  Consider a proper, isometric action by a unimodular, locally compact group
$G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators
that are invertible outside a cocompact subset of $M$, we show that a localised
index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is
well-defined. The approach is based on the use of maximal versions of
equivariant localised Roe algebras, and many of the technical arguments in this
paper are used to handle the ways in which they differ from their reduced
versions.
  By using the maximal group $C^*$-algebra instead of its reduced counterpart,
we can apply the trace given by integration over $G$ to recover an index
defined earlier by the last two authors, and developed further by Braverman, in
terms of sections invariant under the group action. As a very special case,
this allows one to refine numerical obstructions to positive scalar curvature
on a noncompact $\operatorname{Spin}$ manifold $X$ defined via Callias index
theory, to obstructions in the $K$-theory of the maximal $C^*$-algebra of the
fundamental group $\pi_1(X)$.
  As a motivating application in another direction, we prove a version of
Guillemin and Sternberg's quantisation commutes with reduction principle for
equivariant indices of $\operatorname{Spin}^c$ Callias-type operators.
