Abstract

The Daugavet property and the diameters of relatively weakly open subsets of unit balls in Banach lattices XX on measure spaces are studied. It is shown that under mild assumptions the subspace XaX_a of order continuous elements inherits the Daugavet property from XX. This is applied to prove that if XX has the Daugavet property and the Köthe dual X′X' is strictly monotone (resp. order continuous) then X′X' contains a lattice isometric (resp. isomorphic) copy of L1(0,1)L_1(0,1). These results yield that a large class of r.i. spaces including several interpolation sums fail the Daugavet property and also that any r.i. space over a finite atomless measure space with the Daugavet property coincide to either L1L_1 or L∞L_{\infty}. Applications are shown for Orlicz, Lorentz, Marcinkiewicz spaces as well for Nakano spaces. It is established that in most cases these spaces do not enjoy the Daugavet property. However, it is proved that in a large class of Orlicz or Nakano spaces (variable exponent spaces), in particular those induced by fast growing Orlicz functions, all non-empty relatively weakly open subsets of their unit balls have diameter two.

Author information

Contact details are reproduced from the original publication and may be historical.

María D. Acosta

Dep. de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain

dacosta@ugr.es

Anna Kamińska

Dept. of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, U.S.A.

kaminska@memphis.edu

Mieczysław Mastyło

Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
and: Institute of Mathematics, Polish Academy of Sciences, Umultowska 87, 61-614 Poznań, Poland

mastylo@amu.edu.pl

Keywords

  • Daugavet property
  • weak neighborhoods
  • Banach lattices
  • Orlicz spaces
  • Lorentz spaces
  • Marcinkiewicz spaces
  • Nakano spaces
  • variable exponent spaces

Mathematics Subject Classification

46B20, 46E30

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Suggested citation

M. D. Acosta, A. Kamińska, M. Mastyło. “The Daugavet Property and Weak Neighborhoods in Banach Lattices.” Journal of Convex Analysis 19 (2012), No. 3, 875–912. https://doi.org/10.68381/jca19046

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.