C*-algebras of separated graphs

Abstract

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 $K_0(C^*(E,C))$ and $K_1(C^*(E,C))$ 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 $U^{\text{nc}}_{m,n}$, confirming a conjecture of McClanahan. A reduced C*-algebra $\Cstred(E,C)$ 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 $\Cstred(E,C)$ and $C^*(E,C)$, such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan.