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

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 $\mathrm{I}\mathrm{I}{\mathrm{I}}_1$ factor with trivial bicentralizer, then $(M,\phi )$ is selfless for every faithful normal state $\phi \in M_{\ast }$.