On the continuity of convolution semigroups (Q1122668)

Let T be a Hausdorff topological space and \(M_+(T)\) the space of bounded Radon measures on T, with the weak topology. Suppose that \(M_+(T)\) has a bilinear associative operation which is weakly continuous and which preserves probability measures. A convolution semigroup is a homomorphism H: \(s\to \mu_ s\) of the additive half-line [0,\(\infty)\) into \(M_+(T)\). The paper gives two sets of conditions for the map H to be continuous. Each requires H to be continuous at 0 (which means of course, a one-sided continuity). In the first proposition, it is shown that local compactness of T is then a sufficient condition. In the second result, T is taken to be a topological semigroup with identity e, and it is assumed that H(0) is the point mass at e; then the following condition is sufficient: if \((x_ i)\), \((y_ i)\) are nets with \(x_ i\to e\) and \(x_ iy_ i\to z\), then \(y_ i\to z\). Examples are given to illustrate the scope of this condition.