Pressure for automorphisms of exact C*-algebras and a noncommutative variational principle

A notion of pressure with respect to a self-adjoint element is introduced for automorphisms of exact C*-algebras and a number of properties are established, including a generalization of a theorem of N. Brown for entropy asserting that the pressure remains the same upon passing to the extension of an automorphism to the crossed product. A variational inequality bounding the pressure below by the CNT and Sauvageot-Thouvenot free energies is obtained in two stages via a local state approximation entropy, which is shown to be an extension of M. Choda's nuclear entropy. We prove the variational principle for certain asymptotically Abelian automorphisms and introduce the class of weakly AF C*-algebras in order to describe a special subclass of these automorphisms.