By employing some powerful techniques, we establish several characterizations of the usual trace on the algebra \(\mathbb {M}_n\) of complex square matrices. Among them, we prove that any one of (i) the commutator inequality \(\varphi (|{ XY}-{ YX}|)\le \varphi (X^2+Y^2)\) holds for all Hermitian matrices \(X, Y \in \mathbb {M}_n\) , and (ii) the generalized arithmetic–geometric mean inequality \(\varphi (|{ AXB}^*|)\le \frac{\varphi ((|X^*|\,|A|^{2p}|X^*|)^{1/2})}{p}+\frac{\varphi ((|B|^p|X|^2|B|^p)^{1/2})}{q}\) holds for all \(A, B, X \in \mathbb {M}_n\) and all conjugate exponents \(p>0\) and \(q>0\) , characterize the usual trace on \(\mathbb {M}_n\) .
