Searching for linear structures in the failure of the Stone-Weierstrass theorem

Abstract

We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set $\mathcal{C}(L,\mathbb{K})\setminus \overline{\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal{C}(L,\mathbb{K})$ that vanishes nowhere on $L$ but does not necessarily separate the points of $L$. We address the problem of finding the precise codimension of $\overline{\mathcal{A}}$ in a broad setting, which allows us to describe the lineability of $\mathcal{C}(L,\mathbb{K})\setminus \overline{\mathcal{A}}$ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $(\mathcal{C}(\partial D,\mathbb{C})\setminus \overline{\text{Pol}(\partial D)})\cup \{0\}$, where $\text{Pol}(\partial D)$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk. Recent lineability properties are also taken into account.

This updated version of the manuscript includes corrections for typos, results and certain inaccuracies