Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups
type: Item
instance of: scholarly article
title: Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups
zbMATH Open document ID: 0582.22005
DOI: 10.1007/BF02573186
published in: Semigroup Forum
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134797
review text: The following observation in the theory of probability measures on groups was the starting point for our investigations. Let \((\mu_ t)_{t\geq 0}\) be a continuous convolution semigroup of probability measures on a locally compact group G. Then \(\mu_ 0\) is the normed Haar measure \(\omega_ K\) on a compact subgroup K of G. In addition let \(T=(\tau_ t)_{t>0}\) be a continuous automorphism group on G. Then the semigroup \((\mu_ t)_{t\geq 0}\) is said to be T-stable (in the strict sense) if \(\tau_ t(\mu_ 1)=\mu_ t\) for all \(t>0\). Obviously K is T-invariant. Moreover all the measures \(\mu_ t\) are concentrated on the Borel measurable subgroup \(C_ K(T)\) of G where T acts contracting modulo K (cf. Proposition 4.2). Consequently, in order to determine the class of groups G on which there exist T-stable convolution semigroups, it is necessary to study the structure of \(C_ K(T)\). This investigation will be performed in the first part of this paper. Turning matters around we are then going to show that the structure of \(C_ K(T)\) has a strong impact on the structure of the T-stable convolution semigroup \((\mu_ t)_{t\geq 0}\).
Mathematics Subject Classification ID: 22D45
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 3931300
zbMATH Keywords: probability measures on groups
zbMATH Keywords: continuous automorphism group
zbMATH Keywords: stable convolution semigroup
author: Wilfried Hazod
author: Eberhard Siebert
MaRDI profile type: Publication
cites work: Q5639888
cites work: Q3339837
cites work: Q3327409
cites work: Q3897665
cites work: Q5723435
cites work: Q5339349
cites work: Splitting in topological groups
cites work: On some types of topological groups
cites work: Q3229429
cites work: Contractive automorphisms on locally compact groups
cites work: Q5798065
OpenAlex ID: W2070647900
description: scientific article; zbMATH DE number 3931300