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 Mat. Sb., 2002, Volume 193, Number 9, Pages 139–156 (Mi msb682)

Criteria for weak and strong continuity of representations of topological groups in Banach spaces

A. I. Shtern

M. V. Lomonosov Moscow State University

Abstract: Several necessary and sufficient conditions for weak and strong continuity of representations of topological groups in Banach spaces are obtained. In particular, it is shown that a representation $S$ of a locally compact group $G$ in a Banach space is continuous in the strong (or, equivalently, in the weak) operator topology if and only if for some real number $q$, $0\leqslant q<1$, and each unit vector $\xi$ in the representation space of $S$ there exists a neighbourhood $U=U(\xi)\subset G$ of the identity element $e\in G$ such that $\|S(g)\xi-\xi\|\leqslant q$ for all $g\in U$. Versions of this criterion for other classes of groups (including not necessarily locally compact groups) and refinements for finite-dimensional representations are obtained; examples are discussed. Applications to the theory of quasirepresentations of topological groups are presented.

DOI: https://doi.org/10.4213/sm682

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English version:
Sbornik: Mathematics, 2002, 193:9, 1381–1396

Bibliographic databases:

UDC: 512.546+517.987
MSC: Primary 22A25; Secondary 22D12, 47D03

Citation: A. I. Shtern, “Criteria for weak and strong continuity of representations of topological groups in Banach spaces”, Mat. Sb., 193:9 (2002), 139–156; Sb. Math., 193:9 (2002), 1381–1396

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb682
• https://doi.org/10.4213/sm682
• http://mi.mathnet.ru/eng/msb/v193/i9/p139

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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., “Representations of topological groups in locally convex spaces: Continuity properties and weak almost periodicity”, Russ. J. Math. Phys., 11:1 (2004), 81–108
2. A. I. Shtern, “Criteria for the continuity of finite-dimensional representations of connected locally compact groups”, Sb. Math., 195:9 (2004), 1377–1391
3. A. I. Shtern, “Continuity Criterion for Finite-Dimensional Representations of Locally Compact Groups”, Math. Notes, 75:6 (2004), 890–892
4. A. I. Shtern, “Almost periodic functions and representations in locally convex spaces”, Russian Math. Surveys, 60:3 (2005), 489–557
5. A. I. Shtern, “Topological groups with finite von Neumann algebras of type I”, Sb. Math., 196:3 (2005), 447–463
6. A. I. Shtern, “Automatic continuity of pseudocharacters on semisimple Lie groups”, Math. Notes, 80:3 (2006), 435–441
7. A. I. Shtern, “Weak and strong continuity of representations of topologically pseudocomplete groups in locally convex spaces”, Sb. Math., 197:3 (2006), 453–473
8. Shtern A.I., “Van der Waerden continuity theorem for semisimple Lie groups”, Russ. J. Math. Phys., 13:2 (2006), 210–223
9. Shtern A.I., “Continuity conditions for finite-dimensional representations of some locally bounded groups”, Russ. J. Math. Phys., 13:4 (2006), 438–457
10. A. I. Shtern, “Kazhdan–Milman problem for semisimple compact Lie groups”, Russian Math. Surveys, 62:1 (2007), 113–174
11. A. I. Shtern, “Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture”, J. Math. Sci., 159:5 (2009), 653–751
12. Shtern A.I., “Quasisymmetry. II”, Russ. J. Math. Phys., 14:3 (2007), 332–356
13. Shtern A.I., “Stability of the van der Waerden theorem on the continuity of homomorphisms of compact semisimple Lie groups”, Appl. Math. Comput., 187:1 (2007), 455–465
14. 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
15. A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Proc. Steklov Inst. Math., 271 (2010), 212–227
16. Shtern I A., “Continuity Conditions For Finite-Dimensional Locally Bounded Representations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:3 (2018), 345–382
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