  It is shown that the normalized trace of Fedosov star product for quantum
moment map depends only on the path component in the cohomology class of the
symplectic form and the cohomology class of the closed formal 2-form required
to define Fedosov connections (Theorem 1.3). As an application we obtain a
family of obstructions to the existence of closed Fedosov star products
naturally attached to symplectic manifolds (Theorem 1.5) and K\"ahler manifolds
(Theorem 1.6). These obstructions are integral invariants depending only on the
path component of the cohomology class of the symplectic form. Restricted to
compact K\"ahler manifolds we re-discover an obstruction found earlier in [29].
