On isomorphisms of algebras of entire symmetric functions on Banach spaces

Abstract

We consider the problem of isomorphism of Fréchet algebras of symmetric entire functions of bounded type on complex Banach spaces. We show that two such Fréchet algebras are isomorphic if semigroups of symmetries on underlying Banach spaces satisfy some natural conditions. We apply this result to Fréchet algebras of symmetric entire functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power p∈[1,+∞) functions and of Lebesgue measurable essentially bounded functions on [0,1].