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Phys. 7, 45–62 (1990) 14. : Quantization of Kähler Manifolds. II. Trans. Am. Math. Soc. 1, 73–98 (1993) 15. : Quantization of Kähler Manifolds. III. Lett. Math. Phys. 30, 291– 305 (1994) 16. : Quantization of Kähler Manifolds. IV. Lett. Math. Phys. 34, 159– 168 (1995) 17. : Déformations de l’Algèbre des Fonctions d’une Variété Symplectique: Comparaison entre Fedosov et DeWilde, Lecomte. Sel. Math. 4, 667–697 (1995) 18. : Existence of Star-Products and of Formal Deformations of the Poisson Lie Algebra of Arbitrary Symplectic Manifolds.

1. A p, is a Hausdorff locally convex topological vector space. 2. Every class [ f ] ∈ A p, has a unique real-analytic representative f ∈ C ω (Cn ). Therefore, we identify A p, with the corresponding subspace of C ω (Cn ) from now on. 3. e. 11) where : Cn z → (z, z) ∈ Cn × Cn is the diagonal. 4. 12) belongs to A p, . In particular C[z, z] ⊆ A p, . Proof. The first part is clear. For the second, we consider f p, 1,1,0,0 ≥ 1 ( f R! )2 ∂z R ∂z S 4 for all R, S. Thus we obtain ∂ |R+S| f ∂z R ∂z S √ 4 ( p) ≤ f p, 1,1,0,0 √ R!

Of course we can also directly compare the seminorms for different values of . 20) 0,0,R,S ≤ f 0,0,R,S for ≤ whence by induction f as well. 21) this gives the inclusion A ⊆A . 22) 6. The GNS Construction and Coherent States We shall now discuss the GNS construction corresponding to the positive δ-functionals δ p : A −→ C, now in the convergent situation. 1) 46 S. Beiser, H. Römer, S. 1. Let p ∈ Cn . 2) and the GNS pre-Hilbert space D p = A J p is a Fréchet space in the natural way, where the topology of D p is determined by the seminorms [f] p, m, ,R,S = inf f +g p, m, ,R,S g ∈ Jp .