On finite sums of projections and Dixmiers averaging theorem for type ${rm II}_1$ factors

Let $mathcal{M}$ be a type ${rm II_1}$ factor and let $tau$ be the faithful normal tracial state on $mathcal{M}$. In this paper, we prove that given an $X in mathcal{M}$, $X=X^*$, then there is a decomposition of the identity into $N in mathbb{N}$ mutually orthogonal nonzero projections $E_jinmathcal{M}$, $I=sum_{j=1}^NE_j$, such that $E_jXE_j=tau(X) E_j$ 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 $Ain mathcal{M}$ can be written as a finite sum of projections in $mathcal{M}$ if and only if $tau(A)geq tau(R_A)$, 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 $Xin mathcal{M}$, $X=X^*$ and $tau(X)=0$, then there exists a nilpotent element $Z in mathcal{M}$ such that $X$ is the real part of $Z$. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let $X_1,cdots,X_nin mathcal{M}$. Then there exist unitary operators $U_1,cdots,U_kinmathcal{M}$ such that $frac{1}{k}sum_{i=1}^kU_i^{-1}X_jU_i=tau(X_j)I,quad forall 1leq jleq n$. This result is a stronger version of Dixmiers averaging theorem for type ${rm II}_1$ factors.