Commutative Abelian Galois extensions of a Banach algebra

Abstract

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C( X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.