Ring structures in coarse K-theory

The K-theory of the stable Higson corona of a coarse space carries a canonical ring structure. The present thesis covers two aspects of this ring: Chapter 1: The K-theory of the stable Higson corona is the domain of an unreduced version of the coarse co-assembly map of Emerson and Meyer. We show that the target also carries a ring structure and co-assembly is a ring homomorphism, provided that the given coarse space is contractible in a coarse sense. Chapter 2 (pursuing conjectures of John Roe): Applied to a foliated cone over a foliation, we show that the K-theory of the stable Higson corona can be considered as a new model for the K-theory of the leaf space, which is - in contrast to Connes' K-theory model - a ring. We show that Connes' K-theory model is a module over this ring and develop an interpretation of the module structure in terms of twisted longitudinally elliptic operators