Abstract
We study the von Neumann–Jordan (NJ) constant for a class of Day–James spaces, namely the ℓ p −ℓ 1 space, as well as its dual space. In this paper, we only consider the case for p ≥ 2, since the case for 1 ≤ p ≤ 2 has been calculated. If (p−2)22(1/p-1) ≤ 1, we obtain the exact value of the NJ constant; if (p−2)22(1/p-1) > 1, we obtain a formula, from which one can determine the exact value of the NJ constant. As a byproduct, we give a partial answer to the question raised by Kato et al. (Studia Math 144:275–295, 2001).
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Alonso J., Martin P.: A counterexample for a conjecture of G. Zbăganu about the Neumann–Jordan constant. Rev. Roum. Math. Pures Appl. 51, 135–141 (2006)
Alonso J., Martin P., Papini P.L.: Wheeling around von Neumann–Jordan constant in Banach spaces. Studia Math. 188, 135–150 (2008)
Clarkson J.A.: The von Neumann–Jordan constant for the Lebesgue space. Ann. Math. 38, 114–115 (1937)
Dhompongsa S., Piraisangjun P., Saejung S.: Generalized von Neumann–Jordan constants and uniform normal structure. Bull. Aust. Math. Soc. 67, 225–240 (2003)
Gao J., Lau K.S.: On two classes of Banach spaces with uniform normal structure. Studia Math. 99, 41–56 (1991)
Jiménez-Melado A., Llorens-Fuster E., Saejung S.: The von Neumann–Jordan constant, weak orthogonality and normal structure in Banach spaces. Proc. Am. Math. Soc. 134, 355–364 (2006)
Kato M., Maligranda L.: On James and von Neumann–Jordan constants of Lorentz sequence spaces. J. Math. Anal. Appl. 258, 457–465 (2001)
Kato M., Maligranda L., Takahashi Y.: On James and Jordan–von Neumann constants and the normal structure coefficient of Banach spaces. Studia Math. 144, 275–295 (2001)
Kato M., Takahashi Y.: Von Neumann–Jordan constant for Lebesgue–Bochner spaces. J. Inequal. Appl. 2, 89–97 (1998)
Saejung S.: On James and von Neumann–Jordan constants and sufficient conditions for the fixed point property. J. Math. Anal. Appl. 323, 1018–1024 (2006)
Takahashi Y., Kato M.: A simple inequality for the von Neumann–Jordan and James constants of a Banach space. J. Math. Anal. Appl. 359, 602–609 (2009)
Wang F.: On the James and von Neumann–Jordan constants in Banach spaces. Proc. Am. Math. Soc. 138, 695–701 (2010)
Yang C.: An inequality between the James type constant and the modulus of smoothness. J. Math. Anal. Appl. 398, 622–629 (2013)
Yang C., Li H.: An inequality between von Neumann–Jordan constant and James constant. Appl. Math. Lett. 23, 277–281 (2010)
Yang C., Wang F.: On a new geometric constant related to the von Neumann–Jordan constant. J. Math. Anal. Appl. 324, 555–565 (2006)
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Yang, C., Wang, F. The von Neumann–Jordan Constant for a Class of Day–James Spaces. Mediterr. J. Math. 13, 1127–1133 (2016). https://doi.org/10.1007/s00009-015-0539-x
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DOI: https://doi.org/10.1007/s00009-015-0539-x