Theorems of Banach-Stone Type for Algebras of Holomorphic Functions on Infinite Dimensional Spaces

We prove that two domains of holomorphy U and V in separable Fréchet spaces with the bounded approximation property are biholomorphically equivalent if and only if the topological algebras (H(U), τ) and (H(V), τ) are topologically isomorphic for τ = τ₀, τω , τγ. We prove also that given two absolutely convex open subsets U and V of Tsirelson-like spaces, the algebras of holomorphic functions of bounded type Hb(U) and Hb(V) are topologically isomorphic if and only if there is a biholomorphic mapping of a special type between U and V. We obtain similar results for algebras of holomorphic germs H(K) and H(L), where K and L are two compact subsets of Tsirelson-like spaces.