2016/12/09 by Lawson, Jimmie
#28C20 #60B05 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1612.03213
Let C be an open cone in a Banach space equipped with the Thompson metric with closure a normal cone. The main result gives sufficient conditions for Borel probability measures μ,ν on C with finite first moment for which μ≤ ν in the stochastic order induced by the cone to be order approximated by sequences \μn\,\νn\ of uniform finitely supported measures in the sense that μn≤ νn for each n and μn→ μ, νn→ ν in the Wasserstein metric. This result is the crucial tool in developing a pathway for extending various inequalities on operator and matrix means, which include the harmonic, geometric, and arithmetic operator means on the cone of positive elements of a C^*-algebra, to the space P1(C) of Borel measures of finite first moment on C. As an illustrative particular application, we obtain the monotonicity of the Karcher geometric mean on P1(\mathbbA+) for the positive cone \mathbbA+ of a C^*-algebra \mathbbA.