Covering spaces and the Galois theory of commutative Banach algebras

Abstract

The principal result of this paper is that there is a bijective (functorial) correspondence between the projective separable extensions of a comutative Banach algebra A and the finite covering spaces of its maximal ideal space M( A). As a consequence, a full Galois theory for commutative Banach algebras is developed which is analogous to the (unramified) Galois theory of function fields on compact Riemann surfaces. In case M( A) is a reasonably “nice” space, its profinite fundamental group is identified as the automorphism group of the separable closure of A.