Lineability of functions in $C(K)$ with specified range
Artur Bartoszewicz, Szymon Głcab
TL;DR
The paper develops a framework to study lineability and spaceability in function spaces $C(K)$ by translating problems from $\ell_\infty$ to $C(\beta\mathbb N\setminus\mathbb N)$ via the isomorphism $\ell_\infty/c_0 \cong C(\beta\mathbb N\setminus\mathbb N)$ and extends this to idealized settings $C(P_{\mathcal I})$ using $\ell_\infty/c_0(\mathcal I) \cong C(P_{\mathcal I})$. It proves that for suitable compact spaces $K$ resembling $\beta\mathbb N\setminus\mathbb N$, the sets of functions with countable ranges or with range of cardinality $\mathfrak c$ each contain, up to the zero function, an isometric copy of $c_0(\kappa)$ for uncountable $\kappa$, generalizing known results to idealized contexts. Specializing to spaces $P_{\mathcal I}$, the paper derives generalized lineability/spaceability results (including for $\mathcal I$ with the Baire or hereditary Baire property) that recover and extend prior work by Leonetti–Russo–Somaglia and Menet–Papathanasiou. It also shows that the class of functions with interval ranges supports a rich subspace isometric to $C(\beta\mathbb N\setminus\mathbb N)$, implying continuum-range necessity for such substructures.
Abstract
This paper is inspired by the paper of Leonetti, Russo and Somaglia [\textit{Dense lineability and spaceability in certain subsets of $\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised therein. It concerns the properties of $\ell_\infty$ subsets defined by cluster points of sequences. Using the fact that the set of cluster points of a sequence $x$ depends only on its equivalence class in $\ell_\infty/c_0$ and that the quotient space $\ell_\infty/c_0$ is isometrically isomorphic to $C(β\mathbb{N}\setminus\mathbb{N})$, we are able to translate lineability problems from $\ell_\infty$ to $C(β\mathbb{N}\setminus\mathbb{N})$. We prove that for a compact space $K$ with properties similar to those of $β\mathbb{N}\setminus\mathbb{N}$, the sets of continuous functions $f$ in $C(K)$ with $\vert\operatorname{rng}(f)\vert=ω$ and those $f$ with $\vert\operatorname{rng}(f)\vert=\mathfrak c$ contain, up to zero function, an isometric copy of $c_0(κ)$ for uncountable cardinal $κ$. Specializing those results to some closed subspaces $K$ of $β\mathbb{N}\setminus\mathbb{N}$ we are able to generalize known results to their ideal versions.