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Proceedings of the American Mathematical Society

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Automorphisms of commutative Banach algebras

Author: B. E. Johnson
Journal: Proc. Amer. Math. Soc. 40 (1973), 497-499
MSC: Primary 46J05
MathSciNet review: 0317053
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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 [Enhancements On Off] (What's this?)

  • [1] H. Kamowitz and S. Scheinberg, The spectrum of automorphisms of Banach algebras, J. Functional Analysis 4 (1969), 268-276. MR 40 #3316. MR 0250075 (40:3316)
  • [2] L. Schwartz, Théorie des distributions, Tome II, Actualités Sci. Indust., 1122, Hermann, Paris, 1951. MR 12, 833. MR 0041345 (12:833d)
  • [3] A. Zygmund, Trigonometrical series. Vol. I, 2nd ed. reprinted with corrections and some additions, Cambridge Univ. Press, New York, 1968. MR 38 #4882. MR 0076084 (17:844d)

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