Recent contribution to the embedding problem for probability measures on a locally compact group
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 remarks: 1. There is a recent paper by \textit{S. G. Dani} and \textit{M. McCrudden} on the embedding problem for connected semisimple Lie groups with finite center [Parabolic subgroups and factor compactness of measures on semisimple Lie groups. Probability measures on groups. Proc. Oberwolfach 1985, Lect. Notes Math. 1210, 34-40 (1986)]. 2. There are several misprints: On \(122^ 1\) it should read ''Satz 1''; on \(122^{10}\) it should read ''\(Cx_ iCx_ j\cap Cx_{i+j}\neq \emptyset\)''; on \(126_ 5\) it should read ''\(m/k^ n\)''; \(127^{18}\) should read ''\(G_{\alpha}\not\in {\mathcal R}\)''. In particular, for \(\alpha\) purely imaginary, \(G_{\alpha}\not\in {\mathcal R}_ n\) for all'' in \(127^{22}\) it should read ''for not purely imaginary \(\alpha\)''; \(131^{1,2}\) reference [3] should read: ''J. Dixmier (1957). Quelques propriétés des groupes abéliens localement compacts. Bull. Sci. Math. II. Ser. 81, 38-48''.
scientific article; zbMATH DE number 3994625
scientific article