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

Quantum moment map and obstructions to the existence of closed Fedosov star products