The classification of flows on $\mathrm{II}_1$ factors and Connes' bicentralizer problem

Abstract

We solve two long-standing open problems in von Neumann algebras. First, we classify all flows with full Connes spectrum on the hyperfinite $\mathrm{II}_1$ factor up to cocycle conjugacy. Every such flow is cocycle conjugate to an irrational rotation flow on a noncommutative torus. In particular, there is a unique outer flow with full Connes spectrum up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite $\mathrm{II}_1$ factor. To prove this result, we draw on type $\mathrm{III}$ theory. Notably, we develop bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of the Connes--Størmer transitivity theorem and we generalize the Connes--Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of $\mathbb{R}$. We then go back to the type $\mathrm{III}$ world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type $\mathrm{III}_1$ factors.