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

This article is cited in 16 scientific papers (total in 16 papers)

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.


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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
Received: 28.02.2002

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
\by A.~I.~Shtern
\paper Criteria for weak and strong continuity of representations
of topological groups in Banach spaces
\jour Mat. Sb.
\yr 2002
\vol 193
\issue 9
\pages 139--156
\jour Sb. Math.
\yr 2002
\vol 193
\issue 9
\pages 1381--1396

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    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  mathscinet  zmath  isi  elib
    2. A. I. Shtern, “Criteria for the continuity of finite-dimensional representations of connected locally compact groups”, Sb. Math., 195:9 (2004), 1377–1391  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. I. Shtern, “Continuity Criterion for Finite-Dimensional Representations of Locally Compact Groups”, Math. Notes, 75:6 (2004), 890–892  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. I. Shtern, “Almost periodic functions and representations in locally convex spaces”, Russian Math. Surveys, 60:3 (2005), 489–557  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. A. I. Shtern, “Topological groups with finite von Neumann algebras of type I”, Sb. Math., 196:3 (2005), 447–463  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. A. I. Shtern, “Automatic continuity of pseudocharacters on semisimple Lie groups”, Math. Notes, 80:3 (2006), 435–441  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Shtern A.I., “Van der Waerden continuity theorem for semisimple Lie groups”, Russ. J. Math. Phys., 13:2 (2006), 210–223  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    9. Shtern A.I., “Continuity conditions for finite-dimensional representations of some locally bounded groups”, Russ. J. Math. Phys., 13:4 (2006), 438–457  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. A. I. Shtern, “Kazhdan–Milman problem for semisimple compact Lie groups”, Russian Math. Surveys, 62:1 (2007), 113–174  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. A. I. Shtern, “Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture”, J. Math. Sci., 159:5 (2009), 653–751  mathnet  crossref  mathscinet  zmath  elib  elib
    12. Shtern A.I., “Quasisymmetry. II”, Russ. J. Math. Phys., 14:3 (2007), 332–356  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    15. A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Proc. Steklov Inst. Math., 271 (2010), 212–227  mathnet  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
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