<p>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.</p>