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 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].

Strict Comparison of Positive Elements in Multiplier Algebras

“…The same property holds for all von Neumann algebras without a finite type I direct summand with infinite dimensional center [8]. However this property may fail even for C*-algebras of real rank zero (see [16,Proposition 5.1]).…”

“…In [11] and [15] we investigated the notion of PCP in the setting of purely infinite C*-algebras and W*-algebras respectively (see also [12], [13], [14]). Focusing then on finite algebras, we proved in [16,Theorem 6.1] that if A is a simple separable stable σ-unital C*-algebra with real rank zero, stable rank one, strict comparison of projections by traces and has finitely many extremal traces, then a ∈ A + is a PCP if and only if τ (R a ) < ∞ for all τ ∈ T (A), where R a denotes the range projection.…”

Strict comparison of projections and positive combinations of projections in certain multiplier algebras

“…More complex is in C*-algebras of finite types. In [17] under the assumption that A is simple, separable, unital C*-algebras with real rank zero, stable rank one, strict comparison of projections, and finitely many extremal tracial states, we proved that a positive element a ∈ A ⊗ K is a PCP if and only ifτ (R a ) < ∞ for every tracial state τ on A, whereτ denotes the extension of τ ⊗ Tr to a normal semifinite trace on (A⊗K) * * and R a denotes the range projection of a in (A⊗K) * * .…”

“…Aim of the present paper is to investigate Problem (C) for positive elements of the multiplier algebra M(A ⊗ K) for a C*-algebra A of finite type considered in [17]. Of independent interest we also consider (A) and (B) for in the corners of M(A ⊗ K), as necessary steps in investigating (C).…”

“…A key tool in [17] was to the strict comparison of projections in A and in A⊗K. In Section 3 we establish a certain strict comparison of projections in M(A ⊗ K) based on the strict comparison of projections in A.…”