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For any topological group the dual object is defined as the set of equivalence classes of irreducible unitary representations of equipped with the Fell topology. If is compact, is discrete. In an earlier paper we proved that is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when is an almost metrizable precompact group.
Let H be a real Hilbert space. It is well known that a positive-definite function φ on H is the Fourier transform of a Radon measure on the dual space if (and only if) φ is continuous in the Sazonov topology (resp. the Gross topology) on H. Let G be an additive subgroup of H and let (resp. ) be the character group endowed with the topology of uniform convergence on precompact (resp. bounded) subsets of G. It is proved that if a positive-definite function φ on G is continuous in the Gross topology,...
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