  To supplement the already known classification of traces on classical
pseudodifferential operators, we present a classification of traces on the
algebras of odd-class pseudodifferential operators of non-positive order acting
on smooth functions on a closed odd-dimensional manifold. By means of the one
to one correspondence between continuous traces on Lie algebras and
determinants on the associated regular Lie groups, we give a classification of
determinants on the group associated to the algebra of odd-class
pseudodifferential operators with fixed non-positive order. At the end we
discuss two possible ways to extend the definition of a determinant outside a
neighborhood of the identity on the Lie group associated to the algebra of
odd-class pseudodifferential operators of order zero.
