Projective positivity of the function systems

The present paper is devoted to the projective positivity in the category of function systems, which plays a key role in the quantization problems of the operator systems. The main result of the paper asserts that every unital star-normed space can be equipped with the projective positivity. The geometry of the related state spaces is described in the case of Lp-spaces, Schatten matrix spaces, and Lp-spaces of a finite von Neumann algebra.