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
Um,nnc, 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.Comment: 29 pages. Revised version, to appear in J. Functional Analysi