Aron–Berner–type extension in complex Banach manifolds

Lempert, László

doi:10.1090/tran/9092

Aron–Berner–type extension in complex Banach manifolds
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by László Lempert;

Trans. Amer. Math. Soc. 377 (2024), 2169-2203

DOI: https://doi.org/10.1090/tran/9092

Published electronically: January 18, 2024

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Abstract:

Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space $B(S,X)$ of Borel maps. As an application we prove two theorems about bounded holomorphic functions on $C(S,X)$, one reminiscent of the Monodromy Theorem, the other of Liouville’s Theorem.

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