Ordered probability spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:455

Article

Lawson, Jimmie¹ 

[1] Louisiana State Univ, Dept Math, Baton Rouge, LA 70803 USA

关键词: Borel probability measure;    Metric space;    Wasserstein metric;    Barycentric map;    Partially ordered space;   

DOI  :  10.1016/j.jmaa.2017.05.046

来源: Elsevier

【 摘 要 】

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 mu, v on C with finite first moment for which mu <= v in the stochastic order induced by the cone to be order approximated by sequences {mu(n)}, {v(n)} of uniform finitely supported measures in the sense that mu(n) <= v(n) for each n and mu(n) -> mu, v(n) -> v 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 P-1(C) of Borel measures of finite first moment on C. As an illustrative and important particular application, we obtain the monotonicity of the Karcher geometric mean on P-1(A(+)) for the positive cone A(+) of a C*-algebra A. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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