Uniqueness of multiplicative determinants on elliptic pseudodifferential operators

We describe all the multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold that are continuous along continuous paths and the restriction to zero order operators of which is of class C1. This boils down to a description of all traces on zero order classical pseudodifferential operators, which turn out to be linear combinations of the Wodzicki residue and leading symbol traces introduced in previous work of S. Rosenberg and the second author. Both of these are continuous. Consequently, multiplicative determinants are parametrized by the residue determinant of S. Scott and a new ‘leading symbol determinant’, both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator.