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Canonical representation of convolution semigroups on hypergroups. (English) Zbl 0853.22003

Heyer, Herbert (ed.) et al., Infinite dimensional harmonic analysis. Transactions of a German-Japanese symposium, October 3-6, 1995, University of Tübingen, Germany. Bamberg: D. u. M. Gräbner, 188-194 (1996).
On commutative hypergroups \(K\) (in the sense of Jewett), with continuous convolution semigroups \((\mu_t)\) of probability measures on \(K\), Gauss, Poisson and two kinds of Lévy measures are introduced and discussed, e.g. \(\lim \mu_t/t =\) Lévy measure \(\lambda\) for \(0 < t \to 0\) on the complement on \(U\) \((U =\) suitable neighborhood of the neutral element \(e \in K)\). In the Lévy-Khintchine representation \(\gamma *e (\lambda )\) the uniqueness of the Gauss measure \(\gamma\) and the symmetric Lévy measure \(\lambda\) is stated. Finally, for second countable \(K\) and continuous convolution semigroups \((\mu_t)\) on \(K\) with symmetric Lévy measure \(\lambda\), \(\mu_t = \gamma_t* e(t \lambda)\) for \(t > 0\) with Gauss convolution semigroup \((\gamma_t)\). Proofs are to appear elsewhere.
For the entire collection see [Zbl 0840.00033].
Reviewer: H.Günzler (Kiel)

MSC:

22B10 Structure of group algebras of LCA groups
20N20 Hypergroups
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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