ar X iv : m at h / 04 08 27 5 v 2 [ m at h . O A ] 6 S ep 2 00 4 SPECTRAL SYMMETRY IN II 1 - FACTORS

A self-adjoint element in a finite AW*-factor is spectrally symmetric, if its spectral measure under the quasitrace is invariant under the change of variables t 7−→ −t. We show that if A is an AW*-factor of type II1, a self-djoint element in A, without full support, has quasitrace zero, if and only if it can be written as a sum of at most three commuting spectrally symmetric elements. Introduction According to the Murray-von Neumann classification, finite von Neumann factors are either of type Ifin, or of type II1. For the non-expert, the easiest way to understand this classification is by accepting the famous result of Murray and von Neumann (see [9]) which states that every finite von Neumann factor M posesses a unique state-trace τM. Upon accepting this result, the type of M is decided by so-called dimension range: DM = { τM(P ) : P projection in M } as follows. If DM is finite, then M is of type Ifin (more explictly, in this case DM = { k n : k = 0, 1, . . . , n } for some n ∈ N, and M ≃ Matn(C) – the algebra of n × n matrices). If DM is infinite, then M is of type II1, and in fact one has DM = [0, 1]. From this point of view, the factors of type II1 are the ones that are interesting, one reason being the fact that, although all factors of type II1 have the same dimension range, there are uncountably many non-isomorphic ones (by a celebrated result of Connes). In this paper we deal with a very simple problem. We start with a von Neumann II1-factor M, a (self-adjoint) element A ∈ M, and we wish to characterize the condition: τM(A) = 0. The main feature of the trace τM is (1) τM(XY − Y X) = 0, ∀X,Y ∈ M, so a sufficient condition for τM(A) = 0 is that A be expressed as a sum of commutators, i.e. of elements of the form [X,Y ] = XY − Y X with X,Y ∈ M. A remarkable result due to Fack and de la Harpe ([3]) states not only that this condition is sufficient, but if A = A then A can be written as a sum of at most five commutators of the form [X,X]. The aim of this paper is to characterize the condition τM(A) = 0 in a way that is “Hilbert space free.” What we have in mind of course is the purely algebraic setting due to Kaplansky ([5]), who attempted to formalize the theory of von Neumann algebras without any use of pre-duals. What emerged from Kaplansky’s work was the concept of AW*-algebras, which we recall below.