KMS and Ground States on Ultragraph C*-Algebras

  • Gilles Gonçalves de Castro
  • Daniel GonçalvesEmail author


We describe KMS and ground states arising from a generalized gauge action on ultragraph C*-algebras. We focus on ultragraphs that satisfy Condition (RFUM), so that we can use the partial crossed product description of ultragraph C*-algebras recently described by the second author and Danilo Royer. In particular, for ultragraphs with no sinks, we generalize a recent result by Toke Carlsen and Nadia Larsen: Given a time evolution on the C*-algebra of an ultragraph, induced by a function on the edge set, we characterize the KMS states in five different ways and ground states in four different ways. In both cases we include a characterization given by maps on the set of generalized vertices of the ultragraph. We apply this last result to show the existence of KMS and ground states for an ultragraph C*-algebra that is neither an Exel–Laca nor a graph C*-algebra.


KMS states Ultragraph C*-algebras Partial crossed product 

Mathematics Subject Classification

Primary 46L30 Secondary 46L55 



The authors would like to thank Zahra Afsar for valuable discussions regarding the present paper. In particular, the second author would like to thank Zahra for teaching him the theory of KMS states.


  1. 1.
    Afsar, Z., an Huef, A., Raeburn, I.: KMS states on \(C^*\)-algebras associated to local homeomorphisms. Int. J. Math. 25(8), 1450066 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Afsar, Z., Sims, A.: KMS states on the \(C^*\)-algebras of fell bundles over groupoids (2017). arXiv:1708.00629
  3. 3.
    an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on \(C^*\)-algebras associated to higher-rank graphs. J. Funct. Anal. 266(1), 265–283 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beuter, V.M., Gonçalves, D.: Partial crossed products as equivalence relation algebras. Rocky Mt. J. Math. 46(1), 85–104 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. 2. Texts and Monographs in Physics, 2nd edn. Springer, Berlin (1997). Equilibrium states. Models in quantum statistical mechanicsCrossRefGoogle Scholar
  6. 6.
    Carlsen, T.M., Larsen, N.S.: Partial actions and KMS states on relative graph \(C^*\)-algebras. J. Funct. Anal. 271(8), 2090–2132 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    de Castro, G.G., de L. Mortari, F.: KMS states for the generalized gauge action on graph algebras. C. R. Math. Acad. Sci. Soc. R. Can. 36(4), 114–128 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Edgar, G.: Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (2008)CrossRefGoogle Scholar
  9. 9.
    Enomoto, M., Fujii, M., Watatani, Y.: KMS states for gauge action on \(O_{A}\). Math. Japon. 29(4), 607–619 (1984)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Exel, R., Laca, M.: Partial dynamical systems and the KMS condition. Commun. Math. Phys. 232(2), 223–277 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gonçalves, D., Li, H., Royer, D.: Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph \(C^*\)-algebras. Int. J. Math. 27(10), 1650083 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gonçalves, D., Royer, D.: Ultragraphs and shift spaces over infinite alphabets. Bull. Sci. Math. 141(1), 25–45 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gonçalves, D., Royer, D.: Infinite alphabet edge shift spaces via ultragraphs and their \(C^*\)-algebras. Int. Math. Res. Not. (2017). CrossRefzbMATHGoogle Scholar
  14. 14.
    Gonçalves, D., Sobottka, M.: Continuous shift commuting maps between ultragraph shift spaces. Discrete Contin. Dyn. Syst. (2018, to appear). arXiv:1802.04793 [math.DS]
  15. 15.
    Imanfar, M., Pourabbas, A., Larki, H.: The Leavitt path algebras of ultragraphs (2017). arXiv:1701.00323
  16. 16.
    Kajiwara, T., Watatani, Y.: KMS states on finite-graph \(C^*\)-algebras. Kyushu J. Math. 67(1), 83–104 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Katsura, T., Muhly, P.S., Sims, A., Tomforde, M.: Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence. J. Reine Angew. Math. 640, 135–165 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kumjian, A., Renault, J.: KMS states on \(C^*\)-algebras associated to expansive maps. Proc. Am. Math. Soc. 134(7), 2067–2078 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Marrero, A.E., Muhly, P.S.: Groupoid and inverse semigroup presentations of ultragraph \(C^*\)-algebras. Semigroup Forum 77(3), 399–422 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Murphy, G.J.: \(C^*\)-Algebras and Operator Theory. Academic Press, Boston (1990)zbMATHGoogle Scholar
  21. 21.
    Raeburn, I.: Graph Algebras, volume 103 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence (2005)Google Scholar
  22. 22.
    Tomforde, M.: Simplicity of ultragraph algebras. Indiana Univ. Math. J. 52(4), 901–925 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tomforde, M.: A unified approach to Exel–Laca algebras and \(C^\ast \)-algebras associated to graphs. J. Oper. Theory 50(2), 345–368 (2003)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Santa CatarinaFlorianópolisBrazil

Personalised recommendations