# Journal of Operator Theory

Volume 56, Issue 1, Summer 2006 pp. 111-142.

Sums of small number of commutators**Authors**: L.W. Marcoux

**Author institution:**Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

**Summary:**For many $C^*$-algebras $\cA$, techniques have been developed to show that all elements which have trace zero with respect to all tracial states can be written as a sum of finitely many commutators, and that the number of commutators required depends only upon the algebra, and not upon the individual elements. In this paper, we show that if the same holds for $q \cA q$ whenever $q$ is a ``sufficiently small'' projection in $\cA$, then every element that is a sum of finitely many commutators in $\cA$ is in fact a sum of two. We then apply this commutator reduction argument to certain $C^*$-algebras of real rank zero with a unique trace, as well as to a class of approximately homogeneous $C^*$-algebras whose $K_0$ group has large denominators. Finally, we use these results to show that many $C^*$-algebras are linearly spanned by their projections.

**Keywords:**commutators, $C^*$-algebra, real rank zero, approximately homogeneous, projections

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