The strong Nakano property in Banach lattices

Abstract

Abstract We study the strong Nakano property in Banach lattices with a special focus on free Banach lattices. We show that for every finite dimensional Banach space E , the free Banach lattice FBL [ E ] has the strong Nakano property with a constant independent of the dimension. It is also shown that if FBL [ E ] has the strong Nakano property, then E cannot contain subspaces isomorphic to neither $$c_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> nor $$L_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> .