Structure of bicentralizer algebras and inclusions of type $$\mathrm{III}$$ factors

Abstract

We investigate the structure of the relative bicentralizer algebra $$\mathrm{B}(N \subset M, \varphi )$$ for inclusions of von Neumann algebras with normal expectation where N is a type $$\mathrm{III}_1$$ subfactor and $$\varphi \in N_*$$ is a faithful state. We first construct a canonical flow $$\beta ^\varphi : \mathbf {R}^*_+ \curvearrowright \mathrm{B}(N \subset M, \varphi )$$ on the relative bicentralizer algebra and we show that the W$$^*$$-dynamical system $$(\mathrm{B}(N \subset M, \varphi ), \beta ^\varphi )$$ is independent of the choice of $$\varphi $$ up to a canonical isomorphism. In the case when $$N=M$$, we deduce new results on the structure of the automorphism group of $$\mathrm{B}(M,\varphi )$$ and we relate the period of the flow $$\beta ^\varphi $$ to the tensorial absorption of Powers factors. For general irreducible inclusions $$N \subset M$$, we relate the ergodicity of the flow $$\beta ^\varphi $$ to the existence of irreducible AFD 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.