On binary-type approximations in vector lattices
Let \(X\) be a vector lattice. For \(x\in X\) we set \(\{x\}^ d= \{x\in X: | x|\land | z|= 0\}\) and \({\mathcal S}_ x= \{P(x)\): \(P\) is a band projection on \(X\}\), and we denote by \(I_ x\) the ideal of \(X\) generated by \(x\). Given \(x\in X_ +\) we consider the following two conditions related to Freudenthal's spectral theorem: (A) For every \(y\in X_ +\) with \(y\leq x\) there exist \(x_ i'\), \(x_ i^{\prime\prime}\in {\mathcal S}_ x\), \(i= 1,2,\dots\), with \(0\leq{y- \sum^ k_{i=1} {1\over 2^{i+1}} (x_ i'+ x_ i^{\prime\prime})}\leq {1\over 2^ k} x\), \(k= 1,2,\dots\). (B) For every \(y\in X_ +\) with \(y\leq x\) there exist \(x_ i\in {\mathcal S}_ x\), \(i= 1,2,\dots\), with \(0\leq y- \sum^ k_{i=1} {1\over 2^ i} x_ i\leq {1\over 2^ k} x\), \(k=1,2,\dots\). Condition (A) was established for \(X= C(\Omega)\) with \(\Omega\) compact and zero-dimensional and \(x\) strictly positive by \textit{D. Oates} [Bull. Lond. Math. Soc. 22, No. 5, 467-468 (1990; Zbl 0745.46031)]. Condition (B) appears, in a slightly weaker form, in the author [Arch. Math. 58, No. 3, 288-293 (1992; Zbl 0756.28003)]. We present five conditions equivalent to (A), including the following one: (A)\('\) For all \(y_ 1\), \(y_ 2\in I_ X\) with \(I_ x\subset \{y_ 1\}^ d+ \{y_ 2\}^ d\) there exists a band projection \(P\) of \(X\) with \(P(y_ 1)= y_ 1\) and \(P(y_ 2)= 0\). We also show that (B) is equivalent to (B)\('\) For all \(y_ 1\), \(y_ 2\in I_ x\) with \(y_ 1\land y_ 2= 0\) there exists a band projection \(P\) of \(X\) with \(P(y_ 1)= y_ 1\) and \(P(y_ 2)= 0\). The global version of condition (B)\('\) is due to \textit{A. I. Veksler} [Math. Nachr. 74, 7-25 (1976; Zbl 0287.46007)]. It follows from our results and a theorem of \textit{B. Lavrič} [Indagationes Math. 48, 411- 421 (1986; Zbl 0619.46005)] that conditions (A), (B), (A)\('\) and (B)\('\) are globally mutually equivalent. Moreover, in the case where \(X= C(\Omega)\) with \(\Omega\) completely regular they amount to \(\Omega\) being a strongly zero-dimensional \(F\)-space.
scientific article; zbMATH DE number 687073
scientific article