On ideals generated by positive operators
Let \(E\) be a Riesz space with separating order dual and \(L_{b}(E)\) be the space of all order bounded operators. Let \(E^{\sim}\) denote the order dual of \(E\), let \(Z(E)\) denote the ideal center of \(E\) and \(\text{Orth}(E)\) denote the space of all orthomorphisms of \(E\). Let \(A_{T}\) and \(B_{T}\) denote the ideal generated by \(T\) in \(L_{b}(E)\) and the band generated by \(T\), respectively. A Riesz space \(E\) is said to have topologically full center if for each pair \(x,y\) in \(E\) with \(0 \leq y \leq x\), there exists a net \((\pi_{\alpha})\) in \(Z(E)\) with \(0 \leq {\pi}_{\alpha} \leq I\) for each \(\alpha\), such that \({\pi}_{\alpha}x \rightarrow y\) in \(\sigma(E,E^{\sim})\). In the paper under review, the authors study the algebraic structure of \(A_{T},B_{T}\) and their commutants \(\{A_{T}\}^{c}\) and \(\{B_{T}\}^{c}\). The bilinear map \(\phi: E \times E^{\sim} \rightarrow A_{T}^{\sim}\) is defined by \(\phi(x,f)= \mu_{x,f}\), where \(\mu_{x,f}(a)=f(a(x))\) for \(a \in A_{T}\). Let \(E\) have topologically full center and \(0 < T \in L_{b}(E)\) such that \(A_{T}\) has the Riesz Decomposition Property (R.D.P.). Among other things, using the bi-lattice homomorphism of the bilinear mapping \(\phi: E \times E^{\sim} \rightarrow A_{T}^{\sim}\), the authors show that \(\{A_{T} \}^{c} \subseteq Orth(E)\).
scientific article; zbMATH DE number 2002013