Uniform \(\lambda\)-property in \(L^1\cap L^\infty\) (Q2822642)

The authors establish the uniform \(\lambda\)-property for the intersection of the Lebesgue spaces \(L^1 \cap L^{\infty}\) endowed with the standard norm \(\|x\|_{L^1\cap L^{\infty}}=\max\{\|x\|_{L^1}, \|x\|_{L^{\infty}}\}\) and also with the equivalent norm \(\|x\|'_{L^1\cap L^{\infty}}=\|x\|_{L^1} + \|x\|_{L^{\infty}}\).NEWLINENEWLINELet \(X\) be a real Banach space with norm \(\|\cdot \|_X\), let \(B(X)\) denote the closed unit ball in \(X\), and let \(\operatorname{ext}B(X)\) denote the set of extreme points of \(B(X)\). The function \(\lambda :B(X) \rightarrow [0,1]\) is given by the formula \( \lambda (x)=\sup \{ \lambda \in [0,1]: x =\lambda e +(1-\lambda )y, \;e\in \operatorname{ext}B(X),\; y \in B(X)\}\) if \( \operatorname{ext}B(X)\neq \emptyset\), and \(\lambda (x)=0\) if \(\operatorname{ext}B(X)=\emptyset\).NEWLINENEWLINEThis function is called the \(\lambda\)-function. A point \(x \in B(X)\) is said to be a \(\lambda\)-point of \(B(X)\) if \(\lambda (x) >0\) and the space has the \(\lambda\)-property if each \(x \in B(X)\) is a \(\lambda\)-point or \(\lambda(x)>0\). If \(\lambda_X=\operatorname{ext}\{ \lambda(x): \|x\|=1\}>0\), then \(X\) has the uniform \(\lambda\)-property.NEWLINENEWLINELet \((T, \Sigma, \mu)\) be a measure space with a \(\sigma\)-finite non-atomic and complete measure \(\mu\). The three main results in this paper are:NEWLINENEWLINETheorem. The space \((L^1 \cap L^{\infty}, \|\cdot \|_{L^1\cap L^{\infty}})\) has the \(\lambda\)-property.NEWLINENEWLINETheorem. The space \((L^1 \cap L^{\infty}, \|\cdot \|_{L^1\cap L^{\infty}})\) has the uniform \(\lambda\)-property if and only if \(\mu(T)\leq 1\).NEWLINENEWLINETheorem. The space \((L^1 \cap L^{\infty}, \,\|\cdot \|'_{L^1\cap L^{\infty}})\) has the uniform \(\lambda\)-property if and only if \(\mu(T)< \infty.\)