Uniform separation of points and measures and representation by sums of algebras (Q1093133)

Let X and \(Y_ i\), \(1\leq i\leq k\), be compact metric spaces, and let \(\rho_ i: X\to Y_ i\) be continuous functions. The family \(F=\{\rho_ i\}^ k_{i=1}\) is said to be a measure separating family, if there exists some \(\lambda >0\) such that for every measure \(\mu\) in \(C(X)^*\), \(\| \mu \circ \phi_ i^{-1}\| \geq \lambda \| \mu \|\) holds for some \(1\leq i\leq k\). F is a uniformly (point) separating family if the above holds for the purely atomic measures in \(C(X)^*\). It is known that for \(k\leq 2\) the two concepts are equivalent. In this note we present examples which show that for \(k\geq 3\) measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: let \(A_ 1,A_ 2,...,A_ k\) be closed subalgebras of C(X); when is \(A_ 1+A_ 2+...+A_ k\) equal to or dense in C(X)?