Automorphisms of commutative Banach algebras

Abstract

This paper presents a new proof of the theorem of Kamowitz and Scheinberg which states that if α \alpha is an element of infinite order of the automorphism group of a commutative semisimple Banach algebra then the spectrum of α \alpha contains all complex numbers of absolute value 1. The proof depends on the fact that the only closed translation invariant subalgebras of l ∞ ( − ∞ , + ∞ ) {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.