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

Abstract

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.

v3 is the final version, with minor changes compared to v2, and will appear in the Journal fur die Reine und Angewandte Mathematik

References in corpus (2)

• Classification of type III Bernoulli crossed products
• Spectral gap and strict outerness for actions of locally compact groups on full factors