Control and separating points of modular functions (Q2702790)

Let \(L\) be a lattice, \(X\) a locally convex linear space and \(\mu \:L\to X\) a modular function, i.e., a function satisfying NEWLINE\[NEWLINE \mu (x\vee y)+\mu (x\wedge y)=\mu (x)+\mu (y) . NEWLINE\]NEWLINE Let \(u(\mu)\) denote the \(\mu \)-uniformity, i.e., the weakest uniformity on \(L\) which makes the lattice operations and \(\mu \) uniformly continuous. In the first part of the paper the authors study the problems, (i) when for a modular function \(\mu \:L\to X\) there is a real valued modular function \(\nu \) on \(L\) with \(u(\nu)=u(\mu)\) and (ii) when for a set \(M\) of \(X\)-valued modular functions there is a modular function \(\nu \: L\to X\) with \(u(\nu)\)=sup\(\{u(\mu)\:\mu \in M\}\). Such a function \(\nu \) is called a control for \(\mu \) or for \(M\), respectively. The authors present several results concerning controls on complemented lattices generalizing known results about control measures on Boolean algebras. In the second part of the paper the authors study the problem when a sequence \(\mu_n\) of group-valued modular functions on a complemented lattice \(L\) has a separating point, i.e., a point \(a\in L\) such that \(\mu_n(a)\neq \mu_m(a)\) for \(n\neq m\). The presented results were obtained by \textit{A. Basile} and \textit{H. Weber} [Rad. Mat. 2, 113-125 (1986; Zbl 0596.28015)] in the case of \(L\) being a Boolean algebra.