  We compute the Chern-Connes character (a map from the $K$-theory of a
C$^*$-algebra under the action of a Lie group to the cohomology of its Lie
algebra) for the $L^2$-norm closure of the algebra of all classical zero-order
pseudodifferential operators on the sphere under the canonical action of ${\rm
SO}(3)$. We show that its image is $\mathbb{R}$ if the trace is the integral of
the principal symbol.
