Connes’ bicentralizer conjecture for type III factors

Determine whether, for every type III factor M and every faithful normal state ϕ on M, the bicentralizer Bϕ(M) is trivial, i.e., Bϕ(M) = C1. This is Connes’ bicentralizer problem, which remains unresolved in general beyond the amenable case.

Background

The paper recalls the definitions of the asymptotic centralizer and bicentralizer for a σ-finite von Neumann algebra (M, ϕ), and notes that the triviality of the bicentralizer does not depend on the choice of faithful normal state. Connes introduced the bicentralizer problem in the course of classifying amenable factors and conjectured trivial bicentralizer for type III factors.

Haagerup solved Connes’ bicentralizer problem for amenable type III factors, but the problem is still open for non-amenable factors. This article contributes by proving trivial bicentralizer for mixed q-deformed Araki-Woods algebras under type III assumptions, thereby adding non-amenable examples where the bicentralizer is trivial, but not resolving the conjecture in full generality.