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 θ:A→A a contractive completely positive map. We define a notion of dynamical pressure Pθ(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 Pθ(a) below by the free energies hσ(θ)+σ(a) with respect to the Sauvageot–Thouvenot entropy hσ(θ) 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 Pθ(a) with the supremum of the free energies over all θ -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.