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

Abstract

We settle two long-standing open problems in von Neumann algebras. First, we show that every outer flow with full Connes spectrum on the hyperfinite $\mathrm{II}_1$ factor has the Rokhlin property. By the work of Masuda and Tomatsu, such a flow is therefore unique up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite type $\mathrm{II}_1$ factor. Drawing on type $\mathrm{III}$ theory, we develop a 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 Connes-St\o rmer transitivity theorem and we generalize Connes-Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of $\R$. 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.