On the continuity of convolution semigroups
type: Item
instance of: scholarly article
title: On the continuity of convolution semigroups
zbMATH Open document ID: 0676.22002
DOI: 10.1007/BF02573308
author: Torben Maack Bisgaard
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134997
review text: 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.
Mathematics Subject Classification ID: 22A20
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 4107117
zbMATH Keywords: one parameter semigroup
zbMATH Keywords: measure semigroup
zbMATH Keywords: bounded Radon measures
zbMATH Keywords: convolution semigroup
reviewed by: J. Pym
MaRDI profile type: Publication
cites work: Q3755956
cites work: Q4153376
cites work: Spaces with an abstract convolution of measures
cites work: Idempotent measures on semigroups
description: scientific article; zbMATH DE number 4107117