Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of Diff +(S1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms Ds(S1) for any real s> 3 , and in particular to Ck-diffeomorphisms Diff+k(S1) with k≥ 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S1 is covariant with respect to Ds(S1) , s> 3. Moreover every direct sum of irreducible representations of a conformal net is also Ds(S1) -covariant.