We investigate universal operator algebras of directed graphs. We begin with the constructions of the universal operator algebra and universal C *-algebra of a directed graph. We show that each universal algebra can be decomposed as a free product of algebras associated to subgraphs. We also describe the universal algebras of infinite graphs in terms of inductive limits of finite subgraph algebras. We discuss the K -groups and the maximal ideal space of universal operator algebras of directed graphs. When the graph is finite we use the described invariants to classify the universal operator algebras, up to bounded isomorphism, in terms of the underlying graphs. We also discuss commutative quotients of the universal operator algebra of a graph. Lastly we discuss homology and amenability for universal graph operator algebras.
In addition to the material about universal graph operator algebras we show that the universal C *-algebra of a directed graph fits naturally into the framework of the maximal C *-envelope of Blecher. In the process of describing the universal operator algebra of a directed graph we show that the maximal C *-envelope is continuous with respect to direct limits. And further, we relate the maximal ideal space of an operator algebra to the maximal ideal space of the maximal C *-envelope of the algebra.
