Factor Representations of Diffeomorphism Groups

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 ($N\mathrm{/}V$) limit construction for diffeomorphism group representations.