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 that every positive element of A is a linear combination of projections with positive coefficients. Assume that A is as above but σ−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 ifτ (Ra) < ∞ for every τ ∈ T(A), and if and only if , whereτ denotes the extension of τ to a tracial weight on A * * and Ra ∈ 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 [30].

We prove the following: Let A and B be separable C * -algebras. Suppose that B is a type I C * -algebra such that (i) B has only infinite dimensional irreducible * -representations, and (ii) B has finite decomposition rank.is a unital homogeneous exact sequence with Busby invariant , then the extension is absorbing.In the case of infinite decomposition rank, we provide a counterexample. Specifically, we construct a unital, homogeneous, split exact sequence of the formwhich is not absorbing. In this example, Z is an infinite-dimensional, compact, second countable topological space. This gives a counterexample to the natural infinite-dimensional generalization of the result of Pimsner, Popa and Voiculescu.

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The corona factorization property is a property with connections to extension theory, K-theory and the structure of C * -algebras. This paper is a short survey of the subject, together with some new results and open questions.

In this paper we investigate whether positive elements in the multiplier algebras of certain finite C*-algebras can be written as finite linear combinations of projections with positive coefficients (PCP). Our focus is on the category of underlying C*-algebras that are separable, simple, with real rank zero, stable rank one, finitely many extreme traces, and strict comparison of projections by the traces. We prove that the strict comparison of projections holds also in the multiplier algebra M(A ⊗ K). Based on this result and under the additional hypothesis that M(A ⊗ K) has real rank zero, we characterize which positive elements of M(A ⊗ K) are of PCP.