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Continuity conditions for finite-dimensional locally bounded representations of connected locally compact groups. (English) Zbl 1401.22012
Let \(G\) be a connected locally compact group, let \(\mathcal N\) be the family of all compact normal subgroups \(N\subset G\) such that \(G/N\) is a Lie group, and let \(\pi\) be a locally bounded finite-dimensional representation of \(G\) in a normed space \(E\). Set \(FDG(\pi)=\cap_{N\in\mathcal N}\overline{\pi(N)}\).
It is shown that if there is an \(N\in\mathcal N\) for which \(\pi(N)\) is abelian, then \(\pi\) is continuous on the commutator subgroup \(G'\) of \(G\) in the intrinsic Lie topology of this commutator subgroup, and that if \(\pi(N)\) is non-abelian for every \(N\in\mathcal N\) then the restriction of \(\pi\) to \(G'\) is discontinuous.
It is also shown that if \(FDG(\pi|_{G'})\) is contained in a ball of radius less than \(\sqrt{3}\) centered at the identity operator \(\text{id}_E\) then \(FDG(\pi|_{G'})=\{\text{id}_E\}\), and \(\pi|_{G'}\) is continuous.

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22D12 Other representations of locally compact groups
22B05 General properties and structure of LCA groups
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