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 Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 6, Pages 195–222 (Mi izv4129)

The structure of homomorphisms of connected locally compact groups into compact groups

A. I. Shternab

a M. V. Lomonosov Moscow State University
b Scientific Research Institute for System Studies of RAS, Moscow

Abstract: We obtain consequences of the theorem concerning the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the commutator subgroup of the group and also of an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of soluble Lie groups. In particular, we prove that an almost connected locally compact group admitting a (not necessarily continuous) injective homomorphism into a compact group also admits a continuous injective homomorphism into a compact group, and thus the given group is a finite extension of the direct product of a compact group and a vector group. We solve the related problem of describing the images of (not necessarily continuous) homomorphisms of connected locally compact groups into compact groups. Moreover, we refine the description of the von Neumann kernel of a connected locally compact group and describe the intersection of the kernels of all (not necessarily continuous) finite-dimensional unitary representations of a given connected locally compact group. Some applications are mentioned. We also show that every almost connected locally compact group admitting a (not necessarily continuous) locally bounded injective homomorphism into an amenable almost connected locally compact group is amenable.

Keywords: locally compact group, almost connected locally compact group, Freudenthal–Weil theorem, $[\mathrm{MAP}]$-group, semisimple locally compact group, locally bounded map.

DOI: https://doi.org/10.4213/im4129

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English version:
Izvestiya: Mathematics, 2011, 75:6, 1279–1304

Bibliographic databases:

UDC: 512.546+517.986.6+512.815.1
MSC: Primary 22D10; Secondary 43A65, 46H40
Revised: 23.06.2010

Citation: A. I. Shtern, “The structure of homomorphisms of connected locally compact groups into compact groups”, Izv. RAN. Ser. Mat., 75:6 (2011), 195–222; Izv. Math., 75:6 (2011), 1279–1304

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv4129
• https://doi.org/10.4213/im4129
• http://mi.mathnet.ru/eng/izv/v75/i6/p195

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. I. Shtern, “Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations”, Trans. Moscow Math. Soc., 72 (2011), 79–95
2. A. I. Shtern, “The structure of locally bounded finite-dimensional representations of connected locally compact groups”, Sb. Math., 205:4 (2014), 600–611
3. Bergelson V., Christopherson J.C., Robertson D., Zorin-Kranich P., “Finite products sets and minimally almost periodic groups”, J. Funct. Anal., 270:6 (2016), 2126–2167
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