We establish several characterizations of tracial functionals φ on the finite C ∗ -algebra M n (that is, φ = k tr for some number k > 0) via any one of the inequalities φ(Aα B 1−α ) ≤ αφ(A) + (1 − α)φ(B) and φ(|eA |) ≤ φ(eReA ), which are well-known when φ = tr. In addition, we characterize the trace on M n among all positive linear functionals φ with φ(I) = n through an inequality for determinant. We also establish that such a 1 1 functional is equal to the usual trace if and only if φ((B 2 AB 2 )m ) ≤ φ(A)m φ(B)m for all positive integers m and all A, B ∈ M+ n . Furthermore, we show that there is no state φ on Mn , n ≥ 2 such that φ(B1/2 AB1/2 ) ≤ φ(A)φ(B) for all A, B ∈ M+ n . Finally, we establish that for a positive linear functional φ on a C ∗ -algebra A, the following conditions are equivalent: (i) φ is tracial; (ii) φ(eAB −I) ≥ 0 for all A, B ∈ A+ . ∗A new criterion for the commutativity of C -algebras is also provided.

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