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 Unc m,n, confirming a conjecture of McClanahan. A reduced C∗-algebra C∗red(E,C) 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 C∗red(E,C) and C∗(E,C), such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan. © 2011 Elsevier Inc. All rights reserved.