Algebras of Lorch Analytic Mappings Defined on Uniform Algebras

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 tau(d) which coincides with the topology tau(b) when U = E or U = B-r(z(0)) = {z is an element of E; parallel to z - z(0)parallel to < r} (z(0) is an element of E, r > 0). We consider the case U = E-Omega = {z is an element of E; sigma(z) subset of Omega} where Omega (sic) C is a simply connected domain and we study topological and algebraic properties of (H-L(E-Omega), tau(d)) for special algebras E: A description of the spectrum of (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 (H-L(E-Omega), tau(d)) is semisimple.