
This article is cited in 18 scientific papers (total in 18 papers)
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 finitedimensional 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 socalled 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 finitedimensional 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
finitedimensional 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 Received: 14.12.2006
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
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Citing articles on Google Scholar:
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Related articles on Google Scholar:
Russian articles,
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This publication is cited in the following articles:

Shtern A.I., “Connected Lie groups having faithful locally bounded (not necessarily continuous) finitedimensional representations”, Russ. J. Math. Phys., 16:4 (2009), 566–567

Shtern A.I., “Structure of finitedimensional locally bounded finally precontinuous quasirepresentations of locally compact groups”, Russ. J. Math. Phys., 16:1 (2009), 133–138

A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Proc. Steklov Inst. Math., 271 (2010), 212–227

Shtern A.I., “Almost periodic functions on connected locally compact groups”, Russ. J. Math. Phys., 17:4 (2010), 509–510

Shtern A.I., “Von Neumann kernels of connected Lie groups, revisited and refined”, Russ. J. Math. Phys., 17:2 (2010), 262–266

A. I. Shtern, “The structure of homomorphisms of connected locally compact groups into compact groups”, Izv. Math., 75:6 (2011), 1279–1304

Shtern A.I., “Hochschild kernel for locally bounded finitedimensional representations of a connected Lie group”, Appl. Math. Comput., 218:3 (2011), 1063–1066

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

A. I. Shtern, “Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finitedimensional representations”, Trans. Moscow Math. Soc., 72 (2011), 79–95

Shtern A.I., “Continuity conditions for finitedimensional representations of connected locally compact groups”, Russ. J. Math. Phys., 19:4 (2012), 499–501

A. I. Shtern, “The structure of locally bounded finitedimensional representations of connected locally compact groups”, Sb. Math., 205:4 (2014), 600–611

Shtern A.I., “Corrected Automatic Continuity Conditions for FiniteDimensional Representations of Connected Lie Groups”, Russ. J. Math. Phys., 21:1 (2014), 133–134

A. I. Shtern, “A difference property for functions with bounded second differences on amenable topological groups”, J. Math. Sci., 213:2 (2016), 281–286

Shtern A.I., “a FreudenthalWeil Theorem For ProLie Groups”, Russ. J. Math. Phys., 22:4 (2015), 546–549

Shtern A.I., “Description of locally bounded pseudocharacters on almost connected locally compact groups”, Russ. J. Math. Phys., 23:4 (2016), 551–552

A. I. Shtern, “Specific properties of onedimensional pseudorepresentations of groups”, J. Math. Sci., 233:5 (2018), 770–776

A. I. Shtern, “Locally bounded finally precontinuous finitedimensional quasirepresentations of connected locally compact groups”, Sb. Math., 208:10 (2017), 1557–1576

Shtern A.I., “Irreducible Locally Bounded FiniteDimensional Pseudorepresentations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:2 (2018), 239–240

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