Semitopological Barycentric Algebras

Abstract

Barycentric algebras are an abstraction of the notion of convex sets, defined by a set of equations. We study semitopological and topological barycentric algebras, in the spirit of a previous study by Klaus Keimel on semitopological and topological cones (2008), which are special cases of semitopological and topological barycentric algebras. For example, the space of all continuous valuations (a very close cousin of measures) over a topological space is a topological cone, while probability valuations form a topological barycentric algebra, and subprobability valuations form a pointed topological barycentric algebra. Among other results, we show the existence of free semitopological cones over semitopological barycentric algebras and over pointed semitopological algebras, we investigate which semitopological barycentric algebras embed into semitopological cones and which pointed semitopological barycentric algebras embed strictly into semitopological cones. We study notions of local convexity, which split into weak local convexity, local convexity, local affineness and local linearity. We show that the weakly locally convex topological barycentric algebras are exactly the affine retracts of locally affine topological barycentric algebras. On locally convex barycentric algebras, we show sandwich theorems, extending theorems by Roth and Keimel on cones. A running theme of this paper is the notion of barycenters, which we progressively generalize until we reach a general notion of barycenters of continuous (resp., subprobability, probability) valuations, inspired by a definition of Choquet. We conclude with a general barycenter existence theorem, whose proof relies on the study of the Smyth poweralgebra, namely the topological barycentric algebra of all non-empty convex compact saturated subsets of a topological barycentric algebra.