Characterization of the trace by monotonicity inequalities

Let ϕbe a positive linear functional on the algebra of n × n complex matrices and p be a number greater than 1. The main result of the paper says that if for any pair A, B of positive semi-definite n × n matrices with A ⩽ B the inequality ϕ(Ap) ⩽ ϕ(Bp) holds true, then ϕis a nonnegative scalar multiple of the trace.