Automorphisms of commutative Banach algebras

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.