Symmetrized Trace and Symmetrized Determinant of Odd Class Pseudo-differential Operators

We introduce a new canonical trace on odd class logarithmic pseudo-differential operators on an odd dimensional manifold, which vanishes on a commutators. When restricted to the algebra of odd class classical pseudo-differential operators our trace coincides with the canonical trace of Kontsevich and Vishik. Using the new trace we construct a new determinant of odd class classical elliptic pseudo-differential operators. This determinant is multiplicative whenever the multiplicative anomaly formula for usual determinants of Kontsevich-Vishik and Okikiolu holds. In particular, it is multiplicative for operators whose leading symbols commute. When restricted to operators of Dirac type our determinant provides a sign refined version of the determinant constructed by Kontsevich and Vishik.