Projective positivity of function systems

The present talk 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. We show that every unital $*$-normed space can be equipped with the projective positivity. The geometry of the related state spaces is described in the case of $L^{p}$-spaces, Schatten matrix spaces, and $L^{p}$-spaces of a finite von Neumann algebra. Some key facts on the operator systems are provided in the beginning of the presentation. See https://arxiv.org/abs/2303.12510.