Artur Bartoszewicz[a] and Szymon Głab[b]
Lineability of functions in \( C(K) \) with specified range
Pages: 489-505
Received: 23 January 2024
Accepted in revised form: 26 June 2024
Mathematics Subject Classification: Primary: 46B87, 15A03; Secondary: 40A35, 46E15.
Keywords: Lineability, spaceability, subsets of \( \ell_\infty \), Čech-Stone compactification of natural numbers
Authors address:
[a]: Faculty of Mathematics and Computer Science, Łódź University, Łódź, Poland
[b]: Institute of Mathematics, Lódź University of Technology, Łódź, Poland
Abstract: This paper is inspired by that of Leonetti, Russo and Somaglia [Dense lineability and spaceability in certain subsets of \( \ell_\infty\) , Bull. London Math. Soc. 55 (2023), 2283-2303] 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 \(C(\beta\mathbb N\setminus\mathbb N)\), we are able to translate lineability problems from \(\ell_\infty\) to \(C(\beta\mathbb N\setminus\mathbb N)\). We prove that for a compact space \(K\) with properties similar to those of \(\beta\mathbb N\setminus\mathbb N\), the sets of continuous functions \(f\) in \(C(K)\) with \(\vert\operatorname{rng}(f)\vert=\omega\) and those \(f\) with \(\vert\operatorname{rng}(f)\vert=\mathfrak c\) contain, up to zero function, an isometric copy of \(c_0(\kappa)\) for uncountable cardinal \(\kappa\). Specializing those results to some closed subspaces \(K\) of \(\beta\mathbb N\setminus\mathbb N\) we are able to generalize known results to their ideal versions.