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Biholomorphic equivalence of bounded Reinhardt domains. (English) Zbl 0611.46054
A domain D in $${\mathbb{C}}^ n$$ is Reinhardt if $$0\in D$$ and $$(z_ 1,...,z_ n)\in D$$ if and only if $$(\lambda_ 1z_ 1,...,\lambda_ nz_ n)\in D$$ for all $$| \lambda_ i| =1$$. This notion can be extended straightforwardly to domains in a complex Banach space E with a basis $$(e_ n)_{{\mathbb{N}}}$$. It is natural to ask what kind of Riemann mapping theorem may hold for bounded Reinhardt domains in complex Banach spaces, i.e., what are the equivalence classes of such domains with respect to biholomorphic mappings? An answer to this question is furnished by a generalization of a theorem of T. Sunada [Math. Ann. 235, 111-128 (1978; Zbl 0357.32001)].
Let D and $$\tilde D$$ be bounded Reinhardt domains in E and $$\tilde E$$ with respect to the bases $$(e_ n)_{{\mathbb{N}}}$$ and $$(\tilde e_ n)_{{\mathbb{N}}}$$. $$D$$ (and $$\tilde D$$) has a biholomorphic image in $$E$$ ($$\tilde E$$) which is ”normalized”. D and $$\tilde D$$ are biholomorphically equivalent if and only if there is a surjective linear isomorphism $$T: E\to \tilde E$$ taking the normalized image of D onto that of $$\tilde D,$$ and which takes $$(e_ n)_{{\mathbb{N}}}$$ onto a permutation of $$(\tilde e_ n)_{{\mathbb{N}}}$$. Specializing to those bounded Reinhardt domains which have a nontrivial group of biholomorphic automorphisms, sets of biholomorphic invariants are determined. These invariants are given in terms of the parameters of a description of such domains, which generalizes another result of Sunada, established in a joint work by the author, S. Dineen and R. M. Timoney [Compos. Math. 59, 265- 321 (1986)]. The techniques used are Jordan theoretic, bypassing the difficulty of extending the Lie theoretic techniques used in finite dimensions.
##### MSC:
 46G20 Infinite-dimensional holomorphy 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 46H70 Nonassociative topological algebras
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