Involution in quantized endomorphism bundle and reality of   noncommutative gravity actions
  It is shown that for arbitrary connection in the vector bundle compatible with some Hermitian metric, the corresponding Fedosov trace functional commutes with involution generated by this metric. This result is then used to prove that certain noncommutative gravity actions are real in all powers of deformation parameter. 
Michal Dobrski
math-ph hep-th math.MP