Single elements in some reflexive algebra modules (Q2470793)

Let \(R\) be a ring and \(M\) be an \(R\)-bimodule. An element \(s\in M\) is called single if \(asb=0\) (\(a, b\in R\)) implies that \(as=0\) or \(sb=0\). Suppose that \({\mathcal L}\) is a completely distributive subspace lattice on a real or complex Banach space \({\mathcal X}\) and \(\varphi\) is an order homomorphism from \({\mathcal L}\) into itself, while \(\text{Alg}\,\mathcal L\) denotes the set of operators on \({\mathcal X}\) leaving every member of \({\mathcal L}\) invariant. Then the set \[ U_\varphi=\{X\in {\mathcal B}{\mathcal X}\mid XL\subseteq \varphi(L) \;\forall L\in {\mathcal L}\} \] is an \(\text{Alg}\,\mathcal L\)-module. In the spirit of \textit{W. E. Longstaff} and \textit{O. Panaia} [Proc.\ Am.\ Math.\ Soc.\ 125, No.~10, 2875--2882 (1997; Zbl 0883.47024)], the author studies single elements in the \(\text{Alg}\,\mathcal L\)-module \(U_\varphi\). He also discusses some conditions under which \(U_\varphi\) has single elements of rank \(n\) (or \(+\infty\)).