Full factors, bicentralizer flow and approximately inner automorphisms

Abstract

We show that a factor $M$ is full if and only if the $C^*$-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type $\mathrm{III}_1$ factor is always ergodic. As a consequence, for any type $\mathrm{III}_1$ factor $M$ and any $\lambda \in ]0,1]$, there exists an irreducible AFD type $\mathrm{III}_\lambda$ subfactor with expectation in $M$. Moreover, any type $\mathrm{III}_1$ factor $M$ which satisfies $M \cong M \otimes R_\lambda$ for some $\lambda \in ]0,1[$ has trivial bicentralizer. Finally, we give a counter-example to the characterization of approximately inner automorphisms conjectured by Connes and we prove a weaker version of this conjecture. In particular, we obtain a new proof of Kawahigashi-Sutherland-Takesaki's result that every automorphism of the AFD type $\mathrm{III}_1$ factor is approximately inner.