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

Assume that AA 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)T(A) has a finite number of extremal points. We prove that every self-adjoint element a   in AA with τ(a)=0τ(a)=0 for all τ∈T(A)τ∈T(A) is the sum of two commutators in AA and that every positive element of AA is a linear combination of projections with positive coefficients. Assume that AA is as above but σ-unital and not necessarily unital. Then an element (resp. a positive element) a   of AA is a linear combination (resp. a linear combination with positive coefficients) of projections if and only if τ¯(Ra)<∞ for every τ∈T(A)τ∈T(A), where τ¯ denotes the extension of τ   to a tracial weight on A⁎⁎A⁎⁎ and Ra∈A⁎⁎Ra∈A⁎⁎ denotes the range projection of a  . Assume that AA is unital and as above but T(A)T(A) has infinitely many extremal points. Then AA is not the linear span of its projections. This result settles two open problems of Marcoux in [32].