Small Banach bundles and modules

2024/05/23 by Chirvasitu, Alexandru · 1 citation
#46H10 #46H25 #46J10 #46M20 #54C40 #54D20 #54E35 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2405.14518

Abstract

We characterize those (continuously-normed) Banach bundles E→ X with compact Hausdorff base whose spaces Γ(E) of global continuous sections are topologically finitely-generated over the function algebra C(X), answering a question of I. Gogić's and extending analogous work for metrizable X. Conditions equivalent to topological finite generation include: (a) the requirement that E be locally trivial and of finite type along locally closed and relatively F_σ strata in a finite stratification of X; (b) the decomposability of arbitrary elements in ℓ^p(Γ(E)), 1≤ p<∞ as sums of ≤ N products in ℓ^p(C(X))⋅ Γ(E) for some fixed N; (c) the analogous decomposability requirement for maximal Banach-module tensor products F\widehat⊗_C(X)Γ(E) or (d) equivalently, only for F=ℓ¹(C(X)).

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