Abelian Self-commutators in Finite Factors

An abelian self-commutator in a C*-algebra A is an element of the form A = XX−XX, with X ∈ A, such that XX and XX commute. It is shown that, given a finite AW*-factor A, there exists another finite AW*-factor M of same type as A, that contains A as an AW*-subfactor, such that any self-adjoint element X ∈ M of quasitrace zero is an abelian self-commutator in M. 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 [6]) 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 the problem of characterizing the self-adjoint elements of trace zero, in terms of simpler ones. We wish to carry this investigation in a “Hilbert-space-free” framework, so instead of von Neumann factors, we are going to work within the category of AW*-algebras. Such objects were introduced in the 1950’s by Kaplansky ([4]) in an attempt to formalize the theory of von Neumann algebras without any use of pre-duals. Recall that A unital C*-algebra A is called an AW*-algebra, if for every non-empty set X ⊂ A, the left anihilator set L(X ) = { A ∈ A : AX = 0, ∀X ∈ X } is the principal right ideal generated by a projection P ∈ A, that is, L(X ) = AP . Much of the theory – based on the geometry of projections – works for AW*algebras exactly as in the von Neumann case, and one can classify the finite AW*factors into the types Ifin and II1, exactly as above, but using the following alternative result: any finite AW*-factor A posesses a unique normalized quasitrace qA. Recall that a quasitrace on a C*-algebra A is a map q : A → C with the following properties: (i) if A,B ∈ A are self-adjoint, then q(A+ iB) = q(A) + iq(B); 1991 Mathematics Subject Classification. Primary 46L35; Secondary 46L05.