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.

翻译：重心代数是凸集概念的抽象化，由一组方程定义。我们借鉴Klaus Keimel先前关于半拓扑与拓扑锥的研究（2008），对半拓扑和拓扑重心代数进行了考察，而后者是半拓扑与拓扑重心代数的特例。例如，拓扑空间上所有连续估值（测度的近亲）构成一个拓扑锥，概率估值构成一个拓扑重心代数，而次概率估值则构成一个带基点的拓扑重心代数。在众多结果中，我们证明了在半拓扑重心代数及带基点的半拓扑代数上存在自由半拓扑锥；我们探究了哪些半拓扑重心代数可嵌入半拓扑锥，以及哪些带基点的半拓扑重心代数可严格嵌入半拓扑锥。我们研究了局部凸性的概念，这些概念分为弱局部凸性、局部凸性、局部仿射性与局部线性性。结果表明，弱局部凸的拓扑重心代数恰好是局部仿射拓扑重心代数的仿射收缩核。在局部凸重心代数上，我们证明了夹逼定理，推广了Roth与Keimel关于锥的定理。本文的一个贯穿主题是重心概念，我们逐步将其推广，直至得到连续（分别对应次概率、概率）估值的一般重心定义，该定义受Choquet定义的启发。我们以一个一般重心存在定理作结，其证明依赖于Smyth幂代数的研究，即由拓扑重心代数中所有非空凸紧饱和子集构成的拓扑重心代数。