On finite sums of projections and Dixmier's averaging theorem for type ${\mathrm{I}\mathrm{I}}_1$ factors

Let $\mathcal{M}$ be a type $\mathrm{I}{\mathrm{I}}_1$ factor and let $\tau$ be the faithful normal tracial state on $\mathcal{M}$. In this paper, we prove that given an $X\in \mathcal{M}$, $X=X^{\ast }$, then there is a decomposition of the identity into $N\in \mathbb{N}$ mutually orthogonal nonzero projections $E_j\in \mathcal{M}$, $I={\sum }_{j=1}^N{E}_j$, such that $E_{j}X{E}_j=\tau (X)E_j$ for all $j=1,\cdots ,N$. Equivalently, there is a unitary operator $U \in $\mathcal{M}$$with$U^N=I$and$\frac{1}{N}\sum_{j=0}^{N-1}{U^}^jXU^j=\tau(X)I.$ As the first application, we prove that a positive operator $A\in \mathcal{M}$ can be written as a finite sum of projections in $\mathcal{M}$ if and only if $\tau (A)\ge \tau (R_A)$, where $R_A$ is the range projection of $A$. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if $X\in $\mathcal{M}$$,$X=X^$and$\tau(X)=0$,thenthereexistsanilpotentelement$Z \in $\mathcal{M}$$suchthat$X$istherealpartof$Z$. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let $X_1,\cdots ,X_n\in \mathcal{M}$. Then there exist unitary operators $U_1,\cdots ,U_k\in \mathcal{M}$ such that $\frac{1}{k}{\sum }_{i=1}^k{U}_i^{-1}{X}_j{U}_i=\tau (X_j)I,\mathrm{\forall }1\le j\le n$. This result is a stronger version of Dixmier’s averaging theorem for type ${\rm II}_1$ factors.