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A ZERO- $\!\sqrt{5}/2$ LAW FOR COSINE FAMILIES

Part of: Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  17 April 2017

JEAN ESTERLE
JEAN ESTERLE*
Affiliation:
IMB, UMR 5251, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France email esterle@math.u-bordeaux1.fr
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Abstract

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Let $a\in \mathbb{R}$ , and let $k(a)$ be the largest constant such that $\sup |\text{cos}(na)-\cos (nb)|<k(a)$ for $b\in \mathbb{R}$ implies that $b\in \pm a+2\unicode[STIX]{x1D70B}\mathbb{Z}$ . We show that if a cosine sequence $(C(n))_{n\in \mathbb{Z}}$ with values in a Banach algebra $A$ satisfies $\sup _{n\geq 1}\Vert C(n)-\cos (na).1_{A}\Vert <k(a)$ , then $C(n)=\cos (na).1_{A}$ for $n\in \mathbb{Z}$ . Since $\!\sqrt{5}/2\leq k(a)\leq 8/3\!\sqrt{3}$ for every $a\in \mathbb{R}$ , this shows that if some cosine family $(C(g))_{g\in G}$ over an abelian group $G$ in a Banach algebra satisfies $\sup _{g\in G}\Vert C(g)-c(g)\Vert <\!\sqrt{5}/2$ for some scalar cosine family $(c(g))_{g\in G}$ , then $C(g)=c(g)$ for $g\in G$ , and the constant $\!\sqrt{5}/2$ is optimal. We also describe the set of all real numbers $a\in [0,\unicode[STIX]{x1D70B}]$ satisfying $k(a)\leq \frac{3}{2}$ .

Keywords

cosine sequencescalar cosine sequenceKronecker’s theoremcommutative Banach algebracyclotomic polynomial

MSC classification

Primary: 46J45: Radical Banach algebras

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 2 , April 2018 , pp. 195 - 217
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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