Automorphisms of commutative Banach algebras
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- by B. E. Johnson PDF
- Proc. Amer. Math. Soc. 40 (1973), 497-499 Request permission
Abstract:
This paper presents a new proof of the theorem of Kamowitz and Scheinberg which states that if $\alpha$ is an element of infinite order of the automorphism group of a commutative semisimple Banach algebra then the spectrum of $\alpha$ contains all complex numbers of absolute value 1. The proof depends on the fact that the only closed translation invariant subalgebras of ${l^\infty }( - \infty , + \infty )$ (pointwise multiplication) for which the restriction of the shift has a complex number of absolute value 1 in its resolvent set are certain spaces of periodic sequences.References
- Herbert Kamowitz and Stephen Scheinberg, The spectrum of automorphisms of Banach algebras, J. Functional Analysis 4 (1969), 268–276. MR 0250075, DOI 10.1016/0022-1236(69)90014-7
- Laurent Schwartz, Théorie des distributions. Tome II, Publ. Inst. Math. Univ. Strasbourg, vol. 10, Hermann & Cie, Paris, 1951 (French). MR 0041345
- Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co., New York, 1952. 2nd ed. MR 0076084
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 497-499
- MSC: Primary 46J05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0317053-5
- MathSciNet review: 0317053