  We give a non-commutative Positivstellensatz for CP^n: The (commutative)
*-algebra of polynomials on the real algebraic set CP^n with the pointwise
product can be realized by phase space reduction as the U(1)-invariant
polynomials on C^{1+n}, restricted to the real (2n+1)-sphere inside C^{1+n},
and Schm\"udgen's Positivstellensatz gives an algebraic description of the
real-valued U(1)-invariant polynomials on CP^n that are strictly pointwise
positive on the sphere. In analogy to this commutative case, we consider a
non-commutative *-algebra of polynomials on C^{1+n}, the Weyl algebra, and give
an algebraic description of the real-valued U(1)-invariant polynomials that are
positive in certain *-representations on Hilbert spaces of holomorphic sections
of line bundles over CP^n. It is especially noteworthy that the non-commutative
result applies not only to strictly positive, but to all positive elements. As
an application, all *-representations of the quantization of the polynomial
*-algebra on CP^n, obtained e.g. through phase space reduction or
Berezin--Toeplitz quantization, are determined.
