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 Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 1, Pages 183–224 (Mi izv2599)

A version of van der Waerden's theorem and a proof of Mishchenko's conjecture on homomorphisms of locally compact groups

A. I. Shtern

M. V. Lomonosov Moscow State University

Abstract: van der Waerden proved in 1933 that every finite-dimensional locally bounded representation of a semisimple compact Lie group is automatically continuous. This theorem evoked an extensive literature, which related the assertion of the theorem (and its converse) to properties of Bohr compactifications of topological groups and led to the introduction and study of classes of so-called van der Waerden groups and algebras. In the present paper we study properties of (not necessarily continuous) locally relatively compact homomorphisms of topological groups (in particular, connected locally compact groups) from the point of view of this theorem and obtain a classification of homomorphisms of this kind from the point of view of their continuity or discontinuity properties (this classification is especially simple in the case of Lie groups because it turns out that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup). Our main results are obtained by studying new objects, namely, the discontinuity group and the final discontinuity group of a locally bounded homomorphism, and the new notion of a finally continuous homomorphism from one locally compact group into another.
The notion of local relative compactness of a homomorphism is naturally related to the notion of point oscillation (at the identity element of the group) introduced by the author in 2002. According to a conjecture of A. S. Mishchenko, the (reasonably defined) oscillation at a point of any finite-dimensional representation of a ‘good’ topological group can take one of only three values: $0$, $2$ and $\infty$. We shall prove this for all connected locally compact groups.

DOI: https://doi.org/10.4213/im2599

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English version:
Izvestiya: Mathematics, 2008, 72:1, 169–205

Bibliographic databases:

UDC: 512.546+517.987
MSC: Primary 22D12; Secondary 22E30, 22E45

Citation: A. I. Shtern, “A version of van der Waerden's theorem and a proof of Mishchenko's conjecture on homomorphisms of locally compact groups”, Izv. RAN. Ser. Mat., 72:1 (2008), 183–224; Izv. Math., 72:1 (2008), 169–205

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv/v72/i1/p183

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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. Shtern A.I., “Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations”, Russ. J. Math. Phys., 16:4 (2009), 566–567
2. Shtern A.I., “Structure of finite-dimensional locally bounded finally precontinuous quasirepresentations of locally compact groups”, Russ. J. Math. Phys., 16:1 (2009), 133–138
3. A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Proc. Steklov Inst. Math., 271 (2010), 212–227
4. Shtern A.I., “Almost periodic functions on connected locally compact groups”, Russ. J. Math. Phys., 17:4 (2010), 509–510
5. Shtern A.I., “Von Neumann kernels of connected Lie groups, revisited and refined”, Russ. J. Math. Phys., 17:2 (2010), 262–266
6. A. I. Shtern, “The structure of homomorphisms of connected locally compact groups into compact groups”, Izv. Math., 75:6 (2011), 1279–1304
7. Shtern A.I., “Hochschild kernel for locally bounded finite-dimensional representations of a connected Lie group”, Appl. Math. Comput., 218:3 (2011), 1063–1066
8. Shtern A.I., “Alternative proof of the Hochschild triviality theorem for a connected locally compact group”, Russ. J. Math. Phys., 18:1 (2011), 102–106
9. 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
10. Shtern A.I., “Continuity conditions for finite-dimensional representations of connected locally compact groups”, Russ. J. Math. Phys., 19:4 (2012), 499–501
11. A. I. Shtern, “The structure of locally bounded finite-dimensional representations of connected locally compact groups”, Sb. Math., 205:4 (2014), 600–611
12. Shtern A.I., “Corrected Automatic Continuity Conditions for Finite-Dimensional Representations of Connected Lie Groups”, Russ. J. Math. Phys., 21:1 (2014), 133–134
13. A. I. Shtern, “A difference property for functions with bounded second differences on amenable topological groups”, J. Math. Sci., 213:2 (2016), 281–286
14. Shtern A.I., “a Freudenthal-Weil Theorem For Pro-Lie Groups”, Russ. J. Math. Phys., 22:4 (2015), 546–549
15. Shtern A.I., “Description of locally bounded pseudocharacters on almost connected locally compact groups”, Russ. J. Math. Phys., 23:4 (2016), 551–552
16. A. I. Shtern, “Specific properties of one-dimensional pseudorepresentations of groups”, J. Math. Sci., 233:5 (2018), 770–776
17. A. I. Shtern, “Locally bounded finally precontinuous finite-dimensional quasirepresentations of connected locally compact groups”, Sb. Math., 208:10 (2017), 1557–1576
18. Shtern A.I., “Irreducible Locally Bounded Finite-Dimensional Pseudorepresentations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:2 (2018), 239–240
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