On finite sums of projections and Dixmier's averaging theorem for type II1 factors

Let M be a type II1 factor and let tau be the faithful normal tracial state on M. In this paper, we prove that given an X is an element of M, X = X*, then there is a decomposition of the identity into N is an element of N mutually orthogonal nonzero projections E-j is an element of M, I = Sigma(N)(j=1) E-j, such that Ej X E-j = tau(X)E-j for all j = 1, ... , N. Equivalently, there is a unitary operator U is an element of M with U-N = I and 1/N Sigma(N-1)(j=0) U*X-j U-j = tau(X)I. As the first application, we prove that a positive operator A E M can be written as a finite sum of projections in M if and only if tau(A) >= tau(R-A), where RA is the range projection of A. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if X is an element of M, X = X* and tau(X) = 0, then there exists a nilpotent element Z is an element of M such that Xis the real part of Z. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that for X-1, ... , X-n is an element of M, there exist unitary operators U-1, ... , U-k is an element of M such that 1/k Sigma(k)(i=1) U-i* XjUi = tau(X-j)I, for all(1) <= j <= n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.