Structure of bicentralizer algebras and inclusions of type III factors (Mathematical aspects of quantum fields and related topics)

Abstract

We investigate the structure of the relative bicentralizer algebra B(N ⊂ M, φ) for inclusions of von Neumann algebras with normal expectation where N is a type Ill₁ subfactor and φ ∈ N* is a faithful state. We first construct a canonical flow β[φ] : R*₊ ↷ B(N C M, ip) on the relative bicentralizer algebra and we show that the W*-dynamical system (B(N ⊂ M, φ), β[φ]) is independent of the choice of ip up to a canonical isomorphism. In the case when N = M, we deduce new results on the structure of the automorphism group of B(M, φ) and we relate the period of the flow β[φ] to the tensorial absorption of Powers factors. For general irreducible inclusions N ⊂ M, we relate the ergodicity of the flow β[φ] to the existence of irreducible hyperfinite subfactors in M that sit with normal expectation in N. When the inclusion N ⊂ M is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when N is amenable.