Vol. 30, No. 3, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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