Embeddability of infinitely divisible distributions on linear Lie groups
type: Item
instance of: scholarly article
title: Embeddability of infinitely divisible distributions on linear Lie groups
zbMATH Open document ID: 0771.60007
DOI: 10.1007/BF01231332
author: M. McCrudden
author: S. G. Dani
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144050
review text: A probability measure \(\mu\) on a locally compact group \(G\) is said to be infinitely divisible if it admits an \(n\)-th root (in the sense of convolution) of any order \(n\geq 1\), and it is called embeddable if there exists a continuous convolution semigroup \((\mu_ t)_{t>0}\) of probability measures on \(G\) such that \(\mu=\mu_ 1\). Clearly, on any locally compact group \(G\) every embeddable measure is infinitely divisible. The inverse statement gives rise to the question for which locally compact groups \(G\) the embedding property (EP) holds. There is a long standing conjecture that all connected Lie groups admit (EP). The authors achieve the remarkable result that every connected Lie group \(G\) having a representation \(\rho:G\to GL(d,\mathbb{R})\) for some \(d\geq 1\) such that \(\ker\rho\) is discrete admits (EP). The deep and elaborate proof of this embedding theorem is reduced to the case of almost algebraic groups which by definition are subgroups of finite index of real algebraic groups (Theorem 5.4). Then by considering coverings of such groups it is shown that any semisimple connected Lie group admits (EP) (Corollary 6.3). Finally, proceeding to coverings of linear Lie groups the authors establish (EP) for all simply connected Lie groups (Corollary of 7).
reviewed by: Herbert Heyer
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 167394
zbMATH Keywords: infinitely divisible measures
zbMATH Keywords: convolution semigroup
zbMATH Keywords: locally compact groups
zbMATH Keywords: semisimple connected Lie group
Wikidata QID: Q115393816
MaRDI profile type: Publication
cites work: Zur Charakterisierung sukzessiv unendlich teilbarer Wahrscheinlichkeitsverteilungen auf lokalkompakten Gruppen
cites work: Q5580321
cites work: Groupes reductifs
cites work: Q5842420
cites work: Factors, roots and embeddability of measures on Lie groups
cites work: On the factor sets of measures and local tightness of convolution semigroups over Lie groups
cites work: Q3897665
cites work: Q4153376
cites work: Recent contribution to the embedding problem for probability measures on a locally compact group
cites work: Q5339349
cites work: Q3717277
cites work: Semi-Groups of Measures on Lie Groups
cites work: Fully Reducible Subgroups of Algebraic Groups
cites work: Infinitely divisible probabilities on SL(2,) are continuously embedded
cites work: Factors and roots of large measures on connected Lie groups
cites work: Limit theorems for probability measures on simply connected nilpotent Lie groups
cites work: Some rationality questions on algebraic groups
cites work: Infinitely divisible measures on p-adic groups
cites work: Q4146925
cites work: Q5682517
description: scientific article; zbMATH DE number 167394