We investigate extensions of the standard matrix trace $\\\\\\\\\\\\\\\\operatorname{tr} \\\\\\\\\\\\\\\\colon \\\\\\\\\\\\\\\\mathbb{M}_n \\\\\\\\\\\\\\\\to \\\\\\\\\\\\\\\\mathbb{C}$ to more general settings in operator algebras. This includes modifying the domain $\\\\\\\\\\\\\\\\mathbb{M}_n$, the codomain $\\\\\\\\\\\\\\\\mathbb{C}$, or both, replacing them with suitable operator algebras. We further study the existence (or absence) of faithful tracial states in this framework and discuss recent characterizations of tracial states on certain $C^*$-algebras and von Neumann algebras.

Keywords