Characterization of the trace by young's inequality

Let φ be a positive linear functional on the algebra of n × n complex matrices and p, q be positive numbers such that 1/p + 1/q = 1. We prove that if for any pair A, B of positive semi-definite n × n matrices the inequality φ(|AB|) ≤ φ/(Ap)/p + φ/(Bq)/q holds, then φ is a positive scalar multiple of the trace. © 2005 Victoria University. All rights reserved