Questions tagged [choquet-theory]

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Continuous real action on affine functions over a Choquet simplex

Let $K$ be a metrizable Choquet simplex. We define $A=\operatorname{Aff}(K,\mathbb{R})$ as the set of real-valued affine continuous functions. This has both the structure of a real Banach space as ...
Gabor Szabo's user avatar
4 votes
1 answer
208 views

Reference for Choquet-like theorem

While reading a paper, I encountered the following statement: Let $K$ be a convex compact subset of a locally convex topological vector space. If $\mu \in P(K)$ is a Radon probability measure on $K$, ...
Andromeda's user avatar
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2 votes
0 answers
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Characterization of inverse limits of finite-dimensional convex cones

Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
postdoc's user avatar
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1 vote
1 answer
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On examples of subspaces of $C(X)$ for which state spaces are Choquet Simplices

Let $C(X)$ be the Banach space of all Real valued continuous functions on a compact Hausdorff space $X$. What are examples of uniformly closed subspace $\mathcal{A}$ of $C(X)$ such that $\mathcal{A}$ ...
Tanmoy Paul's user avatar
2 votes
1 answer
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Ergodic decomposition - how does restricting measure effect it? (Choquet Theory)

Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$. Since all the $...
Mariah's user avatar
  • 181
3 votes
1 answer
332 views

Peak sets and Choquet boundary of a function algebra

I have two problems to ask. Let $A$ be a function algebra of $C(K)$. $t\in K$ is said to be a peak point of $A$ if $\exists~f\in A$ s.t. $|f(t)|=\|f\|$ and $|f(s)|<|f(t)|$ for any $s\neq t$. ...
Tanmoy Paul's user avatar
2 votes
0 answers
306 views

Dual of the space of affine functions

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...
Tanmoy Paul's user avatar
1 vote
1 answer
155 views

Characterization of state spaces of Boolean algebras

A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces ...
Miroslav Korbelar's user avatar
1 vote
1 answer
125 views

Realizing certain affine functions on Choquet simplices on dimension groups

This is a question that is a bit outside my usual mathematical comfort zone, but I feel like an expert might know the answer. Recall that a dimension group is an ordered abelian group $G$ with ...
Gabor Szabo's user avatar
6 votes
2 answers
368 views

Norms on $\mathbb{R}^d$ whose linear isometries are the hypercube group

It is a known fact that for any $2\neq p\in[1,\infty]$, the linear isometries for the corresponding norm $\|\cdot\|_p$ on $\mathbb{R}^d$ is the set of all square-matrices with entries in $\{-1,1,0\}$, ...
Ayman Moussa's user avatar
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4 votes
0 answers
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SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, $\...
David Handelman's user avatar
6 votes
1 answer
820 views

Is there a non-compact Poulsen simplex?

A Choquet simplex is a closed, convex and metrizable subset of a locally convex Hausdorff topological vector space in which every point is a barycenter of a unique probability measure supported on the ...
Dominik Kwietniak's user avatar