Continuity conditions for finite-dimensional locally bounded representations of connected locally compact groups
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.
continuous representations on normed spaces
scientific article; zbMATH DE number 6964581
scientific article