Extension of positive projections
Let \(P\) be a positive projection defined on a majorizing subspace \(A\) of a Dedekind complete vector lattice \(E\). For an element \(x_0\notin E\setminus A\) the author finds a necessary and sufficient condition for the existence of a positive extension to \([A,x_0]\) that is also a projection, where \([D]\) denotes the linear space generated by a subset \(D\) of \(E\). If \(x_0\) does not satisfy this condition, then there exists some \(y_0\in E\) such that \(P\) can be extended to a positive projection on \([A,x_0, y_0]\). The fact that \(P\) can be extended to a positive projection on \(E\) was proved earlier by \textit{K. Triki} [J. Math. Anal. Appl. 153, 486-496 (1990; Zbl 0727.47021)]. The proofs offered by the author do not use the axiom of choice.
scientific article; zbMATH DE number 1884312
scientific article