Commutators and certain II 1-factors

Abstract

Recently some progress has been made in the structure theory of commutators in von Neumann algebras [I]-[3]. The theory is far from complete, however, and one of the most intractable of the unsolved problems, is that of determining the commutators in a finite von Neumann algebra. A commutator in a finite von Neumann algebra must, of course, have central trace zero, and is not unreasonable to hope that the commutators in such an algebra are exactly the operators with central trace zero. However, despite considerable effort, this has been proved only in case the algebra is a finite direct sum of algebras of type I, [4]. In this note, we consider a certain class of factors of type II, discovered by Wright [13], and we show that every Hermitian operator with trace zero in such a factor is a commutator in the factor. This is accomplished by first proving that every Hermitian operator with central trace zero in an arbitrary finite von Neumann algebra of type I is a commutator in the algebra. Finally, we turn our attention to the problem of characterizing the linear manifold [GY, ac] p s anned by the commutators in an arbitrary von Neumann algebra 67 of type II, . We give three characterizations; in particular we show that [G?, ol] coincides with the set of all linear combinations C& ariEi where Cbi Qi = 0 and each E4 is equivalent in GZ to I Ei .