TY - JOUR

T1 - Symmetry reduction of states II: A non-commutative Positivstellensatz for CP^n

AU - Schmitt, Philipp Lothar

AU - Schötz, Matthias

N1 - Acknowledgements: The first author was partly supported by the Danish National Research Foundation through the Centre of Symmetry and Deformation (DNRF92). The second author was supported by the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk Onderzoek - Vlaaderen (FWO) under EOS Project n∘30950721

PY - 2022/9/15

Y1 - 2022/9/15

N2 - 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 Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on C^{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 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 (semidefinite) 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.

AB - 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 Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on C^{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 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 (semidefinite) 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.

KW - Non-strict Positivstellensatz

KW - Non-commutative -algebra

KW - Symmetry reduction

KW - Star product

U2 - 10.48550/arXiv.2110.03437

DO - 10.48550/arXiv.2110.03437

M3 - Article

VL - 649

SP - 326

EP - 353

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -