  C denotes either the conformal group in 3+1 dimensions, or in one chiral
dimension. Let U be a unitary, strongly continuous representation of C
satisfying the spectrum condition and inducing, by its adjoint action,
automorphisms of a v.Neumann algebra A. We construct the unique inner
representation U^A of the universal covering group of C implementing these
automorphisms. U^A satisfies the spectrum condition and acts trivially on any
U-invariant vector.
  This means in particular: Conformal transformations of a field theory having
positive energy are weak limit points of local observables. Some immediate
implications for chiral subnets are given. We propose the name
``Borchers-Sugawara construction''.
