Commutators of spectral projections of spin operators

We present a proof that the operator norm of the commutator of certain spectral projections associated with spin operators converges to $\frac{1}{2}$ in the semiclassical limit. The ranges of the projections are spanned by all eigenvectors corresponding to positive eigenvalues. The proof involves the theory of Hankel operators on the Hardy space. A discussion of several analogous results is also included, with an emphasis on the case of finite Heisenberg groups.