Universal Operator Algebras of Directed Graphs

Given a directed graph, there exists a universal operator algebra and universal C∗-algebra associated to the directed graph. For finite graphs this algebra decomposes as the universal free product of some building block operator algebras. For countable directed graphs, the universal operator algebras arise as direct limits of operator algebras of finite subgraphs. Finally, a method for computing the K-groups for universal operator algebras of directed graphs is given. In [11] Muhly associates a non-selfadjoint operator algebra to a directed graph (or quiver). Henceforth we refer to these algebras as Toeplitz quiver algebras. Kribs and Power [8] showed that the graph was a complete unitary invariant for these algebras. Recent work on these Toeplitz quiver algebras by Katsoulis and Kribs, [7] and by Solel [15], has demonstrated that the graph is a complete isomorphism invariant for these algebras. In addition Kribs and Power, [8] and [9] study the structure of these algebras, including a free product result for certain amalgamations of graphs. In another direction [3], and [6] have initiated a study of universal operator algebras (both nonself-adjoint and self-adjoint) associated to combinatorial objects (e.g. groups, monoids, and semigroups). We have continued this study by introducing the universal operator algebra, and the universal C-algebra associated to a directed graph. In what follows we look at the construction of these objects, and using [6], [8], and [16] as models, we find nice decompositions. As a consequence of the decompositions we calculate the K-Theory of these algebras. As a result of theorem 2.1 we are able to write any finite graph as a free product of copies of three “building block graphs”. Using universal properties, we then show that the universal operator algebra will be a 2000 Mathematics Subject Classification. 47L40, 47L55, 47L75, 46L80.