Complex convexity and monotonicity in quasi-Banach lattices
For \(0<p<\infty\), the modulus of \(p\)-monotonicity of a quasi-Banach lattice \(E\) is defined as \(\Pi_p^E(\varepsilon)=\inf\{\| (| x| ^p + | y| ^p)^{1/p}\| - 1: x,y \in E\), \(\| x\| =1\), \(\| y\| \geq \varepsilon\}\). \(E\) is said to be uniformly \(p\)-monotone if \(\Pi_p^E(\varepsilon) > 0\) for \(\varepsilon > 0\) and is said to be uniformly monotone if it is uniformly \(1\)-monotone. It is shown that in an \(\alpha\)-convex quasi-Banach lattice with \(\alpha\)-convexity constant~\(1\), uniform monotonicity is equivalent to complex uniform convexity. A quasi-Banach lattice \(E\) has an equivalent lattice renorming which is complex uniformly convex of power type if and only if for any \(\lambda \geq 1\), \(\ell_\infty\) is not finitely \(\lambda\)-representable in \(E\).
scientific article; zbMATH DE number 5186272
scientific article