On Strong Ergodic Properties of Quantum Dynamical Systems

Abstract. We show that the the shift on the reduced C∗–algebras of RD–groups, including the free group on infinitely many generators, and the amalgamated free product C∗–algebras, enjoys the very strong ergodic property of the convergence to the equilibrium. Namely, the free shift converges, pointwise in the weak topology, to the conditional expectation onto the fixed–point subalgebra. Provided the invariant state is unique, we also show that such an ergodic property cannot be fulfilled by any classical dynamical system, unless it is conjugate to the trivial one–point dynamical system.