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.