  In an earlier paper, the authors introduced partial translation algebras as a
generalisation of group C*-algebras. Here we establish an extension of partial
translation algebras, which may be viewed as an excision theorem in this
context. We apply this general framework to compute the K-theory of partial
translation algebras and group C*-algebras in the context of almost invariant
subspaces of discrete groups. This generalises the work of Cuntz, Lance,
Pimsner and Voiculescu. In particular we provide a new perspective on Pimsner's
calculation of the K-theory for a graph product of groups.
