Topological Barycentric Algebras

• Topological barycentric algebras are algebraic structures that generalize convex combinations by incorporating continuous, parameterized operations in a topological setting.
• They unify concepts from convex geometry, functional analysis, and domain theory, providing frameworks for cancellative embeddings and free semitopological cone constructions.
• Their study emphasizes key attributes such as joint continuity, barycenter uniqueness, and universal mapping properties, which have significant implications for interpolation and separation theorems.

A topological barycentric algebra is an abstract algebraic structure that generalizes the convex combination operations found in compact convex subsets of topological vector spaces (TVS), incorporating both continuity and topological properties. These algebras provide a unified framework for studying the interplay between convexity, topology, and algebraic operations, and have deep connections to domain theory, functional analysis, convex geometry, and the theory of continuous valuations.

1. Algebraic Foundations of Barycentric Algebras

A barycentric algebra consists of a set $A$ (or $B$), together with binary operations parameterized by $p \in (0,1)$ (or $a \in [0,1]$), denoted $p(a,b)$ or $x +_a y$. The axioms governing these operations—idempotence, (skew-)commutativity, and (skew-)associativity—abstract the notion of convex combination as found in real vector spaces:

• Idempotence: $p(a, a) = a$
• Skew–commutativity: $p(a, b) = (1 - p)(b, a)$
• Skew–associativity: $p(r(a, b), c) = (r \odot p)(a, p \odot r(b, c))$ where $r \odot p = r + p - r \cdot p$, with $(0,1), \odot$ an open interval

This structure exactly models the algebra of convex combinations on convex subsets of TVS: $(1-p)x + p y$ for $x, y \in K \subseteq E$, a convex subset in a real TVS $E$. The axioms coincide with the classical Stone–Neumann axioms when formulated as $x +_a y$ for $a \in [0,1]$, including $x +_1 y = x$, $x +_0 y = y$, $x +_a y = y +_{1-a} x$, $$(x +_a y) +_b z = x +_{ab} (y +_{\frac{(1-a)b}{1-ab}} z)$$ for $a, b < 1$; otherwise $(x +_a y) +_b z = x +_a z$ (Zamojska-Dzienio, 1 Jan 2025, Goubault-Larrecq, 14 Dec 2025).

A barycentric algebra is cancellative if $p(a, b) = p(a, c) \implies b = c$. Cancellative barycentric algebras embed into real vector spaces as convex subsets—a result known as the Stone–Kneser embedding (Zamojska-Dzienio, 1 Jan 2025).

2. Topological and Semitopological Barycentric Algebras

A topological barycentric algebra is a barycentric algebra equipped with a $T_0$ topology such that the barycentric operation map $(x, a, y) \mapsto x +_a y$ is jointly continuous. The notion of semitopological barycentric algebra requires only separate continuity in the arguments. On c-spaces and “locally finitary compact” spaces, separate continuity implies joint continuity (Ershov–Lawson principle) (Goubault-Larrecq, 14 Dec 2025).

Variants include:

• Preordered barycentric algebras: the binary operations $+_a$ are monotone with respect to a preorder.
• Pointed barycentric algebras: presence of a least element $\bot$ such that $\bot +_a x = \bot$ for all $x$.

Cones (commutative monoids with a scalar multiplication action of $[0,\infty)$) can be viewed as (pre)ordered barycentric algebras via $x+_a y := a x + (1-a)y$ (Goubault-Larrecq, 14 Dec 2025).

3. Subclasses, Partitions of Unity, and Barycentric Coordinates

Given a compact convex set $K \subseteq E$ (where $E$ is a real TVS) with a finite set of extreme points $V = \{v_1, \ldots, v_n\}$, partitions of unity play a central role. Such a partition is an $n$-tuple of continuous functions $f = (f_1, ..., f_n) \in C(K, [0,1])^n$ with $\sum_{i=1}^n f_i(x) = 1$ for all $x \in K$. Specific subclasses are distinguished:

• $Set_1(K, [0,1]^n)$: all continuous partitions of unity
• $Set_{LP}(K, [0,1]^n)$: partitions with the Lagrange property $f_i(v_j) = \delta_{ij}$
• $Set_{BLP}(K, [0,1]^n)$: partitions that are also barycentric-algebra homomorphisms

Each inclusion $$Set_{BLP} \subseteq Set_{LP} \subseteq Set_{1} \subseteq Set$$ corresponds, under the tautological map $T(f)(x) = \sum_{i=1}^n f_i(x) v_i$, to a class of continuous self-maps on $K$:

• $T(Set_1) = \{h \in C(K, K) : h(K) \subseteq K\}$
• $T(Set_{LP}) = \{h \in C(K, K): h(v_i) = v_i\}$
• $$T(Set_{BLP}) = \{h \in Hom_{bary}(K, K): h(v_i) = v_i\}$$

A point $x \in K$ has a unique barycentric coordinate system if and only if $K$ is a simplex (its vertices are affinely independent). On the standard $2$-simplex $\Delta^2$, the canonical barycentric coordinates $(x, y, z)$ yield $T(\lambda)(x, y, z) = (x, y, z)$, and the barycentric algebra operations reduce to standard convex combinations in $\mathbb{R}^3$ (Zamojska-Dzienio, 1 Jan 2025).

4. Embedding, Universal Properties, and Free Constructions

Given a semitopological barycentric algebra $B$, there exists a free semitopological cone $cone(B)$ generated by $B$, with the cone topology generated by upper sets in $(\mathbb{R} \setminus \{0\})_{\sigma} \times B$. The inclusion $\iota: B \hookrightarrow cone(B)$ is affine-continuous, with the universal property that any semi-concave continuous map from $B$ to a semitopological cone $C$ extends uniquely to a continuous linear map $cone(B) \to C$ (Goubault-Larrecq, 14 Dec 2025).

$B$ is embedded in a semitopological cone if and only if the unit map is a topological embedding. Embedding can be characterized in terms of subbasis opens of the form $h^{-1}(]1, \infty])$ for appropriate semi-concave functions $h$.

Pointed barycentric algebras admit similar free constructions, realized as a restriction of $cone(B)$ or via the telescope/colimit construction. These universality properties generalize Flood’s and Keimel–Plotkin’s theorems for cones (Goubault-Larrecq, 14 Dec 2025).

5. Topological Structure: Convexity, Separation, and Hyperspace Algebras

Several topological notions underpin the theory of topological barycentric algebras:

• Locally convex: base of convex open sets
• Locally linear: subbase of open half-spaces determined by affine/linear lower semicontinuous (LSC) functionals
• Weakly locally convex: every neighborhood contains a convex neighborhood
• Locally convex-compact: every point has a base of compact convex neighborhoods

On c-spaces (domain-theoretic continuous dcpos), these properties coincide. The notion of linear separation (convex–$T_0$) is defined by the existence of affine LSC functionals separating points; in such spaces, generalized Hahn–Banach-type separation theorems hold (Goubault-Larrecq, 14 Dec 2025).

For $A \subseteq B$, the convex hull is the set of all finite barycenters; the topological convex hull is its closure. The saturated convex hull is the upper closure of the convex hull. The hyperspace $B^\sharp$ of nonempty convex, compact saturated subsets of $B$, with operations $Q_1 +^\sharp_a Q_2 = \upc\{x +_a y : x \in Q_1, y \in Q_2\}$, is itself a topological barycentric algebra, compatible with the Vietoris upper topology (Goubault-Larrecq, 14 Dec 2025).

6. Barycenters, Valuations, and Existence-Uniqueness Theorems

The barycenter of a finite formal convex combination $\sum_{i=1}^n a_i \delta_{x_i}$ is $\sum_{i=1}^n a_i x_i$. For a continuous valuation $\nu$, the Choquet barycenter $x_0$ is such that for any affine LSC $\Lambda$, $\Lambda(x_0) = \int \Lambda\, d\nu$; equivalently, in the embedding into the double dual $B^{**}$, the barycenter can be identified via $\widehat\beta(\nu) = **(x_0)$.

In linearly separated $T_0$ barycentric algebras, barycenters for probability or subprobability valuations are unique. In a convenient topological barycentric algebra (linearly separated, locally convex-compact, sober), every continuous valuation (probability or subprobability, as appropriate) admits a unique barycenter, and the barycenter map $\beta: \Val B \to B$ is continuous. Generalizations of the Choquet theorem in this setting subsume earlier results for cones and valuation spaces on Polish or core-compact spaces (Goubault-Larrecq, 14 Dec 2025).

If a valuation is supported on a convex, closed, or compact saturated subset $C \subseteq B$, its barycenter lies in $C$. This is a consequence of separation arguments using affine functionals.

7. Open Problems and Counterexamples

Several structural and topological questions remain unresolved:

• Not all semitopological barycentric algebras are embeddable; e.g., $B^- = [-\infty, 0]$ with the Scott topology and the algebraic structure $x +_a y = a x + (1 - a) y$ is topological but not embeddable, as only constant semi-concave LSC maps exist
• The equivalence of strict and weak local convexity outside c-spaces is open; no example of a non-weakly locally convex algebra is currently known
• The distinction between consistency and strong consistency is unresolved in general, although they coincide for cones and topological barycentric algebras
• Some valuation hyperspaces, such as the hyperspace of step-functions, can fail to admit a barycenter for certain valuations (Goubault-Larrecq, 14 Dec 2025)

The theory of topological barycentric algebras is thus situated at the intersection of convex geometry, algebra, and topology, supporting further research along both categorical and analytic axes, with application domains including interpolation theory, functional analysis, domain theory, and the study of convex sets in topological vector spaces (Zamojska-Dzienio, 1 Jan 2025, Goubault-Larrecq, 14 Dec 2025).