  We show that the groups of finite energy loops and paths (that is, those of
Sobolev class $H^1$) with values in a compact connected Lie group, as well as
their central extensions, satisfy an amenability-like property: they admit a
left-invariant mean on the space of bounded functions uniformly continuous with
regard to a left-invariant metric. Every strongly continuous unitary
representation $\pi$ of such a group (which we call skew-amenable) has a
conjugation-invariant state on $B({\mathcal H}_{\pi})$.
