The Daugavet Property and Weak Neighborhoods in Banach Lattices

Abstract

<jats:p> The Daugavet property and the diameters of relatively weakly open subsets of unit balls in Banach lattices <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> on measure spaces are studied. It is shown that under mild assumptions the subspace <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X_a</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>a</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of order continuous elements inherits the Daugavet property from <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . This is applied to prove that if <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> has the Daugavet property and the Köthe dual <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X'</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mo mathvariant="normal" lspace="0em" rspace="0em">′</mml:mo> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is strictly monotone (resp. order continuous) then <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X'</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mo mathvariant="normal" lspace="0em" rspace="0em">′</mml:mo> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> contains a lattice isometric (resp. isomorphic) copy of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L_1(0,1)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . 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 <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L_1</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L_{\infty}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . 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. </jats:p>