Orderings and non-formal deformation quantization

We propose suitable ideas for non-formal deformation quantization of Fréchet Poisson algebras. To deal with the convergence problem of deformation quantization, we employ Fréchet algebras originally given by Gel'fand-Shilov. Ideas from deformation quantization are applied to expressions of elements of abstract algebras, which leads to a notion of "independence of ordering principle". This principle is useful for the understanding of the star exponential functions and for the transcendental calculus in non-formal deformation quantization.