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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.
UDC:
513.83
MSC: 54C25,
54C10,
18A20,
46A03,
46M15,
57S10 Received: 19.03.1976
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