On Positive Deformations of *-Algebras

Motivated by deformation quantization we consider ${}^{\ast }$-algebras over ordered rings and their deformations: we investigate formal associative deformations compatible with the ${}^{\ast }$-involution and discuss a cohomological description in terms of a Hermitian Hochschild cohomology. As an ordered ring allows for a meaningful definition of positive functionals and as the formal power series with coefficients in an ordered ring are again an ordered ring we define a deformation to be positive if any positive linear functional of the undeformed algebra can be deformed into a positive linear functional of the deformed algebra. We discuss various examples and prove in particular that star products on symplectic manifolds are positive deformations.