On the ideal centre of the space of vector valued integrable functions
Let \((\Omega,\Sigma,\mu)\) be a finite measure space, \(E\) a Banach space and let \(L^1(\mu,E)\) be the space of \(E\)-valued Bochner integrable functions \(f:\Omega\to E\). If \(E\) is a Riesz space, denote by \(L_b(E)\) the space of all order bounded operators on \(E\) and let \(Z(E):=\{\pi\in L_b(E): |\pi|\leq\lambda I\), for some \(\lambda\in\mathbb R^+\), \(\pi\) is an orthomorphism on \(E\}\) (\(I\) is the identity mapping on \(E\)) be the ideal centre of \(E\). Denote also \({\mathcal L}_s^\infty(\mu,Z(E)):=\{h:h\) is essentially order bounded from \(\Omega\) to \(Z(E)\) and the function \(t\rightarrow h(t)x\;\) is\ measurable\ for\ each \(x\in E\}\) and\ let \(L_{s}^{\infty}(\mu,Z(E))\)\ be the set of all equivalence classes with respect to the almost everywhere equality relation in \({\mathcal L}_s^\infty(\mu,Z(E))\). The authors present some order properties of \(L^1(\mu,E)\) and show that \(L_s^\infty(\mu,Z(E))\) is the ideal centre of \(L^{1}(\mu,E))\). If \(E\) is a Banach lattice with order continuous norm one obtains a Radon--Nikodým type theorem for \(B\)-integrable functions.
scientific article; zbMATH DE number 5577226