On simpliciality of function spaces not containing constants
Ondřej F. K. Kalenda, Jiří Spurný
TL;DR
This paper investigates simpliciality for function spaces that do not contain constants, revealing a markedly different landscape from the classical constant-containing theory. It develops two core notions—simpliciality and functional simpliciality—for these spaces and demonstrates that, unlike the constant-inclusive setting, many equivalences break down without constants. The authors construct extensive metrizable and non-metrizable counterexamples, including a consistent CH-based counterexample to the representation theorem, to illustrate these breakdowns. They also extend the Dirichlet framework to the constant-free setting, proving Dirichlet-type results in metrizable or one-to-one theta scenarios while highlighting limitations in general. Overall, the work delineates two favorable regimes where the theory partially recovers the classical picture and maps out rich pathologies that arise when constants are absent.
Abstract
We investigate simpliciality of function spaces without constants. We prove, in particular, that several properties characterizing simpliciality in the classical case differ in this new setting. We also show that it may happen that a given point is not represented by any measure pseudosupported by the Choquet boundary, illustrating so limits of possible generalizations of the representation theorem. Moreover, we address the abstract Dirichlet problem in the new setting and establish some common points and nontrivial differences with the classical case.