Abstract.
Let X be one of the Banach spaces c 0 , ℓ p , 1≤p<∞; Ω⊂X pseudoconvex open, a holomorphic Banach vector bundle with a Banach Lie group G * for structure group. We show that a suitable Runge-type approximation hypothesis on X, G * (which we also prove for G * a solvable Lie group) implies the vanishing of the sheaf cohomology groups H q(Ω, 𝒪E), q≥1, with coefficients in the sheaf of germs of holomorphic sections of E. Further, letting 𝒪Γ (𝒞Γ) be the sheaf of germs of holomorphic (continuous) sections of a Banach Lie group bundle Γ→Ω with Banach Lie groups G, G * for fiber group and structure group, we show that a suitable Runge-type approximation hypothesis on X, G, G * (which we prove again for G, G * solvable Lie groups) implies the injectivity (and for X=ℓ1 also the surjectivity) of the Grauert–Oka map H 1(Ω, 𝒪Γ)→H 1(Ω, 𝒞Γ) of multiplicative cohomology sets.
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Received: 1 March 2002 / Published online: 28 March 2003
Mathematics Subject Classification (2000): 32L20, 32L05, 46G20
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ID="*" Kedves Laci Móhan kisfiamnak.
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ID="*" To my dear little Son
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Patyi, I. On the Oka principle in a Banach space, II. Math. Ann. 326, 443–458 (2003). https://doi.org/10.1007/s00208-003-0430-0
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DOI: https://doi.org/10.1007/s00208-003-0430-0