C*-algebras of separated graphs

P. Ara, K. R. Goodearl

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14 Citations (Scopus)

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 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 Cred*(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 Cred*(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.
Original languageEnglish
Pages (from-to)2540-2568
JournalJournal of Functional Analysis
Volume261
Issue number9
DOIs
Publication statusPublished - 1 Nov 2011

Keywords

  • Amalgamated free product
  • Conditional expectation
  • Graph C -algebra *
  • Ideal lattice
  • Separated graph

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