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

Let $\M$ be a type ${\rm II_1}$ factor and let $τ$ 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 is a finite decomposition of the identity into $N \in \NNN$ mutually orthogonal nonzero projections $E_j\in\M$, $I=\sum_{j=1}^NE_j$, such that $E_jX_iE_j=τ(X_i) E_j$ for all $j=1,\cdots,N$ and $i=1,\cdots,n$. Equivalently, there is a unitary operator $U \in \M$ such that $\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jX_iU^j=τ(X_i)I$ for $i=1,\cdots,n$. This result is a stronger version of Dixmier's averaging theorem for type ${\rm II}_1$ factors. As the first application, we show that all elements of trace zero in a type ${\rm II}_1$ factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [10]. As the second application, we prove that any self-adjoint element in a type ${\rm II}_1$ factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [15]. As the third application, we show that if $(\mathcal{M},τ)$ is a finite factor, $X \in \mathcal{M}$, then there exists a normal operator $N \in \mathcal{M}$ and a nilpotent operator $K$ such that $X= N+ K$. This result answers affirmatively Question 1.1 in [9].