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Mat. Sb., 2006, Volume 197, Number 3, Pages 155–176 (Mi msb1540)  

This article is cited in 6 scientific papers (total in 6 papers)

Weak and strong continuity of representations of topologically pseudocomplete groups in locally convex spaces

A. I. Shtern

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Weak and strong continuity conditions for representations of topological groups in locally convex spaces are considered. In particular, weak continuity conditions of reducible locally equicontinuous representations of a topological group in a locally convex space that define weakly continuous representations in an invariant subspace and in the quotient space by this invariant subspace are investigated. These conditions help one to prove the weak continuity of averages and approximations related to weakly continuous locally equicontinuous quasirepresentations of amenable topological groups. Strong continuity conditions for a representation approximating a quasirepresentation of this kind are related to conditions of automatic strong continuity of weakly continuous representations, and fail to hold for some groups, spaces, and representations. In this connection, strong continuity conditions for weakly continuous representations in quasicomplete barrelled locally convex spaces are indicated for a broad class of topologically pseudocomplete groups (which includes the Čech complete groups and the locally pseudocompact groups). Several examples are discussed, in particular, ones relating to the construction of $\Sigma$-products with distinguished subgroups.
Bibliography: 50 titles.

DOI: https://doi.org/10.4213/sm1540

Full text: PDF file (562 kB)
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English version:
Sbornik: Mathematics, 2006, 197:3, 453–473

Bibliographic databases:

UDC: 512.546+517.987
MSC: Primary 22A25; Secondary 46A03
Received: 25.02.2005

Citation: A. I. Shtern, “Weak and strong continuity of representations of topologically pseudocomplete groups in locally convex spaces”, Mat. Sb., 197:3 (2006), 155–176; Sb. Math., 197:3 (2006), 453–473

Citation in format AMSBIB
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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., “Van der Waerden continuity theorem for semisimple Lie groups”, Russ. J. Math. Phys., 13:2 (2006), 210–223  crossref  mathscinet  zmath  isi  elib
    2. Shtern A.I., “Quasisymmetry. II”, Russ. J. Math. Phys., 14:3 (2007), 332–356  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. 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
    4. 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
    5. 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
    6. 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
  • Математический сборник Sbornik: Mathematics (from 1967)
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