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

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

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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

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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• http://mi.mathnet.ru/eng/msb1540
• https://doi.org/10.4213/sm1540
• http://mi.mathnet.ru/eng/msb/v197/i3/p155

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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
2. Shtern A.I., “Quasisymmetry. II”, Russ. J. Math. Phys., 14:3 (2007), 332–356
3. A. I. Shtern, “Kazhdan–Milman problem for semisimple compact Lie groups”, Russian Math. Surveys, 62:1 (2007), 113–174
4. A. I. Shtern, “Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture”, J. Math. Sci., 159:5 (2009), 653–751
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
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
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