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 m , n n c , confirming a conjecture of McClanahan. A reduced C ⁎ -algebra C r e d ⁎ ( 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 r e d ⁎ ( E , C ) and C ⁎ ( E , C ) , such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan.
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