Ergodic states on type III1 factors and ergodic actions

Abstract

Abstract Since the early days of Tomita–Takesaki theory, it is known that a von Neumann algebra 𝑀 that admits a state 𝜑 with trivial centralizer M φ M_{\varphi} must be a type III1 factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense G δ G_{\delta} set among all faithful normal states on any III1 factor with separable predual. Through Connes’ Radon–Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper.