The Stone--Weierstrass theorem and spaces of measures (Q1810207)

The classical Stone-Weierstrass theorem says: if \(X\) is a compact topological space and a subalgebra in the algebra \(C(X)\) of all real-valued continuous functions on \(X\) contains constants and separates points of \(X\), then this subalgebra is dense in \(C(X)\) with respect to the topology of uniform convergence on \(X\). Here the author considers the lattice form of this theorem, that is the vector lattice \(C_b(X)\), where \(X\) is a completely regular space, is considered with locally convex topologies \(t\) for which \((C_b(X),t)'\) is a space of \(\tau\)-additive functionals. For some classes of \(X\), the author proves analogs of the Stone-Weierstrass theorem.