On finite sums of projections and Dixmier's averaging theorem for type \(\operatorname{II}_1\) factors (Q6585651)

Whether a positive bounded operator \(A\) on a separable Hilbert space \(\mathcal H\) can be expressed as the sum of a finite or infinite collection of projections has been a longstanding problem in the research of operator algebras. \textit{V.~Kaftal} et al. [Proc. Am. Math. Soc. 140, No.~9, 3219--3227 (2012; Zbl 1279.46040)] addressed the problem completely in the setting of type I or type~III factors. If \(\mathcal M\) is a type II factor, \textit{X.~Cao} et al. [J. Funct. Anal. 281, No.~5, Article ID 109088, 11~p. (2021; Zbl 1491.46055)] gave conditions for operators \(A \in \mathcal M\) which can be written as the sum of a finite or infinite collection of projections. Furthermore, \textit{H.~Halpern} et al. [Trans. Am. Math. Soc. 365, No.~5, 2409--2445 (2013; Zbl 1275.47088)] provided some equivalent condition for a positive operator can be written as a finite sum of projections. If \(\mathcal M\) is a type~II\(_1\) factor with trace \(\tau\), they showed that a positive operator \(A \in \mathcal M\) can be written as a sum of finite projections if \(\tau(A) > \tau(R_A)\).\N\NLet \(\mathcal M\) be a type II\(_1\) factor. Diximier's theorem states that the trace of any element \(A \in \mathcal M\) is in the closure of the convex hull of the unitary orbits \(\{U^*AU: U \in \mathcal M\), \(U^*U=UU^* =I \}\). The main result of the present paper is the following: For any self-adjoint element \(X \in \mathcal M\), there exists a unitary \(W \in \mathcal M\) with \(W^N = I\) such that \N\[\N\frac{1}{N} \sum\limits_{0 \le j \le N-1 } (W^*)^j X W^j = \tau(X)I.\N\]\NIt is proved that if \(A \in \mathcal M^+\) and \(\tau(A) > 1\), then \(A\) is a sum of finite projections, implying that \(A \in \mathcal M^+\) is a sum of finite projections if and only if \(\tau(A) \ge \tau(R_A)\), which completely answers the original problem. A stronger version of Dixmier's averaging theorem is given: For any \(X_1, X_2, \dots, X_n \in \mathcal M\), there exist unitary operators \(U_1, U_2, \dots, U_k\) such that \N\[\N\frac{1}{k}\sum\limits_{1 \le i \le k} U_i^*X_j U_i = \tau(X_j)I, \ \forall 1 \le j \le n.\N\]\NSuppose that \(A \in \mathcal M^+\) with \(\tau(A) \ge \tau(R_A)\). The authors first show that if \(A\) is the linear combination of two projections, then it is a sum of finitely many projections. Then, mathematical induction is used to show the same result for the case that \(A\) is the linear combination of finitely many projections. In general, consider the decomposition \N\[\NA = PA + (I-P)A=(1+\mu)\sum\limits_{1 \le i < + \infty} p_i + (1- \lambda)\sum\limits_{1 \le j < + \infty} q_i ,\N\]\Nwhere \(P = \chi_{[1, + \infty)}(A)\). By dealing with how to transfer the sum of the positive coefficients of an infinite number of projections into a finite sum of projections, the main result will be obtained.