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On certain representation of topological groups. (English) Zbl 1006.22004

Given a linear representation \(\rho\) of a topological group \(G\) in a separable topological vector space \(E\) (over the field of complex numbers), let \(E^{G}\) denote the subspace of \(\rho (G)\)-invariant elements in \(E\). A continuous functional \(f\) on \(E\) is said to be \(\rho(G)\)-invariant if \(f\circ \rho (g)=f\circ \rho (g)^{-1}=f\) for all \(g\in G\). In the paper under review the author introduces some class of groups representations of which have either invariant vectors or invariant functionals and studies the relationship between them. The author calls a topological group a \(t\gamma\)-group if every \(\rho(G)\)-invariant subspace in \(E\) has a \(\rho(G)\)-invariant complementary subspace. A topological group \(G\) is said to be a \(t\alpha\)-group (respectively, a \(t\beta\)-group) if for each \(x\in E^{G}\setminus \{0\}\), there exists a continuous \(\rho (G)\)-invariant functional \(f\) on \(E\) such that \(f(x)\not= 0\) (respectively, if for each nontrivial continuous \(\rho (G)\)-invariant functional \(f\) on \(E\), there exists \(x\in E^{G}\) such that \(f(x)\not= 0\)). The following results are obtained: (1) a locally compact group is a \(t\beta\)-group if and only if it is compact; (2) every \(t\gamma\)-group is a \(t\alpha\)-group; (3) a topological group is a \(t\beta\)-group if and only if it is a \(t\gamma\)-group; and (4) if we replace in the definition of \(t\delta\)-group, \(\delta =\alpha , \beta , \gamma\), linear representation by finite-dimensional continuous representation, then the three classes of groups coincide.

MSC:

22A25 Representations of general topological groups and semigroups
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