On holomorphic domination. II. (Q2351828)

Holomorphic domination is possible in (a complex Banach manifold) \(M\) if for every locally upper bounded function \(u : M \to \mathbb{R}\) there is a holomorphic function \(h\) defined on \(M\) and taking values on some complex Banach space \(Z\) such that \(u(x) \leq \| h(x) \|\) for every \(x \in M\). Here some ideas of Lempert are adapted to show that if \(X\) is a separable Banach space with the bounded approximation property, then holomorphic domination is possible in every open pseudoconvex \(\Omega \subset X\). For Part I see [the author, Bull. Sci. Math. 135, No. 3, 303--311 (2011; Zbl 1233.32021)].