Topological and algebraic properties of spaces of Lorch analytic mappings

For a complex Banach algebra E , letH L ( U , E )be the space of the mappings from a connected subset U of E into E that are analytic in the sense of Lorch. We consider the spaceH L ( U , E )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 )and we study topological properties of ( H L ( U , E ) , τ d ) in connection with topological properties of the underlying space E . A description of the spectrum of ( H L ( B r ( z 0 ) , E ) , τ b ) is given and as a consequence, it is showed that the algebraH L ( B r ( z 0 ) , E )is semi‐simple if and only if E is semi‐simple.