Characterizations of the canonical trace on full matrix algebras

We establish that a positive linear functional on the full matrix algebra $M_n$ is a positive multiple of the canonical trace if and only if $\phi \left(A\right)=\phi \left(\right|A\left|\right)$ implies that A is positive semidefinite. Furthermore, we characterize the canonical trace on $M_n$ among all positive linear functionals φ on $M_n$ with $\phi \left(I\right)=n$ via Yang's inequality $\phi {\left(A^{1/2}B{A}^{1/2}\right)}^{1/2}\le \phi (A+B)/2$, where $A,B\in M_n$ are positive semidefinite matrices.