  We prove that there does not exist a nontrivial quantization of the Poisson
algebra of the symplectic manifold S^2 which is irreducible on the subalgebra
generated by the components {S_1,S_2,S_3} of the spin vector. We also show that
there does not exist such a quantization of the Poisson subalgebra P consisting
of polynomials in {S_1,S_2,S_3}. Furthermore, we show that the maximal Poisson
subalgebra of P containing {1,S_1,S_2,S_3} that can be so quantized is just
that generated by {1,S_1,S_2,S_3}.
