On Diximier's averaging theorem for operators in type ${\mathrm{II}}_1$ factors

Shilin Wen, Junsheng Fang, Zhaolin Yao

Let $\M$ be a type $\mathrm{I}{\mathrm{I}}_1$ factor and let $\tau$ be the faithful normal tracial state on $\M$. In this paper, we prove that given finite elements $X_1,\cdots X_n \in \M$, there i…