A Riemann manifold structure of the spectra of weighted algebras of holomorphic functions

In this paper we give general conditions on a countable family VV of weights on an unbounded open set UU in a complex Banach space XX such that the weighted space HV(U)HV(U) of holomorphic functions on UU has a Fréchet algebra structure. For such weights it is shown that the spectrum of HV(U)HV(U) has a natural analytic manifold structure when XX is a symmetrically regular Banach space, and in particular when X=CnX=Cn.