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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Automorphisms of commutative Banach algebras


Author: B. E. Johnson
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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0317053-5
Article copyright: © Copyright 1973 American Mathematical Society