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

An amenability-like property of finite energy path and loop groups