A stronger version of Dixmier's averaging theorem and some applications

Let M be a type II1 1 factor and let tau be the faithful normal tracial state on M . In this paper, we prove that given finite elements X 1 , center dot center dot center dot , Xn n E M , there is a finite decomposition of the identity into integer N E N mutually orthogonal nonzero projections E j E M , I = N j =1 E j , such that EjXiEj j X i E j = tau(Xi)Ej ( X i ) E j for all j = 1 , center dot center dot center dot , N and i = 1 , center dot center dot center dot, , n . Equivalently, there is a unitary operator U E M such that 1 N -1 j =0 U * j X i U j = tau(Xi)I ( X i ) I for i = 1, , center dot center dot center dot, , n . This result is a N stronger version of Dixmier's averaging theorem for type II1 1 factors. As the first application, we show that all elements of trace zero in a type II1 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 [6]. As the second application, we prove that any self-adjoint element in a type II1 1 factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if ( M , tau ) isa finite factor, X is an element of M , then there exists a normal operator N is an element of M and a nilpotent operator K such that X = N + K . (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.