  In this paper we investigate the compatibility of the BRST reduction
procedure with the Hermiticity of star products. First, we introduce the
generalized notion of abstract BRST algebras with corresponding involutions. In
this setting we define adjoint BRST differentials and as a consequence one gets
new BRST quotients. Passing to the quantum BRST setting we show that for
compact Lie groups the new quantum BRST quotient and the quantum BRST
cohomology are isomorphic in zero degree implying that reduction is compatible
with Hermiticity.
