Inequalities for determinants and characterization of the trace

Let $\operatorname{tr}$ be the canonical trace on the full matrix algebra ${\Cal M}_ n$ with unit $I$. We prove that if some analog of classical inequalities for the determinant and trace (or the permanent and trace) of matrices holds for a positive functional $\varphi $ on ${\Cal M}_n$ with $\varphi (I) = n$, then $\varphi = \operatorname{tr}$. Also, we generalize Fischer's inequality for determinants and establish a new inequality for the trace of the matrix exponential.