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Abstract. We introduce and investigate the notion of a quasi-complete group. A group G is quasi-complete if every automorphism f A AutðGÞ, with the property that p and p f are unitarily equivalent for every unitary irreducible representation p of G, is an inner automorphism of G. Our main result is that every connected linear real reductive Lie group is quasi-complete. 2000 Mathematics Subject Classification: 22D45; 22E46, 43A65. Introduction The study of the automorphisms group AutðGÞ of a topological group G and of its distinguished normal subgroups raises many interesting questions. In connection with the normal subgroup of inner automorphisms, it was shown in [DG] that any group G can be realized as the outer automorphism group of some group H, i.e. G G AutðHÞ=InnðHÞ. In another direction, an abstract characterization of inner automorphisms in terms of their extension properties was recently given in [Sc]. Other distinguished normal subgroups of AutðGÞ are the automorphisms AutðGÞC preserving the conjugacy classes of G and the automorphisms AutðGÞG preserving the ^ equivalence classes of (continuous) unitary representations of G. These definitions appear in [Bu] for finite groups, but carry over to more general settings. In general, one has the inclusions InnðGÞ
Forum Mathematicum – de Gruyter
Published: Mar 30, 2004
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