Structure of bicentralizer algebras and inclusions of type III factors

Abstract

We investigate the structure of the relative bicentralizer algebra $\mathrm{B}(N\subset M,\phi )$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type $\mathrm{II}{\mathrm{I}}_1$ subfactor and $\phi \in N_{\ast }$ is a faithful state. We first construct a canonical flow ${\beta }^{\phi }:{\mathbf{R}}_{+}^{\ast }\curvearrowright \mathrm{B}(N\subset M,\phi )$ on the relative bicentralizer algebra and we show that the W${}^{\ast }$-dynamical system $(\mathrm{B}(N\subset M,\phi ),{\beta }^{\phi })$ is independent of the choice of $\phi$ up to a canonical isomorphism. In the case when $N=M$, we deduce new results on the structure of the automorphism group of $\mathrm{B}(M,\phi )$ and we relate the period of the flow ${\beta }^{\phi }$ to the tensorial absorption of Powers factors. For general irreducible inclusions $N\subset M$, we relate the ergodicity of the flow ${\beta }^{\phi }$ to the existence of irreducible hyperfinite subfactors in $M$ that sit with normal expectation in $N$. When the inclusion $N\subset M$ is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when $N$ is amenable.

37 pages. v2: minor modifications, a few results added

References in corpus (1)

• Fullness and Connes' $\mathbf{\tau }$ invariant of type III tensor product factors

Cited by in corpus (4)

• Connes' bicentralizer problem for q-deformed Araki-Woods algebras
• Ergodic states on type III${}_1$ factors and ergodic actions
• On the weak relative Dixmier property
• Unique prime factorization for infinite tensor product factors