Ergodic states on type III₁ 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 𝜑 with trivial centralizer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>M</m:mi> <m:mi>φ</m:mi> </m:msub> </m:math> Mφ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi>δ</m:mi> </m:msub> </m:math> Gδ 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.