  General semifinite factor representations of the diffeomorphism group of
euclidean space are constructed by means of a canonical correspondence with the
finite factor representations of the inductive limit unitary group. This
construction includes the quasi-free representations of the canonical
commutation and anti-commutation relations. To establish this correspondence
requires a non-linear form of complete positivity as developed by Arveson. We
also compare the asymptotic character formula for the unitary group with the
thermodynamic ($N/V$) limit construction for diffeomorphism group
representations.
