Recent contribution to the embedding problem for probability measures on a locally compact group
type: Item
instance of: scholarly article
title: Recent contribution to the embedding problem for probability measures on a locally compact group
zbMATH Open document ID: 0614.60007
DOI: 10.1016/0047-259X(86)90097-7
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: Let \(G\) be a locally compact group and \(M^ 1(G)\) the convolution semigroup of its probability measures. In structural respect as well as in probabilistic respect the following two subsets of \(M^ 1(G)\) are of considerable interest: 1. The set \({\mathcal I}(G)\) of all infinitely divisible measures; and 2. the set \({\mathcal E}(G)\) of all continuously embeddable measures. The group \(G\) is said to have the embedding property if \({\mathcal I}(G)={\mathcal E}(G)\). The characterization of these groups is a main problem in probability theory on groups. Various substantial contributions have been made, notably during the years 1965 to 1975. The state of achievement up to 1977 has been described in detail in Chapter 3 of the author's monograph ''Probability measures on locally compact groups.'' (1977; Zbl 0376.60002). Some years later some important contributions due to M. McCrudden have appeared. The paper under review recalls some basic facts on the embedding problem and gives a survey of recent contributions ''in order to stimulate further interest in the theory''. We formulate some typical results: 1. If \(G\) is a simply-connected solvable Lie group or if \(G=SL(2,{\mathbb{R}})\) or if \(G=SL(2,{\mathbb{C}})\) then every homomorphism \(\phi\) of a real directed semigroup \(S\) into \(M^ 1(G)\) is locally tight (3.5, 3.6). 2. If \(G\) is a connected Lie group and if \(\mu\in {\mathcal I}(G)\) is adapted then \(\mu\in {\mathcal E}(G)\) (restricted embedding property 5.1). 3. Let \(G\) be a noncompact Lie group admitting a maximal compact subgroup \(K\), and let \(M^ 1(K,G,K)=\{\mu \in M^ 1(G): \mu\) is \(K\)-bi- invariant\(\}\). If \(G\) is the semidirect product of a vector group with \(K\) or if \(G\) is connected, semisimple, with finite center, then every \(\mu\in M^ 1(K,G,K)\) that is infinitely divisible in \(M^ 1(K,G,K)\) also is continuously embeddable in \(M^ 1(K,G,K)\) (Section 6). This yields applications to certain Gelfand pairs. Reviewer's remark…
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 3994625
zbMATH Keywords: locally tight homomorphism
zbMATH Keywords: locally compact group
zbMATH Keywords: infinitely divisible measures
zbMATH Keywords: embedding problem
author: Herbert Heyer
reviewed by: Eberhard Siebert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3994625