A stronger version of Dixmier's averaging theorem and some applications

Let $M$ be a type $I{I}_1$ 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 integer $N\in N$ mutually orthogonal nonzero projections $E_j\in M$, $I={\sum }_{j=1}^N{E}_j$, such that $E_j{X}_i{E}_j=\tau \left(X_i\right)E_j$ for all $j=1,\cdots ,N$ and $i=1,\cdots ,n$. Equivalently, there is a unitary operator $U\in M$ such that $\frac{1}{N}{\sum }_{j=0}^{N-1}{U^{⁎}}^j{X}_i{U}^j=\tau \left(X_i\right)I$ for $i=1,\cdots ,n$. This result is a stronger version of Dixmier's averaging theorem for type ${\mathrm{II}}_1$ factors. As the first application, we show that all elements of trace zero in a type ${\mathrm{II}}_1$ factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [6]. As the second application, we prove that any self-adjoint element in a type ${\mathrm{II}}_1$ factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if $(M,\tau )$ is a finite factor, $X\in M$, then there exists a normal operator $N\in M$ and a nilpotent operator K such that $X=N+K$.