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