Deformation quantization with separation of variables of an endomorphism bundle

Given a holomorphic Hermitian vector bundle EE and a star-product with separation of variables on a pseudo-Kähler manifold, we construct a star product on the sections of the endomorphism bundle of the dual bundle E∗E∗ which also has the appropriately generalized property of separation of variables. For this star product we prove a generalization of Gammelgaard’s graph-theoretic formula.