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 ∞.
We shall prove this for all connected locally compact groups.
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. Math., 72:1 (2008), 169–205
\Bibitem{Sht08}
\by A.~I.~Shtern
\paper A version of van der Waerden's theorem and a proof of Mishchenko's
conjecture on homomorphisms of locally compact groups
\jour Izv. Math.
\yr 2008
\vol 72
\issue 1
\pages 169--205
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Linking options:
https://www.mathnet.ru/eng/im2599
https://doi.org/10.1070/IM2008v072n01ABEH002397
https://www.mathnet.ru/eng/im/v72/i1/p183
This publication is cited in the following 35 articles:
A.I. Shtern, “Extension of Characters from the Radical of a Connected Lie Group to a One-Dimensional Pure Pseudorepresentation of the Group Revisited”, Russ. J. Math. Phys., 31:1 (2024), 146
A. I. Shtern, “Automatic continuity of a locally bounded homomorphism of Lie groups on the commutator subgroup”, Sb. Math., 215:6 (2024), 861–868
A. I. Shtern, “Kriterii prodolzhaemosti unitarnogo kharaktera normalnoi podgruppy gruppy-proizvedeniya do chistogo odnomernogo psevdopredstavleniya gruppy”, Fundament. i prikl. matem., 25:1 (2024), 251–254
A.I. Shtern, “Extension of Unitary Characters from the Radical of a Connected Locally Compact Group to a One-Dimensional Pure Pseudorepresentation of the Group”, Russ. J. Math. Phys., 31:4 (2024), 762
A. I. Shtern, “Continuity Criterion for Locally Bounded Endomorphisms of Connected Reductive Lie Groups”, Russ. J. Math. Phys., 30:1 (2023), 126
A. I. Shtern, “The Discontinuity Group of a Locally Bounded Homomorphism of a Connected Lie Group into a Connected Lie Group Is Commutative”, Russ. J. Math. Phys., 30:3 (2023), 397
A. I. Shtern, “Continuity criteria for locally bounded homomorphisms of certain Lie groups”, J. Math. Sci., 284:4 (2024), 554–556
A. I. Shtern, “Continuity Criterion for Locally Bounded Automorphisms of Central Extensions of Perfect Lie Groups with Discrete Center”, Russ. J. Math. Phys., 29:1 (2022), 119
A. I. Shtern, “Continuity Criterion for Locally Bounded Finite-Dimensional Representations of Simply Connected Solvable Lie Groups”, Russ. J. Math. Phys., 29:2 (2022), 238
Shtern A.I., “Continuity Criterion For Locally Bounded Automorphisms of Connected Reductive Lie Groups”, Russ. J. Math. Phys., 28:3 (2021), 356–357
Shtern I A., “Continuity Criteria For Locally Bounded Automorphisms of Central Extensions of Perfect Lie Groups”, Russ. J. Math. Phys., 28:4 (2021), 543–544
Shtern A.I., “Sufficiently Close One-Dimensional Pseudorepresentations Are Equal”, Russ. J. Math. Phys., 28:2 (2021), 263–264
Shtern A.I., “A New Triviality Theorem For Group Pseudorepresentations”, Russ. J. Math. Phys., 27:4 (2020), 535–536
A. I. Shtern, “Irreducible Locally Bounded Finite-Dimensional Pseudorepresentations of Connected Locally Compact Groups Revisited”, Russ. J. Math. Phys., 27:3 (2020), 382
Shtern A.I., “Irreducible Locally Bounded Finite-Dimensional Pseudorepresentations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:2 (2018), 239–240
A. I. Shtern, “Continuity Conditions for Finite-Dimensional Locally Bounded Representations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:3 (2018), 345
Christopher J. Fewster, “An Analogue of the Coleman–Mandula Theorem for Quantum Field Theory in Curved Spacetimes”, Commun. Math. Phys., 357:1 (2018), 353
A. I. Shtern, “Locally bounded finally precontinuous finite-dimensional quasirepresentations of connected locally compact groups”, Sb. Math., 208:10 (2017), 1557–1576
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 one-dimensional pseudorepresentations of groups”, J. Math. Sci., 233:5 (2018), 770–776
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