Characterization of the trace by monotonicity inequalities

Abstract

Let φ{symbol}be 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{symbol}(Ap) ≤ φ{symbol}(Bp) holds true, then φ{symbol}is a nonnegative scalar multiple of the trace. © 2006 Elsevier Inc. All rights reserved.