Fuzzy Sphere and Hyperbolic Space from Deformation Quantization

We explicitly construct noncommutative * products on circularly symmetric two dimensional space by using the technique of Fedosov’s deformation quantization. Especially, on constant curvature spaces i.e., S^2 and H^2, we get su(2) and su(1,1) algebra respectively. These are candidates of * products applicable to noncommutative field theories or noncommutative gauge theories on spaces with nontrivial symplectic structure.