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

2024/05/10 by Caballer, Marc, Dantas, Sheldon, Rodríguez-Vidanes, Daniel L.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2405.06453

Abstract

We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set C(L, \mathbbK) ∖ A, where L is a compact Hausdorff space and A is a self-adjoint subalgebra of C(L, \mathbbK) that vanishes nowhere on L but does not necessarily separate the points of L. We address the problem of finding the precise codimension of A in a broad setting, which allows us to describe the lineability of C(L, \mathbbK) ∖ A in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set (C(∂D, ℂ) ∖ Pol(∂D)) ∪ \0\, where Pol(∂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.

Related

• A general lineability criterion for complements of vector spaces
• Ordered Spaces, Metric Preimages, and Function Algebras
• Inverse Limits and Function Algebras
• A Borel linear subspace of \mathbb R^ω that cannot be covered by countably many closed Haar-meager sets
• Stone-Weierstrass Theorem