Representations of positive projections. I
This paper develops a general theory of representations of positive projections on unital f-algebras and Dedekind complete vector lattices motivated by the work of \textit{D.\,Maharam} [Trans.\ Am.\ Math.\ Soc.\ 65, 279--330 (1949; Zbl 0036.31401)]. Preliminary information is provided as well as a discussion of order separability and a weaker notion of locally order separable. A measurable space \((X,\Sigma)\) is called a representation space for a Dedekind \(\sigma\)-complete Riesz space \(L\) with weak order unit \(w\) if there is an ideal \(\widehat L\) in the measurable functions and containing \(1\) and a \(\sigma\)-order continuous Riesz homomorphism \(\Phi\) from \(\widehat L\) onto \(L\) where the image of \(1\) is the weak order unit. It is shown that there indeed exists a representation space for \(L \). Now a Maharam-type representation space for a positive projection \(P\) on \(L\) to a Riesz subspace of \(L\) containing \(w\) is considered. In particular, if a product of two measurable spaces \((X,\Sigma)\) and \((Y,\Lambda)\) such that the product \((X \times Y)\) with the natural measurable sets is a representation space for \(L\) (via a map \(\Phi\)) and there is an appropriate scalar measure \(\mu\) on \(Y\) and measurable function \(m\) on the product with \(Rf(x,y)=\int_Y m(x,z)f(x,z)\,d\mu(z)\) representing \(P\) in the sense that \(P(\Phi f)=\Phi(Rf)\), then this product space is called a Maharam-type representation space. For the construction of representation spaces for projections, it is established that one need only consider spaces with strong order units. Since these spaces have the structure of \(f\)-algebras, while complete Riesz subspaces containing the unit are \(f\)-subalgebras, the authors analyze projections in this setting. For an appropriate projection onto an \(f\)-subalgebra \(B\) containing \(1\), if there exists a complete \(f\)-subalgebra \(C\) containing \(B\), then there exists a unique positive projection \(Q\) onto \(C\) so that \(P\) can be written as \(P\) composed with \(Q\). Another result demonstrates how families of representations for restrictions of a projection can be used to create a global representation. This can then then be related to separable \(f\)-algebras. These results are used in Part~II of this study. This article is well written and includes pertinent examples. [Part II has appeared ibid., No.\,4, 569--605 (2005; Zbl 1119.47039), see the following review.]
scientific article; zbMATH DE number 5070764