Summary: A construction is given relating a finitely generated free-product of cyclic groups with a certain Cuntz-Krieger algebra, generalizing the relation between the Choi algebra and \({\mathcal O}_ 2\). It is shown that a certain boundary action of such a group yields a Cuntz-Krieger algebra by the crossed-product construction. Certain compact convex spaces of completely positive mappings associated to a crossed-product algebra are introduced. These are used to generalize a problem of J. Anderson regarding the representation theory of the Choi algebra. An explicit computation of these spaces for the crossed-products under study yields a negative answer to this problem.