Continuity of Homomorphisms into Certain Commutative Banach Algebras
نویسندگان
چکیده
1. Introduction Let B be a commutative Banach algebra, and let 6 be a homomorphism from B into a Banach algebra A. In this paper we are concerned with establishing conditions on A and B which ensure that 6 is continuous. If A is semisimple, it is well known that any such homomorphism must be continuous, and therefore we shall be interested in cases where A is not semisimple. Questions of this nature have been considered by Bade and Curtis in [3], where, in particular, homomorphisms from algebras regular in the sense of Silov are discussed. Bade and Curtis also remark that discontinuous monomorphisms can always be constructed whenever there is a maximal ideal M in B such that M 2 is not closed in B. If this latter situation occurs, there exist discontinuous point derivations on B at the maximal ideal M. We shall show that for certain Banach algebras A, a condition on B related to the continuity of all point derivations guarantees the continuity of any homomorphism from B into A. In fact, we shall consider the problem of automatic continuity of homomorphisms when the domain algebra is not required to be a Banach algebra. Particularly, we have in mind the case of homomorphisms from 0(U), the algebra of functions analytic in an open set U in C
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