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

TL;DR

This paper investigates the structure of self-adjoint and positive elements in certain finite C*-algebras, establishing conditions under which these elements can be expressed as sums or linear combinations of projections, and resolving two open problems.

Contribution

It proves that in specific finite C*-algebras, elements in the kernel of all traces are sums of commutators, and positive elements are linear combinations of projections, settling two open problems.

Findings

Self-adjoint elements in the kernel of all traces are sums of two commutators.

Positive elements are linear combinations of projections with positive coefficients.

Algebras with infinitely many extremal traces are not spanned by projections.

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.