# Algebras of Lorch Analytic Mappings Defined on Uniform Algebras

### Guilherme V. S. Mauro

Universidade Federal da Integração Latino-Americana, Foz do Iguaçu, Brazil### Luiza A. Moraes

Universidade Federal do Rio de Janeiro, Brazil

## Abstract

For a unitary commutative complex Banach algebra $E$, let $H_{L}(U)$ 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 $H_{L}(U)$ endowed with a convenient topology $τ_{d}$ which coincides with the topology $τ_{b}$ when $U=E$ or $U=B_{r}(z_{0})={z∈E;∥z−z_{0}∥<r}$ ($z_{0}∈E$, $r>0$). We consider the case $U=E_{Ω}={z∈E;σ(z)⊂Ω}$ where $Ω⊊C$ is a simply connected domain and we study topological and algebraic properties of $(H_{L}(E_{Ω}),τ_{d})$ for special algebras $E.$ A description of the spectrum of $(H_{L}(E_{Ω}),τ_{d})$ is given in the case that $E$ is a uniform algebra. As a consequence we get that in this case the algebra $(H_{L}(E_{Ω}),τ_{d})$ is semisimple.

## Cite this article

Guilherme V. S. Mauro, Luiza A. Moraes, Algebras of Lorch Analytic Mappings Defined on Uniform Algebras. Publ. Res. Inst. Math. Sci. 56 (2020), no. 3, pp. 431–443

DOI 10.4171/PRIMS/56-3-1