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The Lévy continuity theorem for nuclear groups. (English) Zbl 0940.43001

The Lévy continuity theorem belongs to that class of theorems of which Bochner’s theorem is another distinguished member which relates positive definite functions on a locally compact Abelian group (LCA) \(G\) with finite positive Radon measures on its dual group \(\Gamma\). It may be stated as follows: a sequence of Radon probability measures on \(G\) is weakly convergent to a Radon probability measure \(\mu\) if and only if the corresponding sequence of Fourier transforms is pointwise convergent to the Fourier transform of \(\mu\). Nuclear groups were introduced by W. Banaszczyk [Lect. Notes Math. 1466, Berlin etc. (1991; Zbl 0743.46002)] after observing that several of the classical theorems in the analysis of LCA groups have their counterpart in nuclear locally convex spaces or locally convex spaces on p-adic fields. They constitute a class of topological Abelian groups with good permanence properties (it is closed under the formation of subgroups, separated quotients, and direct products) which contains all the groups mentioned above and could serve as a common framework for analyzing the properties shared by them. After introducing this new class of topological groups, Banaszczyk started the program of extending those classical results on LCA groups which also hold for nuclear spaces to the context of nuclear groups [see also W. Banaszczyk and E. Martín-Peinador, J. Pure Appl. Algebra 138, 99-106 (1999; Zbl 0935.22004), Papers on general topology and applications Amsterdam, 34-39 (1994), Ann. New York Acad. Sci., 788, New York Acad. Sci., New York (1996; Zbl 0935.22003) and W. Banaszczyk, Studia Math. 105, 271-282 (1993; Zbl 0815.43004)]. Since the goal of this paper is to provide a proof of the Lévy continuity theorem for nuclear groups, and since P. Boulicaut [Z. Wahrscheinlichkeitstheor. Verw. Geb. 28, 43-52 (1973; Zbl 0276.60003)] proved that the Lévy continuity theorem also holds for nuclear locally convex spaces instead of LCA groups, this paper may be regarded as belonging to that program. The proof for nuclear groups given in this paper is based on Boulicaut’s ideas and follows a structural pattern. The main difficulties are faced and solved for quotients and subgroups of Hilbert spaces. Then, after carefully checking the behaviour with respect to projective limits of the property under study, it suffices to apply a representation theorem [Houston J. Math. 26, 314-334 (2000); per bibliography] which describes nuclear groups as subgroups of projective limits of discrete groups and quotients of Hilbert spaces.

MSC:

43A05 Measures on groups and semigroups, etc.
60B10 Convergence of probability measures
22A05 Structure of general topological groups
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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