Certain Free Products of Graph Operator Algebras

We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the C∗algebras and the non-selfadjoint operator algebras associated to edge-colored directed graphs. These algebras arise as free products of directed graph algebras with amalgamation. We then determine the C∗-envelopes for a large class of the non-selfadjoint algebras. Finally, we relate properties of the edgecolored directed graphs to properties of the associated C-algebra, including simplicity and nuclearity. Using the free product description of these algebras we investigate the K-theory of these algebras. The operator algebras of directed graphs are important for two reasons. First because they give concrete examples of large classes of operator algebras. Second, they are useful because structural properties of the algebras can be related to simple observations about their underlying graphs. In this paper we take this nice class of algebras and look at their universal free products amalgamated over specific subalgebras. Again, the reasons are two-fold, we believe that this large class of “concrete” examples will lead to a better understanding of universal free products, and we also have seen that the defined underlying discrete structure will provide insight into structural questions concerning these algebras. The focus of this paper, besides introducing this class of free products, is to make a case that the second aim of this research is tractable. Graph algebras are a generalization of the Cuntz algebras where we have a collection of projections, and collections of partial isometries with domain and range satisfying natural conditions corresponding to the collection of projections. These relations can be identified via directed graphs with projections corresponding to vertices and arrows between projections corresponding to partial isometries. The source of the arrow corresponds to the domain projection for the partial isometry, and the range of the arrow corresponds to the range projection for the partial isometry. However, in the graph algebra context we require that for any projection corresponding to a vertex the set of edges ending at the vertex the ranges of the associated partial isometries “sum” to the projection. However, when we take free products of directed graph algebras with amalgamation over the subalgebra generated by the projections the restrictions on the range of the partial isometries disappear; we can then allow partial isometries with the same range. By amalgamating over the subalgebra corresponding to the projections 2000 Mathematics Subject Classification. 46L09, 46L05, 46L55.