Ergodic states on type III${}_1$ factors and ergodic actions

Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra $M$ that admits a state $\phi$ with trivial centralizer $M_{\phi }$ must be a type III${}_1$ factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense $G_{\delta }$ set among all faithful normal states on any III${}_1$ 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.