Identifying AF-algebras that are graph C*-algebras

Søren Eilers, Takeshi Katsura, Efren Ruiz, Mark Tomforde

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C*-algebra. We prove that any separable, unital, Type I C*-algebra with finitely many ideals is isomorphic to a graph C*-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C*-algebra if and only if it is a Type I C*-algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph C*-algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to Mk for some natural number k. All of these results provide vast supporting evidence for the conjecture that an AF-algebra is isomorphic to a graph C*-algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the extension problem for graph C*-algebras.

Original languageEnglish
Pages (from-to)3968-3996
Number of pages29
JournalJournal of Functional Analysis
Volume266
Issue number6
DOIs
Publication statusPublished - 2014 Mar 15

Keywords

  • AF-algebras
  • Bratteli diagrams
  • Graph C-algebras
  • Type I C-algebras

ASJC Scopus subject areas

  • Analysis

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