  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.
