  In this paper we define a variant of Roe algebras for spaces with cylindrical
ends and use this to study questions regarding existence and classification of
metrics of positive scalar curvature on such manifolds which are collared on
the cylindrical end. We discuss how our constructions are related to relative
higher index theory as developed by Chang, Weinberger, and Yu and use this
relationship to define higher rho-invariants for positive scalar curvature
metrics on manifolds with boundary. This paves the way for classification of
these metrics. Finally, we use the machinery developed here to give a concise
proof of a result of Schick and the author, which relates the relative higher
index with indices defined in the presence of positive scalar curvature on the
boundary.
