Cover-strict topologies, ideals, and quotients for some spaces of vector-valued functions

Abstract

Let $X$ be a completely regular Hausdorff space, let $\mathcal{D}$ be a cover of $X$, and let $\pi :\mathcal{E}\to X$ be a bundle of Banach spaces (algebras). Let $\Gamma \left(\pi \right)$ be the space of sections of $\pi$, and let ${\Gamma }_b(\pi ,\mathcal{D})$ be the subspace of $\Gamma \left(\pi \right)$ consisting of sections which are bounded on each $D\in \mathcal{D}$. We study the subspace (ideal) and quotient structures of some spaces of vector-valued functions which arise from endowing ${\Gamma }_b(\pi ,\mathcal{D})$ with the cover-strict topology.