Characterizing Schwarz maps by tracial inequalities

Abstract

Let phi be a positive map from the ntimes n matrices mathcal {M}_n to the mtimes m matrices mathcal {M}_m. It is known that phi is 2-positive if and only if for all Kin mathcal {M}_n and all strictly positive Xin mathcal {M}_n, phi (K^*X^{-1}K) geqslant phi (K)^*phi (X)^{-1}phi (K). This inequality is not generally true if phi is merely a Schwarz map. We show that the corresponding tracial inequality {{,textrm{Tr},}}[phi (K^*X^{-1}K)] geqslant {{,textrm{Tr},}}[phi (K)^*phi (X)^{-1}phi (K)] holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.