K-theory and exact sequences of partial translation algebras

TL;DR

This paper extends the theory of partial translation algebras, providing an excision theorem and computing their K-theory, thereby generalizing previous results and offering new insights into the K-theory of group C*-algebras and graph products.

Contribution

It introduces an extension of partial translation algebras and applies it to compute their K-theory, generalizing prior work by Pimsner and others.

Findings

Established an excision theorem for partial translation algebras.

Computed K-theory for partial translation and group C*-algebras.

Provided new perspective on K-theory of graph products of groups.

Abstract

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.