On the existence of exotic Banach-Lie groups

Abstract

Let X be a real or complex Banach space, and let G be a topological group. By a representation of G in X we mean a homomorphism of G into GL(X) the group of all automorphisms ofX. A representation T : G ~ G L ( X ) is called weakly (resp. strongly, uniformly) continuous, when it is continuous in the weak (resp. strong, uniform) topology on GL(X). A representation T : G ~ G L ( X ) is called unitary, when X is a Hilbert space, and all the operators T(g), g~ G, are unitary. For unitary representations the notions of weak and strong continuity coincide. In this paper we shall frequently treat vector spaces as additive groups, and topological vector spaces (especially normed spaces) as additive topological groups. Thus, for example, if E is a normed space, we shall speak of the topological group E, of a subgroup K C E, or of a topological quotient group E/K, without special explanations. An abelian topological group is called exotic, if it does not admit any nontrivial strongly continuous unitary representations, and strongly exotic, if it does not admit any non-trivial weakly continuous representations in Hilbert spaces. An example of an exotic group was given by Herer and Christensen in [4]. The group in the mentioned example was a Polish vector space. In the present paper we exhibit the existence of strongly exotic Banach-Lie groups. More precisely, our aim is to prove the following fact: