The connected component of the group of automorphisms of a~locally compact group

O. V. Mel'nikov

Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 219-229

Résumé

The paper is devoted to the investigation of the group of automorphisms $\operatorname{Aut}G$ of a locally compact group $G$. $\operatorname{Aut}G$ is equipped with a topology which is naturally related to the topology of $G$. The connected component of $\operatorname{Aut}G$ is determined for a group $G$ which can be written as a semidirect product of a vector group and a group possessing an open compact subgroup. For a central group $G$ an explicit representation of $(\operatorname{Aut}G)_0$ is obtained in the form of a product of certain well-defined subgroups of $\operatorname{Aut}G$. The following result is obtained: Theorem. {\it If $G$ is locally compact group whose connected component is compact, then the connected component of $\operatorname{Aut}G$ is also compact.} Bibliography: 11 titles.