. O A ] 1 8 D ec 1 99 9 KMS STATES , ENTROPY AND THE VARIATIONAL PRINCIPLE IN FULL C ∗ – DYNAMICAL SYSTEMS

To any periodic and full C∗–dynamical system (A, α,R) a certain invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius theorem asserts the existence of KMS states at inverse temperatures the logarithm of the inner and outer spectral radii of s. Such KMS states are called extremal. Examples arising from subshifts in symbolic dynamics, self–similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Also, unital completely positive maps σ{xj} implemented by partitions of unity {xj} of grade 1 are considered, resembling the ‘canonical endomorphism’ of the Cuntz algebras. The relationship between the Voiculescu topological entropy of σ{xj} and the topological entropy of the associated subshift is studied. Similarly, the measure–theoretic entropy of σ{xj}, in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure–theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy of subshifts is obtained for those maps σ{xj} for which {xj} generates a Matsumoto C∗–algebra. When {xj} generates a Cuntz–Krieger algebra, an explicit construction of states with maximal entropy from KMS states at maximal inverse temperatures is done. On leave of absence from Dipartimento di Matematica, Università di Roma Tor Vergata, 00133 Roma. Supported by the EU and NATO–CNR. Supported by the Grant–in–aid for Scientific Research of JSPS.