  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.
