The pressure in operator algebras

We introduce two notions of the pressure in operator algebras, one is the pressure P _α (π, T) for an automorphism α of a unital exact C _*-algebra at a self-adjoint element T in with respect to a faithful unital *-representation π, the other is the pressure P _τ,α (T) for an automorphism α of a hyperfinite von Neumann algebra at a self-adjoint element T in with respect to a faithful normal α-invariant state τ. We give some properties of the pressure, show that it is a conjugate invariant, and also prove that the pressure of the implementing inner automorphism of a crossed product ×_α ℤ at a self-adjoint operator T in equals that of α at T. Supported by the NNSF of China (Grant No. A0324614), NSF of Shandong (Grant No. Y2006A03) and NSF of QFNU (Grant No. xj0502)