On the comparison of positive elements of a C*-algebra by lower semicontinuous traces

TL;DR

This paper establishes a precise relationship between positive elements of a C*-algebra and their evaluations on lower semicontinuous traces, linking trace agreement to Cuntz-Pedersen equivalence and characterizing trace functions.

Contribution

It provides a new characterization of positive elements based on their trace evaluations and extends the understanding of trace functions in stable C*-algebras.

Findings

Positive elements agree on all lower semicontinuous traces iff they are Cuntz-Pedersen equivalent.

A similar equivalence holds when elements are comparable via trace values.

Characterization of functions on the cone of traces arising from positive elements.

Abstract

It is shown in this paper that two positive elements of a C*-algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case where the two elements are comparable by their values on the lower semicontinuous traces. This result is used to give a characterization of the functions on the cone of lower semicontinuous traces of a stable C*-algebra that arise from positive elements of the algebra.