Ordered Probability Spaces

Abstract

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 $\mu ,\nu$ on $C$ with finite first moment for which $\mu \le \nu$ in the stochastic order induced by the cone to be order approximated by sequences $\{{\mu }_n\},\{{\nu }_n\}$ of uniform finitely supported measures in the sense that ${\mu }_n\le {\nu }_n$ for each $n$ and ${\mu }_n\to \mu$, ${\nu }_n\to \nu$ 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 $C^{\ast }$-algebra, to the space ${\mathcal{P}}^1(C)$ of Borel measures of finite first moment on $C$. As an illustrative particular application, we obtain the monotonicity of the Karcher geometric mean on ${\mathcal{P}}^1({\mathbb{A}}^{+})$ for the positive cone ${\mathbb{A}}^{+}$ of a $C^{\ast }$-algebra $\mathbb{A}$.

20 pages

References in corpus (1)

• Operator means of probability measures and generalized Karcher equations