On the Oka principle in a Banach space. I
In this well written paper, the author addresses two problems: (i) Runge type approximation of holomorphic functions with values in a Banach Lie group; (ii) the Cousin problem for holomorphic cocycles with values in a Banach Lie group. The main theorem is the following. Let \(\Omega \subset X\) be pseudoconvex open in a complex Banach space \(X\) with a countable unconditional basis, \(G\) a Banach Lie group, and \({\mathcal O}^G\) the sheaf of germs of holomorphic functions \(\Omega \rightarrow G.\) Assuming a technical Runge-type hypothesis on \((X,G)\) the author shows that \noindent (a) any holomorphic cocycle on \(\Omega\) with values in \(G\) which is continuously resolvable can be also resolved analytically, \noindent (b) if in addition \(\Omega\) is contractible, then the multiplicative \(H^1(\Omega,{\mathcal O}^G)\) vanishes. The author proves that the Runge-type hypothesis is satisfied if \(X\) has an unconditional basis and \(G\) is a solvable Lie group, thus providing a large class of examples for the main theorem. In the second paper on the subject [Math. Ann. 326, 443--458 (2002; Zbl 1045.32023)], the author addresses cohomology vanishing for vector bundles and a generalization of the Oka principle of Grauert.
scientific article; zbMATH DE number 1982576