Real Analysis Exchange

KMS States, Entropy, and a Variational Principle for Pressure

Artur O. Lopes and Gilles G. de Castro

Full-text: Open access

Abstract

We relate the concepts of entropy and pressure to that of KMS states for $C^*$-algebras. Several different definitions of entropy are known in our days: the one we present here is quite natural, extending the usual one for Dynamical Systems in Thermodynamic Formalism Theory, being basically obtained from transfer operators (also called Ruelle operators) and having the advantage of being very easily introduced. We also present a concept of pressure as a min-max principle. Later on, we consider the concept of a KMS state as an equilibrium state for a potential, in the context of $C^*$-algebras, and we show that there is a relation between equilibrium measures and KMS states for certain algebras arising from a continuous transformation.

Article information

Source
Real Anal. Exchange, Volume 34, Number 2 (2008), 333-346.

Dates
First available in Project Euclid: 29 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1256835191

Mathematical Reviews number (MathSciNet)
MR2569191

Zentralblatt MATH identifier
1189.37009

Subjects
Primary: 37A35: Entropy and other invariants, isomorphism, classification 37A55: Relations with the theory of C-algebras [See mainly 46L55] 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 47B48: Operators on Banach algebras

Keywords
entropy pressure KMS states transfer operator

Citation

de Castro, Gilles G.; Lopes, Artur O. KMS States, Entropy, and a Variational Principle for Pressure. Real Anal. Exchange 34 (2008), no. 2, 333--346. https://projecteuclid.org/euclid.rae/1256835191


Export citation

References

  • R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 153–170.
  • O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, 2, Equilibrium states. Models in quantum statistical mechanics. Second edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  • J.-P. Conze, J.-P. and A. Raugi, Fonctions harmoniques pour un operateur de transition e applications, Bull. Soc. Math. France, 118(4) (1990), 273–310.
  • D. E. Dutkay and P. E. T. Jorgensen, Iterated function systems, Ruelle operators, and invariant projective measures, Math. Comp., 75(256) (2006), 1931–1970.
  • D. E. Dutkay and P. E. T. Jorgensen, Disintegration of projective measures, Proc. Amer. Math. Soc., 135(1) (2007), 169–179.
  • R. Exel, A new look at the crossed-product of a $C^*$-algebra by an endomorphism, Ergodic Theory Dynam. Systems, 23(6) (2003), 1733–1750.
  • R. Exel, Crossed-products by finite index endomorphisms and KMS states, J. Funct. Anal., 199(1) (2003), 153–188.
  • R. Exel, KMS states for generalized Gauge actions on Cuntz-Krieger Algebras, Bull. Braz. Math. Soc., 35(1) (2004), 1–12.
  • R. Exel and A. O. Lopes, $C^*$-algebras, approximately proper equivalence relations, and Thermodynamic Formalism, Ergodic Theory Dynam. Systems, 24 (2004), 1051–1082.
  • D. Kerr and C. Pinzari, Noncommutative pressure and the variational principle in Cuntz-Krieger-type $C^*$-algebras, J. Funct. Anal., 188 (2002), 156–215.
  • A. Kumjian and J. Renault, KMS states on $C^*$-algebras associated to expansive maps, Proc. Amer. Math. Soc., 134(7) (2006), 2067–2078.
  • B. K. Kwaśniewski, On transfer operators for $C^*$-dynamical systems..
  • A. O. Lopes, An analogy of the charge distribution on Julia sets with the Brownian motion, J. Math. Phys., 30(9) (1989), 2120–2124.
  • A. O. Lopes and E. R. Oliveira, Entropy and variational principles for holonomic probabilities of IFS, Discrete Contin. Dyn. Syst. Ser. A, 23(3) (2009), 937–955.
  • A. O. Lopes and P. Thieullen, Mather measures and the Bowen-Series transformation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 663–682.
  • D. Mauldin and M. Urbanski, Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003.
  • W. de Melo and S. Van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25, Springer-Verlag, Berlin, 1993.
  • S. Neshveyev and E. Størmer, Dynamical Entropy in Operator Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 50, Springer-Verlag, Berlin, 2006.
  • W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, (French Summary), Astérisque, 187-188 (1990).
  • G. K. Pedersen, $C\sp{\ast} $-algebras and their Automorphism Groups,
  • C. Pinzari, Y. Watatani, and K. Yonetani, KMS states, entropy and the variational principle in full $C^*$-dynamical systems, Comm. Math. Phys., 213 (2000), 331–379.
  • J. Renault, KMS states on $C^*$-algebras associated to expansive maps, Proc. Amer. Math. Soc., 134(7) (2006), 2067–2078.
  • D. Ruelle, Statistical Mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267–278.
  • D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota, Encyclopedia of Mathematics and its Applications, 5, Addison-Wesley, Reading, Mass., 1978.
  • D. Ruelle, The Thermodynamic Formalism for expanding maps, Comm. Math. Phys., 125 (1989), 239–262.