2010/05/25 by Chi-Wai Leung, Leung, Chi-Wai, Chi-Keung Ng +5
Mathematics · #46H25 #46H40 #46L08 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Rings, Modules, and Algebras #math.FA #math.OA #msc:46H25 #msc:46H40 #msc:46L08
paper · pdf · doi:10.48550/arxiv.1005.4561
arxiv created 2010/05/25 · openalex publication_date 2010/05/25 · arxiv updated 2010/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be a locally compact Hausdorff space. We show that any local ℂ-linear map (where "local" is a weaker notion than C0(Ω)-linearity) between Banach C0(Ω)-modules are "nearly C0(Ω)-linear" and "nearly bounded". As an application, a local ℂ-linear map θ between Hilbert C0(Ω)-modules is automatically C0(Ω)-linear. If, in addition, Ω contains no isolated point, then any C0(Ω)-linear map between Hilbert C0(Ω)-modules is automatically bounded. Another application is that if a sequence of maps \θn\ between two Banach spaces "preserve c0-sequences" (or "preserve ultra-c0-sequences"), then θn is bounded for large enough n and they have a common bound. Moreover, we will show that if θ is a bijective "biseparating" linear map from a "full" essential Banach C0(Ω)-module E into a "full" Hilbert C0(Δ)-module F (where Δ is another locally compact Hausdorff space), then θ is "nearly bounded" (in fact, it is automatically bounded if Δ or Ω contains no isolated point) and there exists a homeomorphism σ: Δ→ Ω such that θ(e⋅ φ) = θ(e)⋅ φ∘ σ (e∈ E, φ∈ C0(Ω)).