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

Abstract

Let $\M$ be a type $\mathrm{I}{\mathrm{I}}_1$ 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 $E_j\in\M$, $I={\sum }_{j=1}^N{E}_j$, such that $E_j{X}_i{E}_j=\tau (X_i)E_j$ for all $j=1,\cdots ,N$ and $i=1,\cdots ,n$. Equivalently, there is a unitary operator $U \in \M$ such that $\frac{1}{N}{\sum }_{j=0}^{N-1}{U^{\ast }}^j{X}_i{U}^j=\tau (X_i)I$ 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 [10]. 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 [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].

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