Symmetry Reduction of States II: A non-commutative Positivstellensatz for CPn

Abstract

We give a non-commutative Positivstellensatz for È n : The (commutative) * -algebra of polynomials on the real algebraic set È n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on 1+n , restricted to the real (2n + 1)-sphere inside 1+n , and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on 1+n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative * -algebra of polynomials on 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 È 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 È n , obtained e.g. through phase space reduction or Berezin-Toeplitz quantization, are determined.