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

TL;DR

This paper studies the structure of positive elements in multiplier algebras of certain finite C*-algebras, establishing conditions under which they can be expressed as positive combinations of projections.

Contribution

It proves that strict comparison of projections extends to multiplier algebras and characterizes positive elements as PCP under real rank zero conditions.

Findings

Strict comparison of projections holds in the multiplier algebra.

Characterization of positive elements as PCP in multiplier algebras.

Extension of comparison properties to multiplier algebras.

Abstract

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 of the stabilizer algebra. Based on this result and under the additional hypothesis that the multiplier algebra has real rank zero, we characterize which positive elements of the multiplier algebra are PCP.