Characterizations of tracial functionals on matrix and operator algebras

This paper is a comprehensive survey of characterizations of the tracial functionals among all positive linear functionals on the full matrix algebras,\linebreak $C^*$-algebras, and von Neumann algebras. It also includes new characterizations of the tracial property. We explore Thompson's triangle inequality and demonstrate its connection to the characterization of the tracial property. Moreover, we provide rather simple proofs for various characterizations of the standard trace on the full matrix algebra $\mathbb{M}_n$. We establish a new characterization of the tracial functionals in the framework of $C^*$-algebras by showing that the following conditions are equivalent for a state $\varphi$ on a $C^*$-algebra $\mathscr{A}$: (i) $\varphi$ is tracial; (ii) $\varphi (A^{1/2}BA^{1/2}+\lambda (AB+BA)) \geq 0 $ for some number $\lambda \geq -1/2$, $\lambda \neq 0$ and all $A, B \in \mathscr{A}^+$. As a consequence, we introduce a new criterion for the commutativity of $C^*$-algebras.