Inequalities for Determinants and Characterization of the Trace

Abstract

Let 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 φ on $${{\cal M}_n}$$ with φ(I) = n, then φ = tr. Also, we generalize Fischer’s inequality for determinants and establish a new inequality for the trace of the matrix exponential.