Vol. 30, No. 3, 1969

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Galois theory for Banach algebras

David Theodore Brown

Vol. 30 (1969), No. 3, 577–600

This paper deals with the classical Galois theory in the context of the Arens-Hoffman extension B = A[x](α(x)) of a commutative Banach algebra A (with identity over the complex field 𝒞) with respect to a monic polynomial α(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 ΦB and ΦA, where the Φ’s denote the respective carrier spaces of the Banach algebras A and B together with the usual weak  topology.

Mathematical Subject Classification
Primary: 46.55
Secondary: 16.00
Received: 5 April 1967
Revised: 12 December 1968
Published: 1 September 1969
David Theodore Brown