Algebras of Lorch Analytic Mappings Defined on Uniform Algebras

Abstract

For a unitary commutative complex Banach algebra $E$, let ${\mathcal{H}}_L(U)$ be the space of the mappings from an open connected subset $U$ of $E$ into $E$ that are analytic in the sense of Lorch. We consider the space ${\mathcal{H}}_L(U)$ endowed with a convenient topology ${\tau }_d$ which coincides with the topology ${\tau }_b$ when $U=E$ or $U=B_r(z_0)=\{z\in E;\Vert z-z_0\Vert <r\}$ ($z_0\in E$, $r>0$). We consider the case $U=E_{\Omega }=\{z\in E;\sigma (z)\subset \Omega \}$ where $\Omega ⊊\mathbb{C}$ is a simply connected domain and we study topological and algebraic properties of $({\mathcal{H}}_L(E_{\Omega }),{\tau }_d)$ for special algebras $E.$ A description of the spectrum of $({\mathcal{H}}_L(E_{\Omega }),{\tau }_d)$ is given in the case that $E$ is a uniform algebra. As a consequence we get that in this case the algebra $({\mathcal{H}}_L(E_{\Omega }),{\tau }_d)$ is semisimple.