  We obtain an equivariant index theorem, or Lefschetz fixed-point formula, for
isometries from complete Riemannian manifolds to themselves. The fixed-point
set of such an isometry may be noncompact. We build on techniques developed by
Roe. Key new ingredients are a localised functional on operators with bounded
smooth kernels, and an algebra (reminiscent of Yu's localisation algebra) of
`asymptotically local' operators, on which this functional has an asymptotic
trace property. As consequences, we show that some earlier indices used are
special cases of the one we introduce here, and obtain an obstruction to
positive scalar curvature.
