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Published online by Cambridge University Press: 22 January 2016
Let G be a locally compact connected group, and let A (G) be the group of all continuous automorphisms of G. We shall introduce a natural topology into A(G) as previously (i.e. the topology of uniform convergence in the wider sense.) When the component of the identity of A(G) coincides with the group of inner automorphisms, we shall call G complete. The purpose of this note is to prove the following theorem and give some applications of it.
1) See e.g. Nomizu, K. and Goto, M., “On the group of automorphisms of a topological group,” forthcoming in Tôhoku Math. Journ.CrossRefGoogle Scholar
2) See Goto, M., “Faithful representations of Lie groups I,” Mathematica Japonicae, Vol. 1, No. 3, (1949). Referred to as F.R.Google Scholar
3) For the definitions and the structures of (L)-groups, see Iwasawa, K., “On some types of topological groups,” Ann. of Math., Vol. 50 (1949).CrossRefGoogle Scholar
4) See F. R. Lemma 4.
5) For the definitions etc. of semi-simple (L)-groups, see M. Goto: “Linear representations of topological groups,” forthcoming in Proc. Amer, Math, Soc.
6) See F. R. Theorem 2.
7) A locally compact group G contains the uniquely determined maximal connected solvable invariant subgroup R, which is closed in G, Following Iwasawa loc. cit., we shall call R the radical of G.
8) On decompositions of (L)-groups as such, see Y. Matsushima, “On the decomposition of an (L)-group,” forthcoming in Journ. of Math, Soc. Japan.
9) See Iwasawa, loc. cit.
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