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

This article is cited in 8 scientific papers (total in 8 papers)

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
Received: 19.03.1976

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
\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
\jour Russian Math. Surveys
\yr 1976
\vol 31
\issue 5
\pages 198--209

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    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  mathnet  crossref  mathscinet  zmath
    2. M. G. Megrelishvili, “A factorization theorem and universal compactifications for $G$-spaces”, Russian Math. Surveys, 38:6 (1983), 125–126  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. Yu. M. Smirnov, “Shape theory for $G$-pairs”, Russian Math. Surveys, 40:2 (1985), 185–203  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. P. S. Gevorgyan, “On the $G$-movability of $G$-spaces”, Russian Math. Surveys, 43:3 (1988), 203–204  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. S. V. Vlasov, “Universal bicompact $G$-spaces”, Russian Math. Surveys, 49:6 (1994), 221–222  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref
    8. Natalia Jonard-Pérez, “Equivariant absolute extensor property on hyperspaces of convex sets”, Topology and its Applications, 177 (2014), 88  crossref
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