  We show that if a countable discrete group acts properly and isometrically on
a spin manifold of bounded Riemannian geometry and uniformly positive scalar
curvature, then, under a suitable condition on the group action, the maximal
higher index of the Dirac operator vanishes in K-theory of the maximal
equivariant Roe algebra. The group action is not assumed to be cocompact. A key
step in the proof is to establish a functional calculus for the Dirac operator
in the maximal equivariant uniform Roe algebra. This allows us to prove
vanishing of the index of the Dirac operator in K-theory of this algebra, which
in turn yields the result for the maximal higher index.
