Monotonic Properties of the Least Squares Mean

We settle an open problem of several years standing by showing that the least-squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature. Our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity. Moreover, we introduce a weighted least squares means and extend our results to this setting.Comment: 21 page

Ordered probability spaces

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 and important particular application, we obtain the monotonicity of the Karcher geometric mean on P1(A+) for the positive cone A+ of a C⁎-algebra A

Existence and uniqueness of the Karcher mean on unital C\u3csup\u3e⁎\u3c/sup\u3e-algebras

The Karcher mean on the cone Ω of invertible positive elements of the C⁎-algebra B(E) of bounded operators on a Hilbert space E has recently been extended to a contractive barycentric map on the space of L1-probability measures on Ω. In this paper we first show that the barycenter satisfies the Karcher equation and then establish the uniqueness of the solution. Next we establish that the Karcher mean is real analytic in each of its coordinates, and use this fact to show that both the Karcher mean and Karcher barycenter map exist and are unique on any unital C⁎-algebra. The proof depends crucially on a recent result of the author giving a converse of the inverse function theorem

Computation on metric spaces via domain theory

The purpose of this paper is to survey recent approaches to realizing (or embedding) a Polish space as the set of maximal points of a continuous domain. Such realizations provide a convenient framework in which to model certain computational algorithms on the space and a useful alternate approach via the probabilistic power domain to measure theory and integraion on the space. © 1998 Elsevier Science B.V

The upper interval topology, property M, and compactness

The use of intrinsic topology such as interval topology and order topology that were typically symmetric was discussed. The theory of continuous lattices provided strong motivation for the consideration of such topologies such as the Scott topology or the hull-kernel topology which were not symmetric. Another approach to the study of topology on ordered structures was to begin with a set X equipped both with a partial order ≤ and a topology. The results show that these orders were called as closed order and the resulting ordered topological space was called as pospace, when the assumption was satisfied

An inverse function theorem converse

We establish the following converse of the well-known inverse function theorem. Let g:U→V and f:V→U be inverse homeomorphisms between open subsets of Banach spaces. If g is differentiable of class Cp and f is locally Lipschitz, then the Fréchet derivative of g at each point of U is invertible and f must be differentiable of class Cp

The stochastic order of probability measures on ordered metric spaces

The general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first. Motivated by recent work on barycentric maps on spaces of probability measures on ordered Banach spaces, we introduce and study a stochastic order on the space of probability measures $\mathcal{P}(X)$, where $X$ is a metric space equipped with a closed partial order, and derive several useful equivalent versions of the definition. We establish the antisymmetry and closedness of the stochastic order (and hence that it is a closed partial order) for the case of a partial order on a Banach space induced by a closed normal cone with interior. We also consider order-completeness of the stochastic order for a cone of a finite-dimensional Banach space and derive a version of the arithmetic-geometric-harmonic mean inequalities in the setting of the associated probability space on positive matrices.Comment: 25 page