On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors

Abstract

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given an X ∈ M, X = X * , then there is a decomposition of the identity into N ∈ N mutually orthogonal nonzero projections Ej ∈ M, I = N j=1 Ej, such that EjXEj = τ (X)Ej for all j = 1, • • • , N . Equivalently, there is a unitary operator U ∈ M with U N = I andAs the first application, we prove that a positive operator A ∈ M can be written as a finite sum of projections in M if and only if τ (A) ≥ τ (RA), 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 ∈ M, X = X * and τ (X) = 0, then there exists a nilpotent element Z ∈ M such that X is the real part of Z. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let X1, • • • , Xn ∈ M. Then there exist unitary operators U1,This result is a stronger version of Dixmier's averaging theorem for type II1 factors.

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

“…Lemma 2.2. (Proposition 1.5 in [8] or Lemma 3.2 in [7]) Let (M, τ ) be a type II 1 factor, A ∈ M + , τ (A) = 1 and N ∈ N. Then the following conditions are equivalent.…”

“…Theorem 2.1. (Theorem 1.1 in [7]) Let (M, τ ) be a type II 1 factor, X ∈ M, X = X * . Then we have the following equivalent results.…”

“…Let M be a type II 1 factor and let τ be the faithful normal tracial state on M. In [7], the following result is proved. If X = X * ∈ M, then there is a decomposition of the identity into N ∈ N mutually orthogonal nonzero projections E j , I = N j=1 E j , for which E j XE j = τ (X)E j for all j = 1, ..., N, equivalently, there is a unitary Shilin Wen was partly supported by NSFC(Grant No.12001437) and the Fundamental Research Funds of the China West Normal University (21E027).…”