C⁎-algebras of separated graphs

The construction of the C⁎-algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. These C⁎-algebras C⁎(E,C) are analyzed in terms of their ideal theory and K-theory, mainly in the case of partitions by finite sets. The groups K0(C⁎(E,C)) and K1(C⁎(E,C)) are completely described via a map built from an adjacency matrix associated to (E,C). One application determines the K-theory of the C⁎-algebras , confirming a conjecture of McClanahan. A reduced C⁎-algebra is also introduced and studied. A key tool in its construction is the existence of canonical faithful conditional expectations from the C⁎-algebra of any row-finite graph to the C⁎-subalgebra generated by its vertices. Differences between and C⁎(E,C), such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan.