Characteristic Classes and Zeroth Order Pseudodifferential Operators

We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the identity. The leading order Chern classes are nonzero in general, and they detect elements of the de Rham cohomology groups of the classifying space BZ.