The spectra of algebras of Lorch analytic mappings

For a complex Banach algebra EE, let HL(E)HL(E) be the space of the mappings from EE into EE that are analytic in the sense of Lorch. We will show that HL(E)HL(E) is a closed subalgebra of Hb(E,E)Hb(E,E) and we will give a description of its spectrum. As an application we will show that EE is semi-simple if and only if HL(E)HL(E) is semi-simple. We will also give descriptions of the spectra of other algebras of analytic mappings in the sense of Lorch. In particular we will study the spectrum of the Banach algebra HL∞(intBE) of the bounded mappings from intBE into EE that are analytic in the sense of Lorch.