Galois theory for Banach algebras
D. J. A. Brown · Pacific Journal of Mathematics · 1969
This paper deals with the classical Galois theory in the context of the Arens-Hoffman extension B -A[x]/(a(x)) of a commutativeBanach algebra A (with identity over the complex field ^) with respect to a monic polynomial a(x) over A with an invertible discriminant.We show that the fundamental theorem of the Galois theory for commutative rings [S.U.Chase, D. K. Harrison, and A. Rosenberg, Galois theory and cohomology of commutative rings, Memoirs, Amer.Math.Soc.No. 52 (1965)] applies to our situation.The fixed algebras of the subgroups of the Galois group are then characterized for the case where A is semi-simple.The techniques are primarily topological and consist in examining the relationships between Φ B and Φ A , where the Φ's denote the respective carrier spaces of the Banach algebras A and B together with the usual weak * topology.