  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 Fr\'echet 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.
