Representations of positive projections. II
[Part I appeared ibid., No.\,3, 293--325 (2005; Zbl 1119.47038), see the preceding review.] The authors continue the study presented in Part I of this two-part series of papers. The present paper develops the authors' results to facilitate the creation of scalar-valued measures needed for the Maharam-type integral representations (defined in Part I) of positive projections on Dedekind complete Riesz spaces. A vector-valued version of a Radon--Nikodym type theorem due to \textit{D.\,Maharam} [Trans.\ Am.\ Math.\ Soc.\ 69, 142--160 (1950; Zbl 0041.18002 )] is established. It is shown that for a Dedekind complete \(f\)-algebra \(A\) with unit and \(P\) a positive order continuous projection onto a \(f\)-subalgebra \(B\) with unit, that if \(A\) is nowhere full with respect to \(B\) (discussed in Part I), \(e\) is a nonzero component of the unit in \(A\), and \(g\) is a positive element in \(B\) with \(g\leq P(e)\), then there exists a component \(p\) of the unit in \(A\) such that \(P(p)=g\). It follows that for any \(\lambda\) in \([0,1]\) there is a component \(p\) of the unit in \(A\) with \(P(p)=\lambda 1\). Properties of the set all such components and its complete Boolean subalgebras are studied. These lead to the creation of the appropriate measure spaces and measures. Indeed, given the same hypotheses as above with the additional property that \(A\) is order separable and a representation for \(B\), then there exists a representation space for the projection \(P\) and the algebra \(A\) compatible with the representation for \(B\). Consequences for Riesz spaces are presented, including assuming only a locally order separable condition and excluding the assumption of a weak order unit.
scientific article; zbMATH DE number 5070779
scientific article