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

Let \(\M\) be a type \({\rm II_1}\) 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 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 \(E_jX_iE_j=\tau(X_i) E_j\) for all \(j=1,\cdots,N\) and \(i=1,\cdots,n\). Equivalently, there is a unitary operator \(U \in \M\) such that \(\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jX_iU^j=\tau(X_i)I\) 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 \((\mathcal{M},\tau)\) 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].