Characterization of tracial functionals on von Neumann algebras

It is proved that the inequality
$$\varphi(A) \le \varphi(|A +iB|) \text{ for all } A \in \mathscr{A}^{+} \text{ and } B \in \mathscr{A}^{sa}$$
characterizes tracial functionals among all positive normal functionals $\varphi$ on a von Neumann algebra $\mathscr{A}$. This strengthens the L. T. Gardner’s characterization (1979; see [1]). As a consequence, a criterion for commutativity of von Neumann algebras is obtained. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra via this inequality. Every faithful normal semifinite trace $\varphi$ on a von Neumann algebra $\mathscr{A}$ satisfies this relation. Let $|||·|||$ be a unitarily invariant norm on a unital $C^*$-algebra $\mathscr{A}$. Then $|||A||| \le |||A+iB||| \text{ for all } A \in \mathscr{A}^{+} \text{ and } B \in \mathscr{A}^{sa}$. For other trace characterizations see [2]–[7] and references therein.