By employing some powerful techniques, we establish several characterizations of the usual trace on the algebra Mn of complex square matrices. Among them, we prove that any one of (i) the commutator inequality φ(|XY − YX|) ≤ φ(X 2 + Y 2 ) holds for all Hermi- tian matrices X , Y ∈ Mn , and (ii) the generalized arithmetic–geometric mean inequality φ(|AXB∗ |) ≤ φ((|X ∗ | |A| p 2 p |X ∗ |)1/2 ) + φ((|B| p |X q |2 |B| p )1/2 ) holds for all A, B, X ∈ Mn and all conjugate exponents p > 0 and q > 0, characterize the usual trace on Mn .

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