Continuity Criteria for Locally Bounded Automorphisms of Central Extensions of Perfect Lie Groups

We prove that every locally bounded automorphism of a linear connected Lie central extension of a connected perfect Lie group is continuous if and only if it is continuous on the center. We also prove that, if \(Z\) is a connected Abelian group without nontrivial compact subgroups, \(H\) is a connected perfect Lie group and the short sequence of Lie groups \(\{e\}\to Z\to G\to H\to\{e\}\) is exact, then every locally bounded automorphism of \(G\) is continuous if and only if it is continuous on the center of \(G\).