On algebras of holomorphic functions of a given type

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert–Schmidt bounded type, are locally m-convex Fréchet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan–Thullen type theorem.