| This dissertation is concerned with C*-algebras associated with boundary actions obtained from graphs of groups. The graphs considered are in the form of a single vertex together with a single edge (a loop). The stabilizer groups of the vertex and edge are both equal to Zn for some n ∈ Z+ . According to the Bass-Serre theory there is a group associated to the graph, the fundamental group of the graph of groups, which is an Higman-Neuman-Neumann (HNN) extension. Moreover there is a tree on which this group acts freely. An orientation of the edge in the graph induces an orientation of the tree. The boundary of this directed tree consists of all equivalence classes of infinite directed paths, and is a locally compact Hausdorff space. The action of the fundamental group on the boundary gives a dynamical system whose C*-algebra is the object of study. Several situations are considered; depending on the integer n and on the choice of the embeddings &phis; 0 and &phis;1 of the edge stabilizer into the vertex stabilizer. (1) When n = 1 and &phis;0, &phis;1 are given by multiplication by 1 and m, where m > 1, the fundamental group is a solvable Baumslag-Solitar group. The C*-algebra is the crossed product of the stabilized Bunce-Deddens algebra (of type minfinity) by an action of the group of integers. It is a simple, nuclear; purely infinite C *-algebra. Its K-theory is also computed. (2) When n > 1 and &phis;0, &phis;1 are given by the identity and A, where A is an n x n integer matrix with |detA| > 1, the C*-algebra is shown to be simple, nuclear and purely infinite if A satisfies an extra condition. (3) When n = 2, the Smith normal form for powers of A is studied. Using this, some K-theory calculations are made. (4) When n = 1 and &phis;0, &phis;1 are both given by multiplication by integers larger than one, the fundamental group is a non-solvable Baumslag-Solitar group. An example (&phis;0 = 2, &phis;1 = 5) is considered. Groupoid C*-algebras are used to compute K-theory for the reduced C*-algebra of a portion of the dynamical system. |
