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Mesures de Lévy et fonctions de Lévy générales. Applications à la décomposition de Lévy-Khintchine. (Lévy measures and generalized Lévy functions. Applications to the Lévy-Khinchin decomposition). (French) Zbl 0622.43004

Let S be a locally compact abelian semigroup with involution and let F be a *-subsemigroup of the dual \(S^*\) of semicharacters. In the paper under review semigroups of positive definite functions \(\exp (t\psi)_{t\geq 0}\) on S (called semigroups of moments), resp. semigroups of probabilities \((\mu_ t)_{t\geq 0}\) and representation of Lévy- Khinchin type are studied, covering, e.g., Lévy-Khinchin representation of convolution semigroups on locally compact abelian groups and of moment semigroups.
Chap. I contains definitions, existence of Lévy measures and Lévy maps, following H. Buchwalter [Les fonctions de Lévy existent!, Math. Ann. 274, 31-34 (1986; Zbl 0572.43003)]. In Chap. II Lévy- Khinchin-type decompositions of semigroups of moments and of convolution semigroups are proved. In Chap. III problems of the following type are studied: 1. Given a semigroup of moments on S and a *-morphism \(S\to T\). Is the image a semigroup of moments ? 2. Given a *-semigroup S and a hermitean complex function \(\psi\). Under which conditions is \((\exp (t\psi))_{t\geq 0}^ a \)semigroup of moments ?
For details on *-semigroups and semigroups of moments the reader is referred to C. Berg, J. P. R. Christensen and P. Ressel [Harmonic analysis on semi-groups (1984; Zbl 0619.43001)].
Reviewer: W.Hazod

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
22A20 Analysis on topological semigroups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
43A32 Other transforms and operators of Fourier type
43A05 Measures on groups and semigroups, etc.
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