Teoremas do tipo Banach-Stone para algebras de funções holomorfas em espaços de dimensão infinita

Abstract

We derive Banach-Stone theorems for several topological algebras of holomorphic functions on infinite dimensional spaces. To be more specific, if E and F are Frechet spaces, one of them separable with the bounded approximation property and U ⊂ E, V ⊂ F are connected domains of holomorphy, we prove that U and V are biholomorphically equivalent if and only if the algebras (H(U), τ) and (H(V ), τ) are topologically isomorphic, for τ = τ0, τω or τδ. We present two examples: one in which the theorem fails if only one of the open subsets is a domain of holomorphy, and another one in which the results are not valid if one of the spaces is not a Frechet space. We also present similar results for algebras of holomorphic functions of bounded type. In fact, if E and F are reflexive Banach spaces, one of them a Tsirelson-like space, and U ⊂ E, V ⊂ F are absolutelly convex open subsets, then there exists a suitable biholomorphic mapping between U and V if and only if the Frechet algebras Hb(U) and Hb(V ) are topologically isomorphic. Finally we give analogous theorems for topological algebras of germs of holomorphic functions as follows: if E and F are Tsirelson-James-like spaces, K ⊂ E and L ⊂ F are absolutely convex compact subsets, then there exists a biholomorphic mapping from an open neighborhood of K onto an open neighborhood of L, which maps K onto L, if and only if the algebras H(K) and H(L) are topologically isomorphic.