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 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 up to sign whenever the multiplicative anomaly formula for usual determinants of Kontsevich-Vishik and Okikiolu holds. When restricted to operators of Dirac type our determinant provides a sign refined version of the determinant constructed by Kontsevich and Vishik. We discuss some applications of the symmetrized determinant to a non-linear $\sigma$-model in superconductivity.