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

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 étale groupoid ${\mathrm{C}}^{\ast }$-algebras. We also answer the Trace Question positively for a large class of ${\mathrm{C}}^{\ast }$-algebras.