Displaying similar documents to “The dual space of precompact groups”

Homomorphic images of ℝ -factorizable groups

Mihail G. Tkachenko (2006)

Commentationes Mathematicae Universitatis Carolinae

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It is well known that every ℝ -factorizable group is ω -narrow, but not vice versa. One of the main problems regarding ℝ -factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every ω -narrow group is a continuous homomorphic image of an ℝ -factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an ℝ -factorizable...

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

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Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable...

On a variant of Kazhdan's property (T) for subgroups of semisimple groups

Mohammed Bachir Bekka, Nicolas Louvet (1997)

Annales de l'institut Fourier

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Let Γ be an irreducible lattice in a product G of simple groups. Assume that G has a factor with property (T). We give a description of the topology in a neighbourhood of the trivial one dimensional representation of Γ in terms of the topology of the dual space G ^ of G . We use this result to give a new proof for the triviality of the first cohomology group of Γ with coefficients in a finite dimensional unitary representation.

Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, Lewis C. Robertson

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For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly...