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

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 the relationship between the KMS states of the generalized gauge actions on our $C^\ast$ algebras and random walks on the groups.