Noncommutative pressure and the variational principle in Cuntz-Krieger-type C-algebras

Abstract

Let a be a self-adjoint element of an exact C*-algebra A and χ: A → a contractive completely positive map. We define a notion of dynamical pressure Pχ (a) which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu-Brown topological entropy and Neshveyev and Størmer unital-nuclear pressure. A variational inequality bounding Pχ(a) below by the free energies hσ(χ) + σ(a) with respect to the Sauvageot-Thouvenot entropy hσ(χ) is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C*-algebras furnish a framework for investigating the variational principle, which asserts the equality of Pχ(a) with the supremum of the free energies over all χ-invariant states. In one direction we extend Brown's result on the constancy of the Voiculescu-Brown entropy upon passing to the crossed product, and in another we show that the pressure of a self-adjoint element over the Markov subshift underlying the canonical map on the Cuntz-Krieger algebra OA is equal to its classical pressure. The latter result is extended to a more general setting comprising an expanded class of Cuntz-Krieger-type Pimsner algebras, leading to the variational principle for self-adjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz-Krieger algebras. © 2002 Elsevier Science (USA).