
TL;DR
This paper explores semitopological and topological barycentric algebras, generalizing barycenter concepts and establishing foundational results about their structure, embeddings, and local convexity properties.
Contribution
It introduces new results on free semitopological cones, embeddings, and barycenter theorems within the framework of barycentric algebras, extending prior work on cones.
Findings
Existence of free semitopological cones over barycentric algebras
Characterization of embeddings into semitopological cones
General barycenter existence theorem for valuations
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…
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