Larki, Hossein

  • 투고 : 2017.09.05
  • 심사 : 2018.09.11
  • 발행 : 2019.01.01


Let ${\mathcal{G}}$ be an ultragraph and let $C^*({\mathcal{G}})$ be the associated $C^*$-algebra introduced by Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*({\mathcal{G}})$, we approach the quotient $C^*$-algebra $C^*({\mathcal{G}})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-Rørdam) via Fell bundles.


ultragraph $C^*$-algebra;primitive ideal;pure infiniteness


  1. R. Exel, Exact groups and Fell bundles, Math. Ann. 323 (2002), no. 2, 259-266.
  2. J. Dixmier, Sur les C*-algebres, Bull. Soc. Math. France 88 (1960), 95-112.
  3. T. Bates, J. H. Hong, I. Raeburn, and W. Szymanski, The ideal structure of the C*-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159-1176.
  4. T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The C*-algebras of row-finite graphs, New York J. Math. 6 (2000), 307-324.
  5. R. Exel, Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, 224, American Mathematical Society, Providence, RI, 2017.
  6. R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119-172.
  7. N. J. Fowler, M. Laca, and I. Raeburn, The C*-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2319-2327.
  8. J. H. Hong and W. Szymanski, Purely infinite Cuntz-Krieger algebras of directed graphs, Bull. London Math. Soc. 35 (2003), no. 5, 689-696.
  9. J. A. Jeong, S. H. Kim, and G. H. Park, The structure of gauge-invariant ideals of labelled graph C*-algebras, J. Funct. Anal. 262 (2012), no. 4, 1759-1780.
  10. T. Katsura, P. S. Muhly, A. Sims, and M. Tomforde, Ultragraph C*-algebras via topological quivers, Studia Math. 187 (2008), no. 2, 137-155.
  11. E. Kirchberg and M. Rrdam, Non-simple purely infinite C*-algebras, Amer. J. Math. 122 (2000), no. 3, 637-666.
  12. B. K. Kwasniewski and W. Szymanski, Pure infiniteness and ideal structure of C*-algebras associated to Fell bundles, J. Math. Anal. Appl. 445 (2017), no. 1, 898-943.
  13. P. S. Muhly and M. Tomforde, Topological quivers, Internat. J. Math. 16 (2005), no. 7, 693-755.
  14. I. Raeburn and W. Szymanski, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39-59.
  15. M. Rrdam, On the structure of simple C*-algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (1992), no. 2, 255-269.
  16. M. Tomforde, A unified approach to Exel-Laca algebras and C*-algebras associated to graphs, J. Operator Theory 50 (2003), no. 2, 345-368.
  17. M. Tomforde, Simplicity of ultragraph algebras, Indiana Univ. Math. J. 52 (2003), no. 4, 901-925.