Cuntz–Krieger–Pimsner Algebras Associated with Amalgamated Free Product Groups

Abstract

We give a construction of a nuclear $C^{\ast }$-algebra associated with an amalgamated free product of groups, generalizing Spielberg’s construction of a certain Cuntz–Krieger algebra associated with a finitely generated free product of cyclic groups. Our nuclear $C^{\ast }$-algebras can be identified with certain Cuntz–Krieger–Pimsner algebras. We will also show that our algebras can be obtained by the crossed product construction of the canonical actions on the hyperbolic boundaries, which proves a special case of Adams’ result about amenability of the boundary action for hyperbolic groups. We will also give an explicit formula of the $K$-groups of our algebras. Finally we will investigate a relationship between the KMS states of the generalized gauge actions on our $C^{\ast }$-algebras and random walks on the groups.