Selfless W${}^{\ast }$-probability spaces and Connes' bicentralizer problem

Abstract

We introduce the notion of selfless W${}^{\ast }$-probability space and study its connection with Connes' bicentralizer problem. In particular, we show that if $M$ is a separable type $\mathrm{II}{\mathrm{I}}_1$ factor with trivial bicentralizer, then $(M,\ddot{I}†)$ is selfless for every faithful normal state $\ddot{I}†\in M_{\ast }$.

5 pages. To appear in J. Math. Soc. Japan