  In this paper we prove a strengthening of a theorem of Chang, Weinberger and
Yu on obstructions to the existence of positive scalar curvature metrics on
compact manifolds with boundary. They construct a relative index for the Dirac
operator, which lives in a relative K-theory group, measuring the difference
between the fundamental group of the boundary and of the full manifold.
Whenever the Riemannian metric has product structure and positive scalar
curvature near the boundary, one can define an absolute index of the Dirac
operator taking value in the K-theory of the C*-algebra of fundamental group of
the full manifold. This index depends on the metric near the boundary. We prove
that the relative index of Chang, Weinberger and Yu is the image of this
absolute index under the canonical map of K-theory groups. This has the
immediate corollary that positive scalar curvature on the whole manifold
implies vanishing of the relative index, giving a conceptual and direct proof
of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental
groups of the manifold and its boundary into account requires working with
maximal C* completions of the involved *-algebras. A significant part of this
paper is devoted to foundational results regarding these completions.
