(c_0\) to uniformly convex spaces. Israel J. Math. 197, 425–435 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  65. Kim, S.K., Lee, H.J.: Uniform convexity and Bishop-Phelps-Bollobás property. Can. J. Math. 66, 373–386 (2014)

    Article  MATH  Google Scholar 

  66. Lee, H.J.: Banach spaces with polynomial numerical index \(1\). Bull. London Math. Soc. 40, 193–198 (2008)

    Article  MATH  Google Scholar 

  67. Lee, H.J., Martín, M.: Polynomial numerical indices of Banach spaces with \(1\)-unconditional bases. Linear Algebra Appl. 437, 2001–2008 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  68. Lee, H.J., Martín, M., Merí, J.: Polynomial numerical indices of Banach spaces with absolute norm. Linear Algebra Appl. 435, 400–408 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  69. Lindenstrauss, J.: On operators which attain their norm. Israel J. Math. 1, 139–148 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  70. López, G., Martín, M., Merí, J.: Numerical Index of Banach spaces of weakly or weakly-star continuous functions. Rocky Mountain J. Math. 38, 213–223 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  71. Martín, M.: The alternative Daugavet property of \(C^*\)-algebras and \(JB^*\)-triples. Math. Nachr. 281, 376–385 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  72. Martín, M., Merí, J., Popov, M.M.: The polynomial Daugavet property for atomless \(L_1(\mu )\)-spaces. Arch. Math. (Basel) 94, 383–389 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  73. Martín, M., Merí, J., Popov, M.M.: On the numerical index of real \(L_p(\mu )\)-spaces. Israel J. Math. 184, 183–192 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  74. Martín, M., Merí, J., Popov, M.M., Randrianantoanina, B.: Numerical index of absolute sums of Banach spaces. J. Math. Anal. Appl. 375, 207–22 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  75. Martín, M., Oikhberg, T.: The alternative Daugavet property. J. Math. Anal. Appl. 294, 158–180 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  76. Martín, M., Payá, R.: Numerical index of vector-valued function spaces. Studia Math. 142, 269–280 (2000)

    MATH  MathSciNet  Google Scholar 

  77. Martín, M., Payá, R.: On CL-spaces and almost CL-spaces. Ark. Mat. 42, 107–118 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  78. Martín, M., Villena, A.: Numerical index and the Daugavet property for \(L_\infty (\mu , X)\). Proc. Edinb. Math. Soc. 46(2), 415–420 (2003)

    Google Scholar 

  79. Rodríguez-Palacios, A.: Numerical ranges of uniformly continuous functions on the unit sphere of a Banach space. J. Math. Anal. Appl. 297, 472–476 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  80. Rosenthal, H.: The Lie algebra of a Banach space. Lecture Notes in Mathematics, vol. 1166. Springer, Berlin (1985)

    Google Scholar 

  81. Saleh, Y.: Norm attaining bilinear forms on \(L_1 (\mu )\). Int. J. Math. & Math. Sci. 23, 833–837 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  82. Saleh, Y.: Norm attaining multilinear forms on \(L_1 (\mu )\). Int. J. Math. & Math. Sci. 1–6 (2008)

    Google Scholar 

  83. Sánchez Pérez, E., Werner, D.: The \(p\)-Daugavet property for function spaces. Arch. Math. (Basel) 96, 565–575 (2011)

    Google Scholar 

  84. Toeplitz, O.: Das algebraische Analogon zu einem Satze von Fejer. Math. Z. 2, 187–197 (1918)

    Article  MATH  MathSciNet  Google Scholar 

  85. Werner, D.: An elementary approach to the Daugavet equation. In: Kalton, N., Saab, E., Montgomery-Smith, S. (eds.) Interaction between Functional Analysis, Harmonic Analysis and Probability. Lecture Notes in Pure and Applied Mathematics, vol. 175, pp. 449–454. Dekker, New York (1996)

    Google Scholar 

  86. Werner, D.: The Daugavet equation for operators on function spaces. J. Funct. Anal. 143, 117–128 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  87. Werner, D.: Recent progress on the Daugavet property. Irish Math. Soc. Bull. 46, 77–97 (2001)

    Google Scholar 

  88. Wojtaszczyk, P.: Some remarks on the Daugavet equation. Proc. Amer. Math. Soc. 115, 1047–1052 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors were supported by MICINN Project MTM2011-22417 and Prometeo II/2013/013.

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  1. Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100, Burjasot (Valencia), Spain

    Domingo García & Manuel Maestre

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  1. Domingo García
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  2. Manuel Maestre
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Correspondence to Manuel Maestre .

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  1. Center Operations Research Univ. Inst., Universidad Miguel Hernandez, Elche, Alicante, Spain

    Juan Carlos Ferrando

  2. Matematica Aplicada, Universidad Politecnica de Valencia, Valencia, Valencia, Spain

    Manuel López-Pellicer

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García, D., Maestre, M. (2014). Some Non-linear Geometrical Properties of Banach Spaces. In: Ferrando, J., López-Pellicer, M. (eds) Descriptive Topology and Functional Analysis. Springer Proceedings in Mathematics & Statistics, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-05224-3_11

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