Advances in Operator Theory

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

Terje Hõim and D. A. Robbins

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Abstract

‎Let $X$ be a completely regular Hausdorff space‎, ‎and let $\mathcal{D}$ be a‎ ‎cover of $X$ by $C_{b}$-embedded sets‎. ‎Let $\pi‎: ‎\mathcal{E} \rightarrow X$‎ ‎be a bundle of Banach spaces (algebras)‎, ‎and let $\Gamma(\pi)$ be the‎ ‎section space of the bundle $\pi‎.‎$ Denote by $\Gamma _{b}(\pi‎,‎\mathcal{D})$‎ ‎the subspace of $\Gamma (\pi )$ consisting of sections which are bounded on‎ ‎each $D \in \mathcal{D}$. We construct a bundle $\rho ^{\prime }: \mathcal{F}‎^{\prime}\rightarrow \beta X$ such that $\Gamma _{b}(\pi‎ ,‎ \mathcal{D}) ‎$ is topologically and algebraically isomorphic to $\Gamma(\rho^\prime‎‎)‎$, ‎and use this to study the subspaces (ideals) and quotients resulting‎ ‎from endowing $\Gamma _{b}(\pi‎,‎\mathcal{D})$ with the cover topology‎ ‎determined by $\mathcal{D}$‎.

Article information

Source
Adv. Oper. Theory Volume 3, Number 2 (2018), 351-364.

Dates
Received: 10 June 2017
Accepted: 16 October 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1513328635

Digital Object Identifier
doi:10.15352/AOT.1706-1177

Subjects
Primary: 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
Secondary: 46H10: Ideals and subalgebras

Keywords
cover topology bundle of Banach spaces bundle of Banach algebras

Citation

Hõim, Terje; Robbins, D. A. Cover topologies, subspaces, and quotients for some spaces of vector-valued functions. Adv. Oper. Theory 3 (2018), no. 2, 351--364. doi:10.15352/AOT.1706-1177. https://projecteuclid.org/euclid.aot/1513328635


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