Automorphisms of locally compact groups

Abstract

It is proved that for arbitrary locally compact groups G the automorphism group Aut (G) is a complete topological group. Several conditions equivalent to closedness of the group Int (G) of inner automorphisms are given, such as G admits no nontrivial central sequences. It is shown that Aut (G) is topologically embedded in the automorphism group Aut^(G) of the group von Neumann algebra. However, closedness of Int &(G) does not imply closedness of Int (G), nor conversely. l Let G be a locally compact group and Aut (G) the group of all its topological automorphisms with the Birkhoff topology. A neighborhood basis of the identity automorphism consists of sets N(C, V) = {ae Aut (G): a(x) e Vx and cr\x) e Vx, all x e C], where C is compact and V is a neighborhood of the identity e oί G. As is well known, Aut (G) is a Hausdorff topological group but not generally locally compact [1; p. 57]. In this article we are mainly concerned with the topological properties of Aut (G) and its subgroup Int (G) of inner automorphisms. We prove that for G arbitrary locally compact Aut (G) is a complete topological group. In particular, if G is also separable Aut (G) is a Polish group. Furthermore, we give two new characterizations of the topology for Aut(G), (1.1 and 1.6). In §2 we turn to the question of when certain subgroups (among them Int ((?)) are closed in Aut (G), and several equivalent conditions are given; for instance, Int (G) is closed iff G admits no nontrivial central sequences (2.2). Applications to more special classes of groups are also given, as well as to the question of unimodularity of Int (G), (2.7). We remark that there is no separability assumption on the groups before 1.11. LEMMA 1.1. The sets