Irreducible domains in Banach spaces (Q1093845)

The authors mainly prove the following results. Every bounded domain in the complex Banach space E is biholomorphically equivalent to a finite product of irreducible complex Banach manifolds if and only if E does not contain a subspace isomorphic to \(c_ 0\). The Banach space E with open unit B has finite cotype if and only if there exists a function \(\phi\) : [1,\(\infty)\to [1,\infty)\) such that if D is a domain in E, \(B\subset D\subset rB\) and B is biholomorphically equivalent to \(D_ 1\times D_ 2\times...\times D_ n\) where each \(D_ i\) is a complex Banach manifold of positive dimension, then \(n\leq \phi (r)\). These results completely settle the problem which is put forward by \textit{W. Kaup} [cf. Symp. Math. 26, 11-21 (1982; Zbl 0482.32012)].