Symmetric continuous Reinhardt domains

whenever |ζ1|, . . . , |ζn| ≤ 1 . In 1974 [11] Sunada investigated the structure of bounded Reinhardt domains containing the origin from the viewpoint of biholomorphic equivalence. He was able to describe completely the symmetric Reihardt domains which, up to linear isomomorphism, turned to be direct products of Euclidean balls. Our aim in this paper is to study infinite dimensional analogs of symmetric Reihardt domains. There are several ways of extending the definition of classical Reihardt domains. We are however motivated by a recent interesting work of Vigué [12] who considered continuous products of discs (of different radius). He obtained a rather surprising result that such a continuous product is not symmetric automatically. We intend to give a definition of a continuous Reinhardt domain which includes the domains studied by Vigué in such a way that our results can also lead to some continuous mixing of Euclidean balls generalizing the work of Sunada. In the classical definition of a Reinhardt domain n -tuples can be viewed as (continuous) functions from the discrete space Ω := {1, . . . , n} to C . In this terminology, complete Reinhardt domains are closely related with the ordering f ≥ 0 ⇐⇒ f(ω) ≥ 0 for all ω ∈ Ω . Namely D ⊂ C(Ω) is a complete Reinhardt domain if f ∈ D and |g| ≤ |f | implies g ∈ D . This definition can simply be extended to any complex function lattice. In the setting of bounded continuous functions, the Gel’fand-Naimark theorem naturally leads to the lattice C0(Ω) of all continuous functions over a locally compact Hausdorff space vanishing at infinity. We note that in 1978 Braun-Kaup-Upmeier [1] introduced another interesting generalization of Reinhardt domains in infinite dimensional setting. This concept is different from ours and even in their later works their school changed the terminology so that today their concept is called bicircular domains. In the present paper we concentrate on the case of symmetric domains. We prove that continuous symmetric Reinhardt domains are in certain sense made of Euclidean balls but in a more complicated manner than simple direct products. Moreover, for a given domain D , the dimensions of these balls are simultaneously bounded by a finite constant determined by some geometric parameters of D . It particular, in Theorem 1 we show that D can be explicitly described in the form {f : supj∈J ∑ ω∈Ωj m(ω)|f(ω)| < 1 } for a suitable partition {Ωj : j ∈ J} of the