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Fuzzy spheres from inequivalent coherent states quantizations

We present a new procedure which allows a coherent state (CS) quantization of any set with a measure. It is manifest through the replacement of classical observables by CS quantum observables, which acts on a Hilbert space of prescribed dimension $N$. The algebra of CS quantum observables has the finite dimension $N^2$. The application to the 2-sphere provides a family of inequivalent CS quantizations, based on the spin spherical harmonics (the CS quantization from usual spherical harmonics appears to give a trivial issue for the cartesian coordinates). We compare these CS quantizations to the usual (Madore) construction of the fuzzy sphere. The difference allows us to consider our procedures as the constructions of new type of fuzzy spheres. The very general character of our method suggests applications to construct fuzzy versions of a variety of sets.