<p>We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$c_0(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$c_0({\mathbb {N}}),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$c_0(\Gamma ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.</p>

Strictly convex norms and the local diameter two property

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2026年1月7日

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Trond A. Abrahamsen,
Petr Hájek,
Vegard Lima,
Stanimir Troyanski

摘要

We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $c_0(\Gamma )$ $c_{0} (Γ)$ possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of $c_0({\mathbb {N}}),$ $c_{0} (N),$ admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on $c_0(\Gamma ),$ $c_{0} (Γ),$ where $\Gamma $ $Γ$ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.