Graph C^∗-algebras with a T₁ primitive ideal space

2013/02/15 by James Gabe, Gabe, James
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #Primary: 46L35 #math.OA #msc:46L35

paper · pdf · doi:10.48550/arxiv.1302.3670

13 pages

arxiv created 2013/02/15 · openalex publication_date 2013/02/15 · arxiv updated 2013/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give necessary and sufficient conditions which a graph should satisfy in order for its associated C^∗-algebra to have a T₁ primitive ideal space. We give a description of which one-point sets in such a primitive ideal space are open, and use this to prove that any purely infinite graph C^∗-algebra with a T₁ (in particular Hausdorff) primitive ideal space, is a c₀-direct sum of Kirchberg algebras. Moreover, we show that graph C^∗-algebras with a T₁ primitive ideal space canonically may be given the structure of a C(\widetilde\mathbb N)-algebra, and that isomorphisms of their \widetilde\mathbb N-filtered K-theory (without coefficients) lift to E(\widetilde\mathbb N)-equivalences, as defined by Dadarlat and Meyer.

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