Characterization of the trace by monotonicity inequalities (Q869935)

Let \(\phi \) be a positive linear functional on the algebra of \(n\times n\) complex matrices and let \(A,B\) be any pair of \(n\times n\) positive semi-definite matrices such that \(A\leq B.\) The authors prove that if the inequality \(\phi (f(A))\leq \phi (f(B))\) holds true when \(f(x)= x^p, p>1,\) or \(f(x)= (x+\lambda )^p, p>1, \lambda >0,\) or \(f(x)=e^x,\) then \(\phi\) is a nonnegative scalar multiple of the trace. These results supplement a number of inequalities which characterize the trace.