  If $G$ is a compact group acting continuously on a compact metric space $(X,
m)$, we prove two results that generalize Dirichlet's classical theorem on
Diophantine approximation. If $G$ is a noncommutative compact group of
isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply
our general result to the special case of the unitary group $U(N)$ acting on
the complex unit sphere, and obtain a noncommutative result in this setting.
