Self-adjoint traces on the Pedersen ideal of $\mathrm{C}^\ast$-algebras

TL;DR

This paper systematically studies self-adjoint traces on the Pedersen ideal of C*-algebras, establishing duality, computing examples, and positively addressing the Trace Question for many cases.

Contribution

It introduces a framework for analyzing self-adjoint traces on the Pedersen ideal, including duality results and explicit computations for certain groupoid C*-algebras.

Findings

Established Kadison duality for $T_{ ext{R}}(A)$

Computed $T_{ ext{R}}(A)$ for principal twisted étale groupoid C*-algebras

Positively answered the Trace Question for a large class of C*-algebras

Abstract

In order to circumvent a fundamental issue when studying densely defined traces on $\mathrm{C}^\ast$-algebras -- which we refer to as the Trace Question -- we initiate a systematic study of the set $T_{\mathbb R}(A)$ of self-adjoint traces on the Pedersen ideal of $A$. The set $T_{\mathbb R}(A)$ is a topological vector space with a vector lattice structure, which in the unital setting reflects the Choquet simplex structure of the tracial states. We establish a form of Kadison duality for $T_{\mathbb R}(A)$ and compute $T_{\mathbb R}(A)$ for principal twisted \'etale groupoid $\mathrm{C}^\ast$-algebras. We also answer the Trace Question positively for a large class of $\mathrm{C}^\ast$-algebras.