Symmetry reduction of states II: A non-commutative Positivstellensatz for CPn

We give a non-commutative Positivstellensatz for CPn: The (commutative) ⁎-algebra of polynomials on the real algebraic set CPn with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C1+n, restricted to the real (2n+1)-sphere inside C1+n, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on C1+n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative ⁎-algebra of polynomials on C1+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 CPn. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive (semidefinite) elements. As an application, all ⁎-representations of the quantization of the polynomial ⁎-algebra on CPn, obtained e.g. through phase space reduction or Berezin–Toeplitz quantization, are determined.