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 Uspekhi Mat. Nauk, 1976, Volume 31, Issue 5(191), Pages 137–147 (Mi umn3961)

On equivariant embeddings of $G$-spaces

Yu. M. Smirnov

Abstract: We study a functorial dependence $\tilde{\alpha}$ between maps $h\colon X\to Y$, where $X$ is a $G$-space with continuous action $\alpha$ of the group $G$, and maps $\tilde{\alpha}(h)\colon X\to Y^X$, where $Y^X$ is taken with the compact open topology. The functor $\tilde{\alpha}$ preserves the properties of being one-to-one, of being continuous, of being a topological embedding and, in the case of a compact group, of being a topological embedding with a closed image. For fixed $X$, $\alpha$, and $Y$, the functor $\tilde{\alpha}$ is a topological embedding of $\mathscr C(X,Y)$ into $\mathscr C(X,\mathscr C(G,Y))$. (The topology is compact-open.) If $Y$ is a topological vector space, then $\tilde{\alpha}$ is a monomorphism. If $G$ is locally compact, then there is a continuous action of $G$ on $\mathscr C(G,Y)$ and $\tilde{\alpha}(h)$ is equivariant for any $h$. If $V$ is a locally convex space, then there exists a continuous monomorphism of $G$ into the group of all topological linear transformations of the locally convex space $\mathscr C(G,V)$. For a locally compact group $G$ every completely regular $G$-space can be embedded in a topologically equivariant way in the locally convex space $\mathscr C(G,V)$ under the natural action of the group of all topological linear transformations. (This result was recently obtained by de Vries by means of a different construction.) If $G$ is compact, then the embedding can be made to have a closed image.

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English version:
Russian Mathematical Surveys, 1976, 31:5, 198–209

Bibliographic databases:

UDC: 513.83
MSC: 54C25, 54C10, 18A20, 46A03, 46M15, 57S10

Citation: Yu. M. Smirnov, “On equivariant embeddings of $G$-spaces”, Uspekhi Mat. Nauk, 31:5(191) (1976), 137–147; Russian Math. Surveys, 31:5 (1976), 198–209

Citation in format AMSBIB
\Bibitem{Smi76} \by Yu.~M.~Smirnov \paper On equivariant embeddings of $G$-spaces \jour Uspekhi Mat. Nauk \yr 1976 \vol 31 \issue 5(191) \pages 137--147 \mathnet{http://mi.mathnet.ru/umn3961} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=440521} \zmath{https://zbmath.org/?q=an:0362.57021} \transl \jour Russian Math. Surveys \yr 1976 \vol 31 \issue 5 \pages 198--209 \crossref{https://doi.org/10.1070/RM1976v031n05ABEH004197} 

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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. Yu. M. Smirnov, “Shape theory and continuous transformations groups”, Russian Math. Surveys, 34:6 (1979), 138–143
2. M. G. Megrelishvili, “A factorization theorem and universal compactifications for $G$-spaces”, Russian Math. Surveys, 38:6 (1983), 125–126
3. Yu. M. Smirnov, “Shape theory for $G$-pairs”, Russian Math. Surveys, 40:2 (1985), 185–203
4. A. Yu. Lemin, Yu. M. Smirnov, “Isometry groups of metric and ultrametric spaces and their subgroups”, Russian Math. Surveys, 41:6 (1986), 213–214
5. P. S. Gevorgyan, “On the $G$-movability of $G$-spaces”, Russian Math. Surveys, 43:3 (1988), 203–204
6. S. V. Vlasov, “Universal bicompact $G$-spaces”, Russian Math. Surveys, 49:6 (1994), 221–222
7. S.A.. Antonyan, Natalia Jonard-Pérez, Saúl Juárez-Ordóñez, “Hyperspaces of Keller compacta and their orbit spaces”, Journal of Mathematical Analysis and Applications, 2013
8. Natalia Jonard-Pérez, “Equivariant absolute extensor property on hyperspaces of convex sets”, Topology and its Applications, 177 (2014), 88
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