Skip to main content
SpringerLink
Log in

KMS States on the Operator Algebras of Reducible Higher-Rank Graphs

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

We study the equilibrium or KMS states of the Toeplitz \(C^*\)-algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of the real numbers in the torus. Here we use an embedding which leads to a dynamics which has previously been identified as “preferred”, and we scale the dynamics so that 1 is a critical inverse temperature. As with 1-graphs, we study the strongly connected components of the vertices of the graph. The behaviour of the KMS states depends on both the graphical relationships between the components and the relative size of the spectral radii of the vertex matrices of the components. We test our theorems on graphs with two connected components. We find that our techniques give a complete analysis of the KMS states with inverse temperatures down to a second critical temperature \(\beta _c<1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II, 2nd edn. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  2. Carlsen, T.M., Larsen, N.S.: Partial actions and KMS states on relative graph \(C^*\)-algebras. J. Funct. Anal. 271, 2090–2132 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chlebovec, C.: KMS states for quasi-free actions on finite-graph algebras. J. Oper. Theory 75, 119–138 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Christensen, J., Thomsen, K.: Finite digraphs and KMS states. J. Math. Anal. Appl. 433, 1626–1646 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Davidson, K.R., Yang, D.: Periodicity in rank 2 graph algebras. Can. J. Math. 61, 1239–1261 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Enomoto, M., Fujii, M., Watatani, Y.: KMS states for gauge action on \(\cal{O}_A\). Math. Jpn. 29, 607–619 (1984)

    MATH  Google Scholar 

  7. Exel, R., Laca, M.: Partial dynamical systems and the KMS condition. Commun. Math. Phys. 232, 223–277 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fowler, N.J., Raeburn, I.: The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48, 155–181 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hazlewood, R., Raeburn, I., Sims, A., Webster, S.B.G.: Remarks on some fundamental results about higher-rank graphs and their \(C^*\)-algebras. Proc. Edinburgh Math. Soc. 56, 575–597 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. an Huef, A., Kang, S., Raeburn, I.: Spatial realisations of KMS states on the \(C^*\)-algebras of higher-rank graphs. J. Math. Anal. Appl. 427, 977–1003 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the \(C^*\)-algebras of finite graphs. J. Math. Anal. Appl. 405, 388–399 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on \(C^*\)-algebras associated to higher-rank graphs. J. Funct. Anal. 266, 265–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the \(C^*\)-algebra of a higher-rank graph and periodicity in the path space. J. Funct. Anal. 268, 1840–1875 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the \(C^*\)-algebras of reducible graphs. Ergodic Theory Dyn. Syst. 35, 2535–2558 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. an Huef, A., Raeburn, I.: Equilibrium states on graph algebras. In: Proceedings of 2015 Abel Symposium Operator Algebras and Applications, pp. 171–183. Springer, Berlin (2016)

  16. Ionescu, M., Kumjian, A.: Hausdorff measures and KMS states. Indiana Univ. Math. J. 62, 443–463 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kajiwara, T., Watatani, Y.: KMS states on finite-graph \(C^*\)-algebras. Kyushu J. Math. 67, 83–104 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kumjian, A., Pask, D.: Higher-rank graph \(C^*\)-algebras. N. Y. J. Math. 6, 1–20 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Laca, M., Raeburn, I.: Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225, 643–688 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Laca, M., Raeburn, I., Ramagge, J.: Phase transitions on Exel crossed products associated to dilation matrices. J. Funct. Anal. 261, 3633–3664 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. McNamara, R.: KMS States of Graph Algebras with Generalised Gauge Dynamics, PhD thesis, Univ. of Otago (2015)

  22. Raeburn, I.: Graph Algebras, CBMS Regional Conference Series in Math, vol. 103. American Mathematical Society, Providence (2005)

  23. Raeburn, I., Sims, A.: Product systems of graphs and the Toeplitz algebras of higher-rank graphs. J. Oper. Theory 53, 399–429 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Raeburn, I., Sims, A., Yeend, T.: Higher-rank graphs and their \(C^*\)-algebras. Proc. Edinburgh Math. Soc. 46, 99–115 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Raeburn, I., Sims, A., Yeend, T.: The \(C^*\)-algebras of finitely aligned higher-rank graphs. J. Funct. Anal. 213, 206–240 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Seneta, E.: Non-negative Matrices and Markov Chains, 2nd edn. Springer, New York (1981)

    Book  MATH  Google Scholar 

  27. Sims, A.: Gauge-invariant ideals in the \(C^*\)-algebras of finitely aligned higher-rank graphs. Can. J. Math. 58, 1268–1290 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sims, A., Whitehead, B., Whittaker, M.F.: Twisted \(C^*\)-algebras associated to finitely aligned higher-rank graphs. Doc. Math. 19, 831–866 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Thomsen, K.: KMS weights on groupoid and graph \(C^*\)-algebras. J. Funct. Anal. 266, 2959–2988 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, 9054, New Zealand

    Astrid an Huef

  2. Department of Mathematics, Sungkyunkwan University, Seobu-ro 2066, Jangan-gu, Suwon, 16419, Republic of Korea

    Sooran Kang

  3. Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, 9054, New Zealand

    Iain Raeburn

Authors
  1. Astrid an Huef
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sooran Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Iain Raeburn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Astrid an Huef.

Additional information

This research was supported by the Marsden Fund of the Royal Society of New Zealand.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

an Huef, A., Kang, S. & Raeburn, I. KMS States on the Operator Algebras of Reducible Higher-Rank Graphs. Integr. Equ. Oper. Theory 88, 91–126 (2017). https://doi.org/10.1007/s00020-017-2356-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-017-2356-z

Mathematics Subject Classification

Keywords

Search

Navigation