Arkiv för Matematik

  • Ark. Mat.
  • Volume 40, Number 2 (2002), 323-333.

On M-structure, the asymptotic-norming property and locally uniformly rotund renormings

Eduardo Nieto and Migdalia Rivas

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Abstract

Letr, s ∈ [0, 1], and let X be a Banach space satisfying the M(r, s)-inequality, that is, $\parallel x^{***} \parallel \geqslant r\parallel \pi _X x^{***} \parallel + s\parallel x^{***} - \pi _X x^{***} \parallel for x^{***} \in X^{***} ,$ where πX is the canonical projection from X*** onto X*. We show some examples of Banach spaces not containing c0, having the point of continuity property and satisfying the above inequality for r not necessarily equal to one. On the other hand, we prove that a Banach space X satisfying the above inequality for s=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition, X satisfies $\mathop {\lim \sup }\limits_\alpha \parallel u^* + sx_\alpha ^* \parallel \leqslant \mathop {\lim \sup }\limits_\alpha \parallel v^* + x_\alpha ^* \parallel $ whenever u*, v*X* with ‖u*‖≤‖v*‖ and (x ${}_{α}^{*}$ ) is a bounded weak* null net in X*, then X can be renormed to satisfy the, M(r, 1) and the M(1, s)-inequality such that X* has the weak* asymptotic-norming property I with respect to BX.

Article information

Source
Ark. Mat., Volume 40, Number 2 (2002), 323-333.

Dates
Received: 25 August 2000
Revised: 12 December 2001
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898773

Digital Object Identifier
doi:10.1007/BF02384539

Mathematical Reviews number (MathSciNet)
MR1948068

Zentralblatt MATH identifier
1034.46010

Rights
2002 © Institut Mittag-Leffler

Citation

Nieto, Eduardo; Rivas, Migdalia. On M -structure, the asymptotic-norming property and locally uniformly rotund renormings. Ark. Mat. 40 (2002), no. 2, 323--333. doi:10.1007/BF02384539. https://projecteuclid.org/euclid.afm/1485898773


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References

  • Cabello, J. C. and Nieto, E., On properties of M-ideals, Rocky Mountain J. Math. 28 (1998), 61–93.
  • Cabello, J. C. and Nieto, E., An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property, Studia Math. 129 (1998), 185–196.
  • Cabello, J. C. and Nieto, E., On M-type structures and the fixed point property, Houston J. Math. 26 (2000), 549–560.
  • Cabello, J. C., Nieto, E. and Oja, E., On ideals of compact operators, satisfying the M(r, s)-inequality, J. Math. Anal. Appl. 220 (1998), 334–348.
  • Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renormings in Banach Spaces, Longman, Harlow, 1993.
  • Diestel, J., Geometry of Banach Spaces—Selected Topics, Lecture Notes in Math. 485, Springer-Verlag, Berlin-Heidelberg, 1975.
  • van Dulst, D., Reflexive and Superreflexive Banach Spaces, Math. Centre Tracts 102, Mathematisch Centrum, Amsterdam, 1978.
  • van Dulst, D., Characterizations of Banach Spaces not Containing l1, CWI Tract 59, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.
  • Ghoussoub, N. and Maurey, B., Gσ-embeddings in Hilbert spaces I, J. Funct. Anal. 61 (1985), 72–97.
  • Godefroy, G., Kalton, N. J. and Saphar, P. D., Unconditional ideals in Banach spaces, Studia Math. 104 (1993), 13–59.
  • Hagler, J. and Sullivan, F., Smoothness and weak* sequential compactness, Proc. Amer. Math. Soc. 78 (1980), 497–503.
  • Harmand, P., Werner, D. and Werner, W., M-ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer-Verlag, Berlin-Heidelberg, 1993.
  • Hu, Z. and Lin, B.-L., On the asymptotic-norming property of Banach spaces, in Function Spaces (Edwardsville, Ill., 1990) (Jarosz, K., ed.), Lecture Notes in Pure and Appl. Math. 136, pp. 281–294, Marcel Dekker, New York, 1992.
  • Hu, Z. and Lin, B.-L., Smoothness and the asymptotic-norming properties of Banach spaces, Bull. Austral. Math. Soc. 45 (1992), 285–296.
  • James, R. C. and Ho, A., The asymptotic-norming and the Radon-Nikodým properties for Banach spaces, Ark. Mat. 19 (1981), 53–70.
  • John, K. and Zizler, V., Smoothness and its equivalents in weakly compactly generated Banach spaces, J. Funct. Anal. 15 (1974), 1–11.
  • Lindenstrauss, L. and Stegall, C., Examples of separable Banach spaces which do not contain l1 and whose duals are non-separable, Studia Math. 54 (1975), 81–105.
  • Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, Berlin-Heidelberg, 1989.
  • Raja, M., Locally uniformly rotund norms, Mathematika 46 (1999), 343–358.