Abelian self-commutators in finite factors

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An abelian self-commutator in a C*-algebra A is an element A that can be written as A=X*X-XX*, with X such that X*X 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 of quasitrace zero is an abelian self-commutator in M.