  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.
