K-theory for subspaces of groups

Abstract

We show that a subspace of a group carries a natural partial translation structure which gives rise to a C*-algebra similar to that of a reduced C*-algebra of a group. We investigate the question: Does the inclusion of a subspace into an ambient group induce a homomorphism of C*-algebras? We prove that the answer is affirmative for subspaces of groups with non-coarsely dense complement. We show that under this condition there exists an exact sequence of C*-algebras which is analogous to the Pimsner-Voiculescu extension