Commutators and linear spans of projections in certain finite C*-algebras

Abstract

Assume that A is a unital separable simple C*-algebra with real rank zero, stable rank one, strict comparison of projections, and that its tracial simplex T(A) has a finite number of extremal points. We prove that every self-adjoint element a in A in the kernel of all tracial states is the sum of two commutators in A and that every positive element of A is a linear combination of projections with positive coefficients. Assume that A is as above but \sigma-unital. Then an element (resp. a positive element) a of A is a linear combination (resp. a linear combination with positive coefficients) of projections if and only if for all \tau in T(A), the extension \bar\tau to the enveloping von Neumann algebra has a finite value for the range projection of a. Assume that A is unital and as above but T(A) has infinitely many extremal points. Then A is not the linear span of its projections. This result settles two open problems of Marcoux.