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 with τ ( a ) = 0 for all τ ∈ T ( A ) 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 σ -unital and not necessarily 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 τ ¯ ( R a ) ∞ for every τ ∈ T ( A ) , where τ ¯ denotes the extension of τ to a tracial weight on A ⁎ ⁎ and R a ∈ A ⁎ ⁎ denotes 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 in [32] .