Some theorems on $(\textrm {CA})$ analytic groups

Abstract:

An analytic group $G$ is called $(CA)$ if the group of inner automorphisms of $G$ is closed in the Lie group of all (bicontinuous) automorphisms of $G$. We show that each non-$(CA)$ analytic group $G$ can be written as a semidirect product of a $(CA)$ analytic group and a vector group. This decomposition yields a natural dense immersion of $G$ into a $(CA)$ analytic group $H$, such that each automorphism of $G$ can be extended to an automorphism of $H$. This immersion and extension property enables us to derive a sufficient condition for the normal part of a semidirect product decomposition of a $(CA)$ analytic group to be $(CA)$.