On differentiable vectors for representations of infinite dimensional Lie groups

Abstract

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π : G → GL ( V ) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V ∞ of smooth vectors. If G is a Banach–Lie group, we define a topology on the space V ∞ of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V ∞ is a Fréchet space. This applies in particular to C ∗ -dynamical systems ( A , G , α ) , where G is a Banach–Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function 〈 π ( g ) v , v 〉 is smooth. The second class of results concerns criteria for C k -vectors in terms of operators of the derived representation for a Banach–Lie group G acting on a Banach space V. In particular, we provide for each k ∈ N examples of continuous unitary representations for which the space of C k + 1 -vectors is trivial and the space of C k -vectors is dense.