  We construct Chern-Weil classes on infinite dimensional vector bundles with
structure group contained in the algebra $\cl[\leq 0](M, E)$ of non-positive
order classical pseudo-differential operators acting on a finite rank vector
bundle $E$ over a closed manifold $M$. Mimicking the finite dimensional
Chern-Weil construction, we replace the ordinary trace on matrices by linear
functionals on $\cl[\leq 0] (M, E)$ built from the leading symbols of the
operators. The corresponding Chern classes vanish for loop groups, but a
weighted trace construction yields a non-zero class perviously constructed by
Freed. For loop spaces, the structure group reduces to a gauge group of bundle
automorphisms, and we produce non-vanishing universal Chern classes in all
degrees, using a universal connection theorem for these bundles.
